Search: keyword:new
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A293899
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Number of proper divisors of form 3k+1 minus number of proper divisors of from 3k+2.
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+0
0
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0, 1, 1, 0, 1, 0, 1, 1, 1, -1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 2, -1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, -1, 1, 1, 1, 1, 2, -1, 1, 0, 1, 1, 0, -1, 1, 1, 2, 1, 0, 1, 1, 0, -1, 1, 2, -1, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 0, -1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, 1, 2, -1, 1, 0, -1, 1, 0, 3, 1, 2, -1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0
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OFFSET
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1,21
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..20000
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FORMULA
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When n = 3k, a(n) = A002324(n), when n = 3k+1, a(n) = A002324(n) - 1, when n = 3k+2, a(n) = A002324(n) + 1.
a(n) ≡ A002324(n) - A010872(n) (mod 3).
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MATHEMATICA
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Table[DivisorSum[n, 1 &, And[Mod[#, 3] == 1, # != n] &] - DivisorSum[n, 1 &, And[Mod[#, 3] == 2, # != n] &], {n, 105}] (* Michael De Vlieger, Nov 08 2017 *)
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PROG
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(PARI)
A293895(n) = sumdiv(n, d, (d<n)*(1==(d%3)));
A293896(n) = sumdiv(n, d, (d<n)*(2==(d%3)));
A293899(n) = (A293895(n) - A293896(n));
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CROSSREFS
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Cf. A002324, A293895, A293896.
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KEYWORD
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sign,new
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AUTHOR
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Antti Karttunen, Nov 06 2017
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STATUS
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approved
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A293898
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Sum of proper divisors of the form 3k+2.
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+0
0
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0, 0, 0, 2, 0, 2, 0, 2, 0, 7, 0, 2, 0, 2, 5, 10, 0, 2, 0, 7, 0, 13, 0, 10, 5, 2, 0, 16, 0, 7, 0, 10, 11, 19, 5, 2, 0, 2, 0, 35, 0, 16, 0, 13, 5, 25, 0, 10, 0, 7, 17, 28, 0, 2, 16, 24, 0, 31, 0, 27, 0, 2, 0, 42, 5, 13, 0, 19, 23, 56, 0, 10, 0, 2, 5, 40, 11, 28, 0, 35, 0, 43, 0, 16, 22, 2, 29, 65, 0, 7, 0, 25, 0, 49, 5, 42, 0, 16, 11, 77, 0, 19, 0, 36, 40
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OFFSET
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1,4
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sums of divisors
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FORMULA
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a(n) = A078182(n) - ([n == 2 (mod 3)]*n).
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MATHEMATICA
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Table[DivisorSum[n, # &, And[Mod[#, 3] == 2, # != n] &], {n, 105}] (* Michael De Vlieger, Nov 08 2017 *)
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PROG
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(PARI) A293898(n) = sumdiv(n, d, (d<n)*(2==(d%3))*d);
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CROSSREFS
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Cf. A078182, A293896, A293897.
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KEYWORD
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nonn,new
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AUTHOR
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Antti Karttunen, Nov 06 2017
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STATUS
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approved
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A293897
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Sum of proper divisors of the form 3k+1.
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+0
0
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0, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 8, 1, 5, 1, 1, 1, 15, 8, 1, 1, 5, 1, 14, 1, 12, 1, 11, 1, 21, 1, 1, 8, 5, 1, 20, 14, 15, 1, 8, 1, 27, 1, 1, 1, 21, 8, 36, 1, 18, 1, 1, 1, 40, 20, 1, 1, 15, 1, 32, 8, 21, 14, 23, 1, 39, 1, 18, 1, 5, 1, 38, 26, 24, 8, 14, 1, 71, 1, 1, 1, 40, 1, 44, 1, 27, 1, 11, 21, 51, 32, 1, 20, 21, 1, 57, 1, 40, 1, 35, 1, 70, 8
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OFFSET
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1,8
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sums of divisors
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FORMULA
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a(n) = A078181(n) - ([n == 1 (mod 3)]*n).
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MATHEMATICA
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Table[DivisorSum[n, # &, And[Mod[#, 3] == 1, # != n] &], {n, 105}] (* Michael De Vlieger, Nov 08 2017 *)
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PROG
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(PARI) A293897(n) = sumdiv(n, d, (d<n)*(1==(d%3))*d);
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CROSSREFS
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Cf. A078181, A293901, A293895, A293898.
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KEYWORD
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nonn,new
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AUTHOR
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Antti Karttunen, Nov 06 2017
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STATUS
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approved
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A293896
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Number of proper divisors of the form 3k+2.
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+0
0
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0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 2, 0, 2, 0, 2, 1, 1, 0, 2, 0, 2, 0, 2, 1, 2, 1, 1, 0, 1, 0, 4, 0, 2, 0, 2, 1, 2, 0, 2, 0, 2, 1, 2, 0, 1, 2, 3, 0, 2, 0, 3, 0, 1, 0, 3, 1, 2, 0, 2, 1, 4, 0, 2, 0, 1, 1, 2, 1, 2, 0, 4, 0, 2, 0, 2, 2, 1, 1, 4, 0, 2, 0, 2, 0, 2, 1, 3, 0, 2, 1, 4, 0, 2, 0, 3, 2
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OFFSET
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1,10
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..20000
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FORMULA
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a(n) = A001822(n) - [n == 2 (mod 3)].
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MATHEMATICA
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Table[DivisorSum[n, 1 &, And[Mod[#, 3] == 2, # != n] &], {n, 105}] (* Michael De Vlieger, Nov 08 2017 *)
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PROG
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(PARI) A293896(n) = sumdiv(n, d, (d<n)*(2==(d%3)));
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CROSSREFS
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Cf. A001822, A293895, A293898, A293899.
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KEYWORD
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nonn,new
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AUTHOR
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Antti Karttunen, Nov 06 2017
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STATUS
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approved
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A293895
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Number of proper divisors of the form 3k+1.
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+0
0
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0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 1, 3, 1, 1, 1, 3, 2, 3, 1, 3, 1, 1, 1, 4, 2, 1, 1, 3, 1, 2, 2, 3, 2, 2, 1, 3, 1, 3, 1, 2, 1, 2, 2, 3, 2, 2, 1, 5, 1, 1, 1, 4, 1, 2, 1, 3, 1, 2, 3, 3, 2, 1, 2, 3, 1, 3, 1, 4, 1, 2, 1, 4, 2
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OFFSET
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1,8
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..20000
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FORMULA
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a(n) = A001817(n) - [n == 1 (mod 3)].
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MATHEMATICA
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Table[DivisorSum[n, 1 &, And[Mod[#, 3] == 1, # != n] &], {n, 105}] (* Michael De Vlieger, Nov 08 2017 *)
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PROG
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(PARI) A293895(n) = sumdiv(n, d, (d<n)*(1==(d%3)));
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CROSSREFS
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Cf. A001817, A293451, A293896, A293897, A293899.
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KEYWORD
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nonn,new
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AUTHOR
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Antti Karttunen, Nov 06 2017
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STATUS
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approved
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A293450
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Restricted growth sequence transform of (3*A293225(n) + A010872(n)), a filter combining (n mod 3) with two products, the other formed from the 1-digits (A293221) and the other from the 2-digits (A293222) present in the ternary expansions of proper divisors of n.
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+0
0
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1, 2, 3, 4, 2, 5, 6, 7, 8, 9, 2, 10, 6, 11, 12, 13, 2, 14, 6, 15, 16, 17, 2, 18, 19, 20, 21, 22, 2, 23, 6, 24, 25, 26, 27, 28, 6, 29, 30, 31, 2, 32, 6, 33, 34, 35, 2, 36, 37, 38, 14, 39, 2, 40, 41, 42, 43, 44, 2, 45, 6, 46, 47, 48, 49, 50, 6, 51, 52, 53, 2, 54, 6, 55, 56, 57, 58, 59, 6, 60, 61, 62, 2, 63, 64, 65, 66, 67, 2
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OFFSET
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1,2
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..59049
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FORMULA
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For all i, j: a(i) = a(j) => A002324(i) = A002324(j).
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PROG
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(PARI)
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset, vec, bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A019565(n) = {my(j, v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A289813(n) = { my (d=digits(n, 3)); from digits(vector(#d, i, if (d[i]==1, 1, 0)), 2); };
A289814(n) = { my (d=digits(n, 3)); from digits(vector(#d, i, if (d[i]==2, 1, 0)), 2); };
A293221(n) = { my(m=1); fordiv(n, d, if(d < n, m *= A019565(A289813(d)))); m; };
A293222(n) = { my(m=1); fordiv(n, d, if(d < n, m *= A019565(A289814(d)))); m; };
Anot_submitted(n) = (1/2)*(2 + ((A293222(n) + A293221(n))^2) - A293222(n) - 3*A293221(n)); \\ Eq.class-wise equal to A293225.
Anot2submitted(n) = ((3*Anot_submitted(n))+(n%3));
write_to_bfile(1, rgs_transform(vector(59049, n, Anot2submitted(n))), "b293450.txt");
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CROSSREFS
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Cf. A002324, A010872, A293221, A293222, A293225, A293226.
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KEYWORD
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nonn,new
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AUTHOR
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Antti Karttunen, Nov 06 2017
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STATUS
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approved
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A294785
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E.g.f. A(x) satisfies: A(x) = A(x^2) * exp( Integral A(x^2) dx ).
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+0
0
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1, 1, 3, 9, 57, 297, 2187, 15921, 181233, 1731249, 20741139, 241294329, 3524256297, 49123306521, 781173645723, 12522002462433, 247000850880993, 4516315005395169, 92648539990208547, 1886480713319540841, 43524900326040674841, 986331301183882645641, 24094409085348757028523, 596222660659090240456209, 16242798073806940474325457, 438933088683325211888103057, 12586136448791084548892537907
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OFFSET
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0,3
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LINKS
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Paul D. Hanna, Table of n, a(n) for n = 0..400
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FORMULA
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E.g.f. satisfies:
(1) exp( Integral A(x^2) dx ) = 1 + Integral A(x) dx.
(2) A(x)/A(x^2) = 1 + Integral A(x) dx.
(3) A(x) = Product_{n>=0} B( x^(2^n) ) where B(x) = 1 + Integral A(x) dx.
(4) A'(x)/A(x) = A(x^2) + 2*x * A'(x^2)/A(x^2).
(5) A'(x)/A(x) = Sum_{n>=1} 2^n * x^(2^n-1) * A( x^(2^(n+1) ).
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 3*x^2/2! + 9*x^3/3! + 57*x^4/4! + 297*x^5/5! + 2187*x^6/6! + 15921*x^7/7! + 181233*x^8/8! + 1731249*x^9/9! + 20741139*x^10/10! + 241294329*x^11/11! + 3524256297*x^12/12! + 49123306521*x^13/13! + 781173645723*x^14/14! + 12522002462433*x^15/15! + 247000850880993*x^16/16! +...
such that A(x) = A(x^2) * exp( Integral A(x^2) dx ).
Also,
A(x) = B(x) * B(x^2) * B(x^4) * B(x^8) * B(x^16) *...* B(x^(2^n)) *...
where B(x) = 1 + Integral A(x) dx.
Further,
A'(x)/A(x) = A(x^2) + 2*x*A(x^4) + 4*x^3*A(x^8) + 8*x^7*A(x^16) + 16*x^15*A(x^32) 32*x^31*A(x^64) +...+ 2^n * x^(2^n-1) * A(x^(2^(n+1))) +...
RELATED SERIES.
E.g.f. A(x) as a series with reduced fractional coefficients begins:
A(x) = 1 + x + 3/2*x^2 + 3/2*x^3 + 19/8*x^4 + 99/40*x^5 + 243/80*x^6 + 1769/560*x^7 + 20137/4480*x^8 + 192361/40320*x^9 + 2304571/403200*x^10 + 8936827/1478400*x^11 + 43509337/5913600*x^12 + 1819381723/230630400*x^13 + 3214706361/358758400*x^14 + 51530874331/5381376000*x^15 + 277217565523/23482368000*x^16 +...
The logarithm of the e.g.f. begins:
log(A(x)) = x + x^2 + 1/3*x^3 + x^4 + 3/10*x^5 + 1/3*x^6 + 3/14*x^7 + x^8 + 19/72*x^9 + 3/10*x^10 + 9/40*x^11 + 1/3*x^12 + 243/1040*x^13 + 3/14*x^14 + 1769/8400*x^15 + x^16 + 20137/76160*x^17 + 19/72*x^18 + 192361/766080*x^19 + 3/10*x^20 + 2304571/8467200*x^21 + 9/40*x^22 + 8936827/34003200*x^23 + 1/3*x^24 + 43509337/147840000*x^25 + 243/1040*x^26 + 1819381723/6227020800*x^27 + 3/14*x^28 + 3214706361/10403993600*x^29 + 1769/8400*x^30 + 51530874331/166822656000*x^31 + x^32 +...
The logarithmic derivative of the e.g.f. begins:
A'(x)/A(x) = 1 + 2*x + x^2 + 4*x^3 + 3/2*x^4 + 2*x^5 + 3/2*x^6 + 8*x^7 + 19/8*x^8 + 3*x^9 + 99/40*x^10 + 4*x^11 + 243/80*x^12 + 3*x^13 + 1769/560*x^14 + 16*x^15 + 20137/4480*x^16 +...
where A'(x)/A(x) = A(x^2) + 2*x * A'(x^2)/A(x^2).
The following series demonstrates an important property of the e.g.f.:
A(x)/A(x^2) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 57*x^5/5! + 297*x^6/6! + 2187*x^7/7! + 15921*x^8/8! + 181233*x^9/9! +...+ a(n)*x^(n+1)/(n+1)! +...
where A(x)/A(x^2) = 1 + Integral A(x) dx.
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PROG
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(PARI) {a(n) = my(A=1); for(i=1, #binary(n+1), A = subst(A, x, x^2) * exp( intformal( subst(A, x, x^2) +x*O(x^n))) ); n!*polcoeff(H=A, n)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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Cf. A294638.
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KEYWORD
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nonn,new
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AUTHOR
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Paul D. Hanna, Nov 08 2017
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STATUS
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approved
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A294730
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Smallest average >= 6 of a twin prime pair that has exactly 2*n divisors, 0 if no such pair exists.
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+0
0
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6, 12, 30, 0, 60, 192, 270, 180, 240, 0, 420, 0, 2112, 1620, 1320, 0, 2340, 786432, 3120, 4800, 15360, 0, 3360, 388962, 724992, 6300, 29760, 0, 12240, 0, 7560, 617472, 47382528, 81648, 21600, 0, 651952128, 995328, 21840, 0, 33600, 0, 138240
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OFFSET
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2,1
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COMMENTS
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It is conjectured that a(n)=0 for prime n with the only exceptions given by n=A181490(k)+1, i.e. a(2)=6, a(3)=12, a(7)=192 and a(19)=786432 are the only currently known exceptions.
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LINKS
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Table of n, a(n) for n=2..44.
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CROSSREFS
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Cf. A014574, A068525, A075590, A181490.
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KEYWORD
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nonn,new
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AUTHOR
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Hugo Pfoertner, Nov 08 2017
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STATUS
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approved
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A294775
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Number A(n,k) of partitions of 1 into exactly k*n+1 powers of 1/(k+1); square array A(n,k), n>=0, k>=0, read by antidiagonals.
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+0
0
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 2, 4, 5, 1, 1, 1, 1, 2, 4, 7, 9, 1, 1, 1, 1, 2, 4, 8, 13, 16, 1, 1, 1, 1, 2, 4, 8, 15, 25, 28, 1, 1, 1, 1, 2, 4, 8, 16, 29, 48, 50, 1, 1, 1, 1, 2, 4, 8, 16, 31, 57, 92, 89, 1, 1, 1, 1, 2, 4, 8, 16, 32, 61, 112, 176, 159, 1
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OFFSET
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0,14
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LINKS
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Alois P. Heinz, Antidiagonals n = 0..140, flattened
Christian Elsholtz, Clemens Heuberger, Helmut Prodinger, The number of Huffman codes, compact trees, and sums of unit fractions, arXiv:1108.5964v1 [math.CO], Aug 30, 2011
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EXAMPLE
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A(4,1) = 3: [1/4,1/4,1/4,1/8,1/8], [1/2,1/8,1/8,1/8,1/8], [1/2,1/4,1/8,1/16,1/16].
A(5,2) = 7: [1/9,1/9,1/9,1/9,1/9,1/9,1/9,1/9,1/27,1/27,1/27], [1/3,1/9,1/9,1/9,1/9,1/27,1/27,1/27,1/27,1/27,1/27], [1/3,1/9,1/9,1/9,1/9,1/9,1/27,1/27,1/81,1/81,1/81], [1/3,1/3,1/27,1/27,1/27,1/27,1/27,1/27,1/27,1/27,1/27], [1/3,1/3,1/9,1/27,1/27,1/27,1/27,1/27,1/81,1/81,1/81], [1/3,1/3,1/9,1/9,1/27,1/81,1/81,1/81,1/81,1/81,1/81], [1/3,1/3,1/9,1/9,1/27,1/27,1/81,1/81,1/243,1/243,1/243].
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 2, 2, 2, 2, 2, 2, 2, ...
1, 3, 4, 4, 4, 4, 4, 4, 4, ...
1, 5, 7, 8, 8, 8, 8, 8, 8, ...
1, 9, 13, 15, 16, 16, 16, 16, 16, ...
1, 16, 25, 29, 31, 32, 32, 32, 32, ...
1, 28, 48, 57, 61, 63, 64, 64, 64, ...
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MAPLE
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b:= proc(n, r, k) option remember;
`if`(n<r, 0, `if`(r=0, `if`(n=0, 1, 0), add(
b(n-j, k*(r-j), k), j=0..min(n, r))))
end:
A:= (n, k)-> `if`(k=0, 1, b(k*n+1, 1, k+1)):
seq(seq(A(n, d-n), n=0..d), d=0..14);
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CROSSREFS
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Columns k=0-9 give (offsets may differ): A000012, A002572, A176485, A176503, A194628, A194629, A194630, A194631, A194632, A194633.
Main diagonal gives A011782(n-1) for n>0.
Cf. A294746.
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KEYWORD
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nonn,tabl,new
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AUTHOR
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Alois P. Heinz, Nov 08 2017
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STATUS
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approved
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A294717
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Numbers n such that 2^((n-1)/3) = 1 mod n and (2*n-1)*(2^((n-1)/6)) = 1 mod n.
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+0
0
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1, 43, 109, 157, 229, 277, 283, 307, 397, 499, 643, 691, 733, 739, 811, 997, 1021, 1051, 1069, 1093, 1459, 1579, 1597, 1627, 1699, 1723, 1789, 1933, 2179, 2203, 2251, 2341, 2347, 2731, 2749, 2917, 2971, 3061, 3163, 3181, 3229, 3259, 3277, 3331, 3373, 3541, 4027
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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Most of the elements of this sequence are prime. The "pseudoprimes" of these sequence are part of A244626.
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LINKS
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Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv:1608.00862 [math.GM], 2016.
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PROG
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(PARI) is(n)=n%6==1 && Mod(2, n)^(n\3)==1 && (2*n-1)*Mod(2, n)^(n\6)==1 \\ Charles R Greathouse IV, Nov 08 2017
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CROSSREFS
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Cf. A001133, A244626.
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KEYWORD
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nonn,new
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AUTHOR
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Jonas Kaiser, Nov 07 2017
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STATUS
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approved
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