1,1

If there are no odd perfect numbers, then the terms give all solutions n > 1 to A323244(n) = 0.

Table of n, a(n) for n=1..8.

a(n) = A062457(A000043(n)).

A323244(a(n)) = 0.

a(n) = A005940(1+A000396(n)). [Provided no odd perfect numbers exist]

Cf. A000043, A000396, A005940, A062457, A323244, A324200.

nonn

Antti Karttunen, Feb 18 2019

approved

1,1

Sum of n-th even perfect number and n-th even superperfect number.

Also, sum of n-th perfect number and n-th superperfect number, if there are no odd perfect and odd superperfect numbers, then the n-th perfect number is the difference between a(n) and the n-th superperfect number (see A135652, A135653, A135654 and A135655).

Table of n, a(n) for n=1..10.

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

a(n) = 2^(2*A000043(n) - 1). Also, a(n) = 2^A133033(n), if there are no odd perfect numbers. Also, a(n) = A000396(n) + A019279(n), if there are no odd perfect and odd superperfect numbers. Also, a(n) = A000396(n) + A061652(n), if there are no odd perfect numbers, then we can write: perfect number A000396(n) = a(n) - A061652(n).

a(n) = A061652(n)*(A000668(n)+1) = A061652(n)*A072868(n). - Omar E. Pol, Apr 13 2008

a(5)=33554432 because A000043(5)=13 and 2^(2*13 - 1) = 2^25 = 33554432.

Also, if there are no odd perfect and odd superperfect numbers then we can write a(5) = A000396(5) + A019279(5) = A000396(5) + A061652(5) = 33554432.

Cf. A000079, A000396, A019279, A061645, A061652, A133033, A135652, A135653, A135654, A135655, A139286, A139294, A139307.

Cf. A000043, A000668, A072868.

nonn

Omar E. Pol, Apr 13 2008

approved

1,1

Table of n, a(n) for n=1..47.

a(n) = A010872(A000043(n)). - Michel Marcus, Aug 12 2014

Array[Mod[MersennePrimeExponent@ #, 3] &, 45] (* Michael De Vlieger, Apr 07 2018 *)

(PARI) forprime(p=1, 1e3, if(isprime(2^p-1), print1(p%3, ", "))) \\ Felix Fröhlich, Aug 12 2014

Cf. A010872 (n mod 3), A126044-A126059.

nonn

Artur Jasinski, Dec 17 2006

a(45)-a(47) from Ivan Panchenko, Apr 08 2018

approved

1,1

Table of n, a(n) for n=1..47.

Array[Mod[MersennePrimeExponent@ #, 19] &, 45] (* Michael De Vlieger, Apr 10 2018 *)

Cf. A126043-A126058.

nonn,more

Artur Jasinski, Dec 17 2006

a(45)-a(47) from Ivan Panchenko, Apr 09 2018

approved

1,1

The second element of the spectral basis of A330817 is A330820.

Walter Kehowski, Table of n, a(n) for n = 1..12

a(n) = A000668(n)^2*(A000668(n)+2)^2.

If p=2, then M_2=3, and a(1) = 3^2(3+2)^2 = 15^2 = 225.

A330819:=[]:

for www to 1 do

for i from 1 to 31 do

  #ithprime(31)=127

  p:=ithprime(i);

  q:=2^p-1;

  if isprime(q) then x:=2^(2*p+1)*q^2; A330819:=[op(A330819), x]; fi;

od;

od;

A330819;

(m = 2^MersennePrimeExponent[Range[9]] - 1)^2 * (m + 2)^2 (* Amiram Eldar, Jan 03 2020 *)

Cf. A000043, A000668, A132794, A133049, A330817, A330818, A330820.

nonn

Walter Kehowski, Jan 01 2020

approved

1,1

Sequence is well-defined also in case there are only a finite number of Mersenne primes.

Antti Karttunen, Table of n, a(n) for n = 1..3217

For all applicable n >= 1, a(A000043(n)) = A000668(n).

For p in A000043: 2, 3, 5, 7, 13, 17, 19, ..., a(p) = (2^p)-1, thus a(2) = 2^2 - 1 = 3, a(3) = 7, a(5) = 31, a(7) = 127, a(13) = 8191, a(17) = 131071, etc., with the rest of positions filled by the least unused prime:

1, 2, 3, 4,  5,  6,   7,  8,  9, 10, 11, 12,   13, 14, 15, 16, 17, ...

2, 3, 7, 5, 31, 11, 127, 13, 17, 19, 23, 29, 8191, 37, 41, 43, 131071, ...

(PARI)

up_to = 127;

A332211list(up_to) = { my(v=vector(up_to), xs=Map(), i=1, q); for(n=1, up_to, if(isprime(q=((2^n)-1)), v[n] = q, while(mapisdefined(xs, prime(i)), i++); v[n] = prime(i)); mapput(xs, v[n], n)); (v); };

v332211 = A332211list(up_to);

A332211(n) = v332211[n];

\\ For faster computing of larger values, use precomputed values of A000043:

v000043 = [2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217];

up_to = v000043[#v000043];

A332211list(up_to) = { my(v=vector(up_to), xs=Map(), i=1, q); for(n=1, up_to, if(vecsearch(v000043, n), q = (2^n)-1, while(mapisdefined(xs, prime(i)), i++); q = prime(i)); v[n] = q; mapput(xs, q, n)); (v); };

Cf. A000040, A000043, A000668, A332210 (inverse permutation of primes), A332220.

Used to construct permutations A332212, A332214.

nonn

Antti Karttunen, Feb 09 2020

approved

1,1

Mersenne numbers (with exception first two) are congruent to 31, 127, 271, 607 mod 6!. This sequences is subset of A000043.

Table of n, a(n) for n=1..17.

p = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 43112609}; a = {}; Do[If[Mod[2^p[[n]] - 1, 6! ] == 31, AppendTo[a, p[[n]]]], {n, 1, Length[p]}]; a (*Artur Jasinski*)

Cf. A000043, A000668, A124477, A139484, A145038, A112633, A145041, A145042, A145044, A145045, A145046.

nonn

Artur Jasinski, Sep 30 2008

approved

1,1

Mersenne numbers (with exception first two) are congruent to 31, 127, 271, 607 mod 6!. This sequences is subset of A000043.

Table of n, a(n) for n=1..13.

p = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 43112609}; a = {}; Do[If[Mod[2^p[[n]] - 1, 6! ] == 127, AppendTo[a, p[[n]]]], {n, 1, Length[p]}]; a (*Artur Jasinski*)

Cf. A000043, A000668, A124477, A139484, A145038, A112633, A145041, A145042, A145044, A145045, A145046.

nonn

Artur Jasinski, Sep 30 2008

approved

1,1

If there are no odd perfect numbers then these are the positions of zeros in A324185.

The next term has 314 digits:

11781361728633673532894774498354952494238773929196300355071513798753168641589311119865182769801300280680127783231251635087526446289021607771691249214388576215221396663491984443067742263787264024212477244347842938066577043117995647400274369612403653814737339068225047641453182709824206687753689912418253153056583680.

Table of n, a(n) for n=1..4.

a(n) = ((2^A000720(A000668(n)))-1) * 2^(A000043(n)-1) = ((2^A059305(n)) - 1) * 2^(A000043(n)-1).

a(n) = A243071(A156552(A324201(n))) = A243071(A156552(A062457(A000043(n)))).

If no odd perfect numbers exist, then a(n) = A243071(A000396(n)), and thus A007814(a(n)) = A007814(A000396(n)).

(PARI) A324200(n) = (2^(A000043(n)-1))*((2^primepi(A000668(n)))-1);

Subsequence of A023758 and A324199.

Cf. A000043, A000396, A000668, A000720, A007814, A023758, A059305, A156552, A243071, A324185.

nonn

Antti Karttunen, Feb 18 2019

approved

1,1

Also the first factor of A330817, 2^(2*p+1)*M_p^2. The second factor of A330817 is A133049.

Walter Kehowski, Table of n, a(n) for n = 1..12

a(n) = 2^(2*A000043(n)+1).

a(1) = 2^(2*2+1) = 32. Since M_2=3, the number 2^5*3^2 has power-spectral basis {225,64}.

A330818:=[]:

for www to 1 do

for i from 1 to 31 do

  #ithprime(31)=127

  p:=ithprime(i);

  q:=2^p-1;

  if isprime(q) then x:=2^(2*p+1); A330818:=[op(A330818), x]; fi;

od;

od;

A330818;

2^(2 * MersennePrimeExponent[Range[10]] + 1) (* Amiram Eldar, Jan 03 2020 *)

Cf. A000043, A000668, A132794, A133049, A330817, A330819, A330820.

nonn

Walter Kehowski, Jan 01 2020

approved