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Search: keyword:new
Displaying 1-10 of 267 results found. page 1 2 3 4 5 6 7 8 9 10 ... 27
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A363281 Numbers which are the sum of 4 squares of distinct primes. +0
0
87, 159, 183, 199, 204, 207, 231, 247, 252, 303, 319, 324, 327, 343, 348, 351, 364, 367, 372, 399, 423, 439, 444, 463, 468, 471, 484, 487, 492, 495, 511, 516, 532, 535, 540, 543, 556, 559, 564, 567, 583, 588, 591, 604, 607, 612, 628, 655, 660, 663, 676, 679, 684, 700, 703, 708 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..56.
EXAMPLE
87 is a term as 87 = 2^2 + 3^2 + 5^2 + 7^2.
MATHEMATICA
Select[Range@1000,
Length[PowersRepresentations[#, 4, 2] // Select[AllTrue@PrimeQ] //
Select[DuplicateFreeQ]] > 0 &]
PROG
(Python）
from itertools import combinations as comb
ps=[p**2 for p in [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31]]
a=[n for n in range(1001) if n in [sum(n) for n in list(comb(ps, 4))]]
print(a)
CROSSREFS
Cf. A133524, A051395.
KEYWORD
easy,nonn,new
AUTHOR
Zhining Yang, May 25 2023
STATUS
approved
A364162 Number of chordless cycles (of length > 3) in the complement of the n X n queen graph. +0
0
0, 0, 1, 180, 2263, 13280, 48772, 139880, 335746, 717172, 1394367, 2530308, 4334037, 7090956, 11152386 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Table of n, a(n) for n=1..15.
Eric Weisstein's World of Mathematics, Chordless Cycle
Eric Weisstein's World of Mathematics, Queen Graph
KEYWORD
nonn,more,new
AUTHOR
Eric W. Weisstein, Jul 11 2023
STATUS
approved
A363923 a(n) = n^npf(n) / rad(n), where npf(n) is the number of prime factors with multiplicity of n. +0
0
1, 1, 1, 8, 1, 6, 1, 256, 27, 10, 1, 288, 1, 14, 15, 32768, 1, 972, 1, 800, 21, 22, 1, 55296, 125, 26, 6561, 1568, 1, 900, 1, 16777216, 33, 34, 35, 279936, 1, 38, 39, 256000, 1, 1764, 1, 3872, 6075, 46, 1, 42467328, 343, 12500, 51, 5408, 1, 1417176, 55, 702464 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = n^A001222(n) / A007947(n).
a(n) = 1 <=> n term of A008578.
MAPLE
with(NumberTheory): a := n -> n^NumberOfPrimeFactors(n) / Radical(n):
seq(a(n), n = 1..56);
MATHEMATICA
Array[#^PrimeOmega[#]/(Times @@ FactorInteger[#][[All, 1]]) &, 56] (* Michael De Vlieger, Jul 11 2023 *)
PROG
(PARI) a(n) = my(f=factor(n)); n^bigomega(f)/factorback(f[, 1]); \\ Michel Marcus, Jul 11 2023
CROSSREFS
Cf. A001222, A007947, A205959, A008578.
KEYWORD
nonn,new
AUTHOR
Peter Luschny, Jul 11 2023
STATUS
approved
A364146 Numbers k such that k! belongs to A038040. +0
0
0, 1, 3, 4, 5, 6, 10, 11, 12, 13, 14, 15, 16, 17, 21, 22, 25, 26, 28, 29, 31, 32, 35, 36, 37, 38, 39, 40, 41, 49, 50, 52, 53, 54, 55, 57, 58, 59, 64, 66, 67, 70, 71, 76, 77, 78, 79, 80, 81, 82, 83, 85, 90, 91, 92, 95, 96, 97, 98, 99, 101, 103, 106, 108, 115, 121, 122, 123, 124, 125, 126, 127 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Numbers k such that k! = m * number of divisors of m, for some integer m.
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..500
EXAMPLE
For k = 10, we have k! = m * A000005(m) for m = 37800 or m = 43200. Hence, 10 is a term.
CROSSREFS
Cf. A000005, A000142, A036438, A038040.
KEYWORD
nonn,new
AUTHOR
Max Alekseyev, Jul 10 2023
STATUS
approved
A363240 Number of distinct resistances that can be produced from a circuit that is a 2-connected loopless multigraph with n edges and each edge having a unit resistor. +0
0
1, 2, 5, 12, 32, 88, 260, 819, 2680, 8642, 27976, 88946, 281541, 893028, 2841344, 9092174, 29176634, 93854841, 302611365 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
COMMENTS
The resistances between any two nodes of the graph are counted.
All resistances in A337517 can be obtained by serial combinations of resistances of one or more 2-connected loopless multigraphs.
LINKS
Table of n, a(n) for n=2..20.
EXAMPLE
a(2)=1 since the only multigraph with 2 edges is a double edge graph which forms resistance 1/2.
For n=4, there are a quadruple edge graph (resistance 1/4), a triangle graph with one double edge (2/5 between double edge and 3/5 between single edge) and square graph (3/4 between neighbor nodes and 1 between opposite nodes) so a(4)=5.
CROSSREFS
Cf. A010357, A337517.
KEYWORD
nonn,more,new
AUTHOR
Zhao Hui Du, May 23 2023
STATUS
approved
A363274 a(n) is the smallest prime such that a(1)*a(2)*...*a(n) +/- 1 is prime. +0
0
2, 2, 2, 2, 2, 3, 2, 2, 2, 11, 3, 3, 2, 2, 5, 2, 2, 3, 3, 5, 2, 2, 5, 2, 2, 7, 3, 3, 2, 2, 23, 47, 2, 2, 2, 29, 2, 53, 13, 5, 17, 29, 5, 3, 3, 3, 5, 17, 2, 3, 3, 79, 31, 11, 7, 23, 2, 7, 3, 7, 2, 13, 5, 7, 5, 7, 23, 17, 23, 13, 31, 29, 7, 67, 47, 7, 2, 23, 13, 23, 23, 31, 73, 13, 23, 3, 17 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..87.
MATHEMATICA
a[1] = 2; a[n_] := a[n] = Module[{r = Product[a[k], {k, 1, n - 1}], p = 2}, While[! PrimeQ[r*p - 1] && ! PrimeQ[r*p + 1], p = NextPrime[p]]; p]; Array[a, 100] (* Amiram Eldar, Jul 08 2023 *)
CROSSREFS
Cf. A036014, A236465.
KEYWORD
nonn,new
AUTHOR
Dmitry Kamenetsky, Jul 08 2023
STATUS
approved
A363377 Largest positive integer having n holes that can be made using the fewest possible digits. +0
0
7, 9, 8, 98, 88, 988, 888, 9888, 8888, 98888, 88888, 988888, 888888, 9888888, 8888888, 98888888, 88888888, 988888888, 888888888, 9888888888, 8888888888, 98888888888, 88888888888, 988888888888, 888888888888, 9888888888888, 8888888888888, 98888888888888, 88888888888888, 988888888888888 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Each decimal digit has 0, 1 or 2 holes so that n holes requires A065033(n) digits.
LINKS
Table of n, a(n) for n=0..29.
Index entries for sequences related to holes in digits.
Index entries for linear recurrences with constant coefficients, signature (1,10,-10).
FORMULA
From Nathan L. Skirrow, Jun 26 2023: (Start)
a(n) = (89*(10^((n-1)/2))-8)/9 for odd n; a(n) = 8*(10^(n/2)-1)/9 for even n >= 2.
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3), for n >= 4.
G.f.: (7+2*x-71*x^2+70*x^3)/((1-x)*(1-10*x^2)).
E.g.f.: (80*cosh(sqrt(10)*x) + 89*sqrt(10)*sinh(sqrt(10)*x) - 80*e^x)/90 + 7. (End)
EXAMPLE
For n=0, the largest integer with no holes in it that is as short as possible is 7 (9 is larger, but has 1 hole; 11 is larger and has no holes, but is longer at length 2 > length 1).
For n=1, the largest integer with 1 hole that is as short as possible is 9 (following the same kind of reasoning as with n=0).
MATHEMATICA
CoefficientList[Series[(7 + 2 x - 71 x^2 + 70 x^3)/((1 - x) (1 - 10 x^2)), {x, 0, 30}], x] (* Michael De Vlieger, Jul 05 2023 *)
PROG
(Python)
A363377=lambda n: (8+n%2*81)*10**(n>>1)//9 if n else 7
print([A363377(n) for n in range(30)]) # Nathan L. Skirrow, Jun 26 2023
CROSSREFS
Cf. A002281 and A002282 (number of holes), A065033 (digits required).
Cf. A249572 and A250256 (smallest number).
Cf. A337099 (largest 7-segment).
KEYWORD
nonn,base,easy,new
AUTHOR
Julia Zimmerman, May 29 2023
STATUS
approved
A363256 Number of length n strings on the alphabet {0,1,2,3} with digit sum at most 4. +0
0
1, 4, 13, 32, 66, 121, 204, 323, 487, 706, 991, 1354, 1808, 2367, 3046, 3861, 4829, 5968, 7297, 8836, 10606, 12629, 14928, 17527, 20451, 23726, 27379, 31438, 35932, 40891, 46346, 52329, 58873, 66012, 73781, 82216, 91354, 101233, 111892, 123371, 135711 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..40.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
a(n) = (((n + 10)*n + 35)*n + 26)*n/24 + 1.
G.f.: -(x^4 - 3*x^3 + 3*x^2 - x + 1)/(x - 1)^5.
a(n) = 1 + A005718(n-1) for n>=1.
EXAMPLE
For n=2, the 13 strings are all possible 2-character strings of '0', '1', '2' and '3' except the four strings '33', '32', '23'.
MATHEMATICA
f[n_, r_, l_] := If[r < 0, 0, If[r==0, 1, If[l < 0, 0, If[l == 0, 1, Sum[f[n, r-j, l-1], {j, 0, n}]]]]]; Table[f[3, 4, x], {x, 0, 40}]
CROSSREFS
Cf. A227259 (the same for {0,1,2} with digit sum <= 4).
Cf. A105163 (the same for {0,1,2} with digit sum <= 3, shifted by 2).
Cf. A005718.
KEYWORD
nonn,easy,new
AUTHOR
Daniel T. Martin, May 23 2023
STATUS
approved
A362466 First occurrence of n in A362465 or -1 if n is not a term in A362465. +0
1
1, 0, 3, 29, -1, 521, -1, 31751, -1, 47973321, -1 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,3
COMMENTS
All terms a(n) for even n > 4 are conjectures and conjectured to be -1. This follows from the reason derived from Polignac's conjecture that is described in A362465.
The requirements for the truth of the above conjecture about this sequence seem to be notably weaker than for Polignac's conjecture.
LINKS
Table of n, a(n) for n=2..12.
EXAMPLE
a(5) = 29 because it is the least number that cannot be expressed as a sum of more than 1 and fewer than 5 consecutive signed primes. The example in A362465 shows that there is a solution for 29 with 5 consecutive signed primes, but not with more than 1 and fewer than 5.
CROSSREFS
Cf. A362465.
KEYWORD
more,sign,new
AUTHOR
Karl-Heinz Hofmann and Peter Munn, Apr 24 2023
STATUS
approved
A363168 Balanced primes of order 100. +0
0
27947, 111337, 193283, 197341, 197621, 347063, 809821, 955193, 1029803, 1184269, 1292971, 1609163, 1630859, 1656019, 1752449, 1883381, 1935517, 1969661, 2120221, 2156383, 2238959, 2287133, 2548631, 2592089, 2750903, 2866403, 3165769, 3257941, 3590299, 3889423 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A prime p is in this sequence if the sum of the 100 consecutive primes just less than p, plus p, plus the sum of the 100 consecutive primes just greater than p, divided by 201 equals p.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..350 (* All terms up to the 4 millionth prime. *)
MATHEMATICA
Module[{bal=100, nn=300000}, Select[Partition[Prime[Range[nn]], 2bal+1, 1], Mean[#]== #[[bal+1]]&]] [[;; , 101]]
CROSSREFS
Cf. Balanced primes of order b: A006562 (b=1), A082077 (b=2), A082078 (b=3), A082079 (b=4), A096697 (b=5), A096698 (b=6), A096699 (b=7), A096700 (b=8), A096701 (b=9), A096702 (b=10), A096703 (b=11), A096704 (b=12), A300364 (b=13), A300365 (b=14).
KEYWORD
nonn,new
AUTHOR
Harvey P. Dale, Jul 07 2023
STATUS
approved
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Last modified July 12 01:53 EDT 2023. Contains 363813 sequences. (Running on oeis4.)