A046063 -id:A046063 - cp's OEIS frontend

A285218 Indices of primes in A001156.

4, 5, 6, 7, 8, 12, 21, 25, 28, 32, 34, 36, 44, 51, 58, 68, 71, 73, 76, 84, 85, 105, 117, 131, 132, 148, 150, 160, 162, 163, 170, 172, 188, 190, 216, 226, 233, 249, 252, 253, 264, 273, 284, 307, 338, 356, 358, 372, 378, 383, 390, 424, 435, 449, 456, 468, 479

Offset: 1

Vaclav Kotesovec, Apr 14 2017

nonn

			51 is in the sequence because A001156(51) = 107 is prime.

Vaclav Kotesovec, Table of n, a(n) for n = 1..2119

Cf. A001156, A046063.

A285219 Indices of primes in A003105.

Original entry on oeis.org

5, 6, 7, 8, 9, 11, 13, 22, 28, 29, 31, 38, 47, 53, 56, 59, 63, 64, 76, 85, 88, 91, 110, 111, 124, 135, 165, 202, 210, 214, 234, 243, 256, 262, 280, 322, 335, 346, 438, 443, 458, 463, 508, 580, 590, 696, 790, 865, 903, 951, 993, 996, 1004, 1163, 1338, 1396

Offset: 1

Vaclav Kotesovec, Apr 14 2017

nonn

			28 is in the sequence because A003105(28) = 47 is prime.

Vaclav Kotesovec, Table of n, a(n) for n = 1..373

Cf. A003105, A035359, A046063, A265835.

A285220 Indices of primes in A006128.

Original entry on oeis.org

2, 30, 49, 54, 56, 85, 91, 104, 135, 175, 195, 203, 266, 303, 343, 409, 421, 427, 445, 448, 523, 538, 607, 622, 780, 1006, 1109, 1186, 1338, 1341, 1453, 1591, 1629, 1707, 1826, 1852, 1884, 1949, 2033, 2039, 2056, 2154, 2408, 2444, 2495, 2559, 3134, 3227, 3240

Offset: 1

Vaclav Kotesovec, Apr 14 2017

nonn

			30 is in the sequence because A006128(30) = 54563 is prime.

Vaclav Kotesovec, Table of n, a(n) for n = 1..257

Cf. A006128, A046063.

A295291 Indices of primes in sequence A000700.

Original entry on oeis.org

8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 20, 24, 29, 32, 35, 38, 39, 51, 56, 61, 77, 82, 88, 90, 91, 92, 107, 118, 119, 123, 139, 148, 161, 162, 166, 185, 189, 190, 194, 208, 214, 333, 346, 355, 373, 401, 402, 493, 543, 567, 578, 603, 629, 653, 665, 666, 678

Offset: 1

Vaclav Kotesovec, Nov 19 2017

nonn

			51 is in the sequence because A000700(51) = 107 is prime.

Vaclav Kotesovec, Table of n, a(n) for n = 1..1755

Cf. A000700, A035359, A046063, A265835, A285217.

A113518 Numbers n such that P(13*n) is prime, where P(n) is the unrestricted partition number.

Original entry on oeis.org

1, 42, 122, 224, 1665, 1861, 2504, 2530, 4750, 4765, 7831, 9589, 9932, 12141, 15574, 15749, 22629, 23492, 24350, 25819, 29837, 29940, 31106, 43589, 44496, 47526, 47751, 48020, 49216, 49304, 49637, 58051, 62381, 64112, 66710, 67047, 69244, 73954, 76985, 77664, 82824, 91694, 92749, 99625

Offset: 1

Parthasarathy Nambi, Jan 12 2006

nonn

Integer n belongs to this sequence if and only if 13*n belongs to A046063.

			If n=224 then P(13*n) is a prime with 56 digits.

Max Alekseyev, Table of n, a(n) for n = 1..379

Cf. A000041, A049575, A046063, A114165, A111389, A111045, A114166, A111036, A114167, A114168, A114169, A114170, A113499, A115214.

Select[ Range@27272, PrimeQ@ PartitionsP[13# ] &] (* Robert G. Wilson v, Jan 17 2006; corrected by Harvey P. Dale, Aug 06 2019 *)

a(5)-a(20) from Robert G. Wilson v, Jan 17 2006

Terms a(21) onward from Max Alekseyev, Dec 18 2011

A114171 Least number m, a multiple of n, such that P(m*n) is prime, where P(k) is the number of partitions of k.

Original entry on oeis.org

2, 1, 1, 1, 1, 1, 11, 21, 4, 44, 7, 3, 1, 12, 35, 81, 1429, 2, 132, 22, 8, 6, 154, 7, 21, 21, 8, 6, 29, 434, 6, 91, 4, 900, 15, 1, 126, 66, 14, 11, 122, 4, 477, 3, 481, 77, 4, 27, 15, 612, 600, 56, 4, 4, 8, 3, 44, 71, 310, 217, 6, 3, 74, 132, 333, 2, 1146, 450, 1655, 186, 58, 3

Offset: 1

Robert G. Wilson v, Nov 14 2005

nonn

Cf. A000041, A046063, A114165, A111389, A111045, A114166, A111036, A114167, A114168, A114169, A114170, A114171.

Do[ PartitionsP[n], {n, 200000}]; f[n_] := Block[{k = 1}, While[ !PrimeQ[ PartitionsP[k*n]], k++ ]; k]; Array[f, 72]

A257662 Least prime q such that p(q*n) is prime, where p(.) is the partition function given by A000041.

Original entry on oeis.org

2, 2, 2, 47, 1481, 31, 11, 557, 277, 1847, 7, 3, 1861, 47, 1451, 557, 1429, 2, 18367, 2069, 13411, 463, 26731, 7, 50119, 61, 101, 877, 29, 11261, 2971, 421, 298589, 32633, 31, 55933, 5521, 7307, 22349, 11, 641, 13, 47881, 3, 2309, 51673, 94309, 186679, 136207, 1301

Offset: 1

Zhi-Wei Sun, Jul 12 2015

nonn

Conjecture: a(n) exists for any n > 0.

This implies the conjecture that the sequence p(n) (n = 1,2,3,...) contains infinitely many primes.

			a(1) = 2 since p(2*1) = 2 is prime.
a(4) = 47 since 47 and p(47*4) = p(188) = 1398341745571 are both prime.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..88
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A000041, A046063, A049575, A259764.

Do[k=0;Label[bb];k=k+1;If[PrimeQ[PartitionsP[Prime[k]*n]],Goto[aa],Goto[bb]]; Label[aa];Print[n, " ", Prime[k]];Continue,{n,1,50}]

a(n)={my(r=1); while(!isprime(numbpart(prime(r)*n)), r++); return(prime(r));}
main(size)={return(vector(size,n,a(n)));} /* Anders Hellström, Jul 12 2015 */

A118653 Least number of squares that add up to the partition number A000041(n).

Original entry on oeis.org

1, 1, 2, 3, 2, 4, 3, 4, 3, 3, 3, 3, 3, 2, 4, 3, 4, 3, 3, 2, 3, 3, 3, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 2, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 2, 2, 4, 2, 3, 3, 4, 3, 4, 3, 3, 4, 3, 2, 4, 4, 4, 3, 3, 2, 3, 3, 3, 2, 2, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 0

Jonathan Vos Post, May 17 2006

			a(20) = 3 because P(20) = 627 = 25^2 + 1^2 + 1^2.
a(36) = 2 because P(36) = 17977 = 124^2 + 51^2, which is prime.
a(66) = 2 because P(66) = 2323520 = 1504^2 + 248^2.
a(67) = 2 because P(67) = 2679689 = 1205^2 + 1108^2.
a(100) = 3 because P(100) = 190569292 = 13730^2 + 1434^2 + 6^2.

Cf. A000009, A000041, A000203, A008284, A002828, A001318, A046063, A103266, A118487.

a(n) = A002828(A000041(n)).

A121061 Numbers k such that k-th partition number A000041(k) is a 3-almost prime (A014612).

Original entry on oeis.org

9, 10, 16, 18, 20, 22, 25, 29, 31, 34, 37, 42, 45, 48, 50, 56, 57, 60, 63, 69, 71, 72, 73, 83, 85, 91, 101, 102, 112, 113, 114, 119, 139, 144, 148, 151, 155, 156, 172, 175, 183, 185, 190, 192, 195, 202, 208, 217, 238, 242, 244, 245, 247, 256, 257, 272, 275, 291, 293

Offset: 1

Jonathan Vos Post, Aug 09 2006

			a(1) = 9 because P(9) = 30 = 2 * 3 * 5.
a(2) = 10 because P(10) = 42 = 2 * 3 * 7.
a(3) = 16 because P(16) = 231 = 3 * 7 * 11.

Cf. A000041, A046063, A065728, A065729.

Select[Range[300],PrimeOmega[PartitionsP[#]]==3&] (* James C. McMahon, Oct 12 2024 *)

A000041(a(n)) in A014612.

112 and 113 inserted by R. J. Mathar, Dec 22 2010

A355728 Indices k of partition function where consecutive p(k) and p(k+1) are prime.

Original entry on oeis.org

2, 3, 4, 5, 1085

Offset: 1

Serge Batalov, Jul 15 2022

Because asymptotically the size of the partition number function p(n) is ~ O(exp(sqrt(n))), and the probability of primality of p(n) is ~ O(1/sqrt(n)) and the combined probability of primality of p(n) and p(n+1) is ~ O(1/n), the sum of the prime probabilities is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.

a(6) > 10^8.

			5 is in the sequence because A000041(5) = 7 and A000041(6) = 11 are prime.

Cf. A000041, A046063, A072213, A284594, A285086, A285087, A285088, A355704, A355705, A355706.

for(k=1, 5000, if(ispseudoprime(numbpart(k))&&ispseudoprime(numbpart(k+1)), print1(k, ", ")))

cp's OEIS Frontend

A285218 Indices of primes in A001156.

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A285219 Indices of primes in A003105.

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A285220 Indices of primes in A006128.

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A295291 Indices of primes in sequence A000700.

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A113518 Numbers n such that P(13*n) is prime, where P(n) is the unrestricted partition number.

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A114171 Least number m, a multiple of n, such that P(m*n) is prime, where P(k) is the number of partitions of k.

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A257662 Least prime q such that p(q*n) is prime, where p(.) is the partition function given by A000041.

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PARI

A118653 Least number of squares that add up to the partition number A000041(n).

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A121061 Numbers k such that k-th partition number A000041(k) is a 3-almost prime (A014612).

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Mathematica

Formula

Extensions

A355728 Indices k of partition function where consecutive p(k) and p(k+1) are prime.

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