A049575 -id:A049575 - cp's OEIS frontend

A065728 Partition numbers (A000041) that are semiprimes (A001358).

15, 22, 77, 1255, 2012558, 2679689, 9289091, 18004327, 38887673, 56634173, 72533807, 82010177, 104651419, 2056148051, 2552338241, 20390982757, 27517052599, 118159068427, 749474411781, 5134205287973, 18028182516671

Offset: 1

Patrick De Geest, Nov 18 2001

nonn

Enoch Haga asks if this is a finite sequence. The larger these numbers get, the more opportunity for more factors.

			E.g., the 808th partition number 8151756509675604512522473567 = 5963320232189 * 1366982853893003.

Harry J. Smith, Table of n, a(n) for n=1,...,100
Hisanori Mishima, Factorization results
Eric Weisstein's World of Mathematics, Semiprime.

Intersection of A001358 and A000041.

Cf. A049575, A065729.

Select[PartitionsP[Range[0,450]],PrimeOmega[#]==2&] (* Harvey P. Dale, Sep 19 2016 *)

{ n=0; for (m=1, 10^9, p=numbpart(m); if (bigomega(p) == 2, write("b065728.txt", n++, " ", p); if (n==100, return)) ) } \\ Harry J. Smith, Oct 28 2009

A064911(a(n))*A167392(a(n)) = 1. [From Reinhard Zumkeller, Nov 03 2009]

OFFSET changed from 0,1 to 1,1 by Harry J. Smith, Oct 28 2009

A065729 Numbers k such that the k-th partition number (A000041(k)) is a semiprime.

Original entry on oeis.org

7, 8, 12, 23, 65, 67, 76, 81, 87, 90, 92, 93, 95, 121, 123, 143, 146, 161, 181, 203, 218, 220, 235, 241, 251, 252, 287, 321, 330, 388, 406, 423, 437, 438, 443, 455, 456, 507, 555, 594, 603, 626, 646, 661, 665, 672, 685, 707, 708, 715, 720, 755, 808, 837, 856

Offset: 1

Patrick De Geest, Nov 18 2001

nonn

			The 808th partition number 8151756509675604512522473567 = 5963320232189 * 1366982853893003.

Harry J. Smith, Table of n, a(n) for n = 1..100
Hisanori Mishima, Factorization results
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A000041, A001358, A049575, A065728.

Select[Range[900],PrimeOmega[PartitionsP[#]]==2&] (* Harvey P. Dale, Nov 27 2024 *)

isok(k) = { bigomega(numbpart(k))==2 } \\ Harry J. Smith, Oct 28 2009

Offset changed from 0 to 1 by Harry J. Smith, Oct 28 2009

Missing a(38) = 507 added by Harry J. Smith, Oct 28 2009

A115214 Numbers n such that p(12n) is prime, where p(n) is the number of partitions of n.

Original entry on oeis.org

3, 11, 14, 18, 108, 178, 209, 214, 264, 704, 1085, 1354, 1523, 2550, 2770, 2831, 3709, 6055, 8241, 9011, 10590, 11360, 11780, 15358, 18305, 23576, 23628, 24331, 25589, 25620, 32435, 40219, 41373, 48204, 50239, 53174, 55984, 57521, 78831, 84136

Offset: 1

Robert G. Wilson v, Nov 14 2005

nonn

n belongs to this sequence if and only if 12n belongs to A046063.

Max Alekseyev, Table of n, a(n) for n = 1..410

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

Select[ Range@34000, PrimeQ@ PartitionsP[12# ] &]

More terms from Max Alekseyev, Dec 18 2011

A038753 Nonprime partition numbers.

Original entry on oeis.org

1, 15, 22, 30, 42, 56, 77, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173525

Offset: 1

Henry Bottomley, May 03 2000

nonn

Cf. A000041, A049575.

A005171(a(n))*A167392(a(n)) = 1. [From Reinhard Zumkeller, Nov 03 2009]

A091689 Smallest partition number with n-th prime as factor.

Original entry on oeis.org

2, 3, 5, 7, 11, 3718, 386155, 627, 8349, 2436, 75175, 34262962, 14883, 3010, 526823, 281589, 386155, 1064144451, 124754, 63261, 105558, 2552338241, 4565, 1958, 75175, 101, 12132164, 118114304, 37274405776748077, 1505499, 37338, 6185689, 2323520, 966467, 90175434980549623

Offset: 1

Reinhard Zumkeller, Jan 29 2004

nonn

Erdős conjectured that every prime divides at least one value of the partition function, see Ahlgren and Ono link.

			For n = 10, A000040(10) = 29: a(10) = A000041(26) = 2436 = 29*7*3*2*2, as 29 does not divide smaller partition numbers.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Scott Ahlgren and Ken Ono, Addition and Counting: The Arithmetic of Partitions, Notices of the AMS, 48 (2001) pp. 978-984. See p. 982.
Gerard P. Michon, Table of partition function p(n) (n=0 through 4096).
Eric Weisstein's World of Mathematics, Partition Function P Congruences.

Cf. A051143, A049575, A046063.

Cf. A000041, A091690.

a(n) = A000041(A091690(n)).

More terms from Amiram Eldar, May 16 2025

A094699 Number of prime partition numbers <= n-th partition number.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

Reinhard Zumkeller, May 20 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000041, A000720, A046063, A049575, A094700.

Accumulate[Boole[PrimeQ[Table[PartitionsP[n], {n, 0, 100}]]]] (* Amiram Eldar, May 15 2025 *)

a(n)=sum(i=1,n,ispseudoprime(numbpart(i))) \\ Charles R Greathouse IV, May 28 2015

Offset corrected by Amiram Eldar, May 15 2025

A094700 Number of partition numbers that are smaller than and coprime to the n-th partition number.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 5, 6, 4, 4, 6, 7, 13, 9, 8, 5, 7, 5, 6, 8, 5, 8, 17, 7, 10, 6, 7, 10, 10, 10, 10, 10, 13, 12, 11, 36, 14, 13, 5, 9, 8, 12, 12, 31, 26, 14, 17, 19, 18, 14, 15, 10, 21, 10, 19, 30, 17, 9, 9, 59, 7, 16, 36, 11, 37, 23, 67, 19, 47, 19, 25, 39, 70, 13, 10, 52, 77, 24

Offset: 0

Reinhard Zumkeller, May 20 2004

nonn

a(n) = n iff n <= 1 or A000041(n) is prime. [corrected by Amiram Eldar, May 15 2025]

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000041, A046063, A049575, A094699.

With[{p = Table[PartitionsP[n], {n, 0, 100}]}, Table[Count[p[[1;;k-1]], ?(CoprimeQ[#,p[[k]]]&)], {k, 1, Length[p]}]] (* _Amiram Eldar, May 15 2025 *)

Offset corrected by Amiram Eldar, May 15 2025

A096371 Arithmetic derivative of n-th partition number.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 8, 13, 31, 41, 92, 18, 1, 162, 368, 131, 324, 167, 483, 299, 1788, 841, 256, 1905, 1179, 3680, 2607, 2769, 1383, 7484, 4065, 4664, 10101, 8627, 8030, 1, 5135, 10538, 55107, 42077, 25514, 31443, 90990, 33270, 46823, 89849, 106449, 70151

Offset: 0

Reinhard Zumkeller, Jul 19 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..3000

a:= n->(p->p*add(i[2]/i[1], i=ifactors(p)[2]))(combinat[numbpart](n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 04 2015

a[n_] := If[n<2, 0, Function[p, p*Sum[i[[2]]/i[[1]], {i, FactorInteger[p]}]][PartitionsP[n]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 04 2025, after Alois P. Heinz *)

a(n) = A003415(A000041(n)); a(A049575(n)) = 1.

A239232 a(n) = |{0 < k <= n: p(n+k) + 1 is prime}|, where p(.) is the partition function (A000041).

Original entry on oeis.org

1, 0, 0, 1, 3, 3, 3, 2, 1, 1, 1, 1, 2, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 3, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 3, 4, 5, 5, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 7, 6, 6, 6, 6, 7, 8, 8, 9, 9, 9, 10

Offset: 1

Zhi-Wei Sun, Mar 13 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 3.
(ii) If n > 15, then p(n+k) - 1 is prime for some k = 1, ..., n.
(iii) If n > 38, then p(n+k) is prime for some k = 1, ..., n.
The conjecture implies that there are infinitely many positive integers m with p(m) + 1 (or p(m) - 1, or p(m)) prime.

			a(4) = 1 since p(4+4) + 1 = 22 + 1 = 23 is prime.
a(8) = 2 since p(8+1) + 1 = 31 and p(8+2) + 1 = 43 are both prime.
a(11) = 1 since p(11+8) + 1 = 491 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Cf. A000040, A000041, A049575, A234470, A234569, A238393, A238457, A238509, A238516, A239207, A239209, A239214.

p[n_]:=PartitionsP[n]
a[n_]:=Sum[If[PrimeQ[p[n+k]+1],1,0],{k,1,n}]
Table[a[n],{n,1,80}]

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

cp's OEIS Frontend

A065728 Partition numbers (A000041) that are semiprimes (A001358).

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A065729 Numbers k such that the k-th partition number (A000041(k)) is a semiprime.

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A115214 Numbers n such that p(12n) is prime, where p(n) is the number of partitions of n.

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A038753 Nonprime partition numbers.

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A091689 Smallest partition number with n-th prime as factor.

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A094699 Number of prime partition numbers <= n-th partition number.

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A094700 Number of partition numbers that are smaller than and coprime to the n-th partition number.

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A096371 Arithmetic derivative of n-th partition number.

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