A046063 -id:A046063 - cp's OEIS frontend

A038601 Prime numbers p such that the number of partitions of p is also a prime.

2, 3, 5, 13, 157, 491, 863, 1621, 2633, 5347, 8117, 13513, 35227, 62311, 76367, 84017, 141637, 170537, 189353, 192667, 201821, 216617, 251677, 269257, 288203, 293621, 353807, 366103, 367621, 372023, 441703, 444167, 478571, 518657, 582371, 626333, 780707, 816521

Offset: 1

Jeff Burch

nonn

			5 = (1+1+1+1+1+1,1+1+1+2,1+1+3,1+4,1+2+2,2+3,5), so partition(5) = 7; 5 and 7 are primes.

Hisanori Mishima, Partition Number (n = 0 to 1000): Factorizations.
Hisanori Mishima, Partition Number (n = 1001 to 2000): Factorizations.
Hisanori Mishima, Partition Number (n = 2001 to 3000); Factorizations.
Hisanori Mishima, Partition Number (n = 3001 to 4000): Factorizations.
Hisanori Mishima, Partition Number (n = 4001 to 5000): Factorizations.
Hisanori Mishima, Partition Number (n = 5001 to 6000): Factorizations.
Hisanori Mishima, Partition Number (n = 6001 to 7000): Factorizations.
Hisanori Mishima, Partition Number (n = 7001 to 8000): Factorizations.
Hisanori Mishima, Partition Number (n = 8001 to 9000): Factorizations.
Hisanori Mishima, Partition Number (n = 9001 to 10000): Factorizations.
Hisanori Mishima, Factorization of Partition Number.

Cf. A046063, A000041, A070177.

Do[ If[ PrimeQ[n] && PrimeQ[ PartitionsP[n]], Print[n]], {n, 1, 10^5} ]

More terms from Simon Plouffe
More terms from Robert G. Wilson v, Aug 29 2001
a(17)-a(36) from Jacques Tramu, Jun 26 2005
Corrected by T. D. Noe, Oct 31 2006
Offset changed and a(37)-a(38) from Michael S. Branicky, Jun 24 2025

A046064 Not a product of partition numbers (A000041).

Original entry on oeis.org

13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, 41, 43, 46, 47, 51, 52, 53, 57, 58, 59, 61, 62, 65, 67, 68, 69, 71, 73, 74, 76, 78, 79, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 102, 103, 104, 106, 107, 109, 111, 113, 114, 115, 116, 117, 118, 119, 122, 123, 124, 127

Offset: 1

Eric W. Weisstein

nonn

Robin Jones, Table of n, a(n) for n = 1..10000
G. K. Patil, Ramanujan's Life And His Contributions In The Field Of Mathematics, International Journal of Scientific Research and Engineering Studies (IJSRES), 1(6) (2014), ISSN: 2349-8862.
Eric Weisstein's World of Mathematics, Abelian Group.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.

Cf. A000041, A046056, A046063, A033637.

with(combinat): A000041:=proc(n) options remember: RETURN(numbpart(n)): end: partdiv:=proc(m,i) local j,q,f: f:=0: for j from i by -1 to 2 while(f=0) do if(irem(m, A000041(j))=0) then q:=iquo(m, A000041(j)): if(q=1) then RETURN(1) else f:=partdiv(q,j) fi fi od: RETURN(f): end: for i from 2 to 14 do for n from A000041(i) to A000041(i+1)-1 do m:=partdiv(n,i): if m=0 then printf("%d, ",n) fi od od: # C. Ronaldo

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

A285216 Indices of primes in A000219.

Original entry on oeis.org

2, 4, 11, 30, 32, 40, 50, 85, 100, 237, 381, 733, 805, 882, 1015, 1650, 2439, 3163, 3335, 3506, 3675, 4152, 4446, 4576, 5010, 5101, 6045, 6760, 7412, 8178, 8562, 10026, 10527, 10888, 12406, 12693, 13479, 16109, 16978, 17962, 20696, 22483, 25383, 31458, 38956

Offset: 1

Vaclav Kotesovec, Apr 14 2017

nonn

			11 is in the sequence because A000219(11) = 859 is prime.

Vaclav Kotesovec, Table of n, a(n) for n = 1..82

Cf. A000219, A035359, A046063.

A000219 = Cases[Import["https://oeis.org/A000219/b000219.txt", "Table"], {, }][[All, 2]];
Flatten[Position[A000219, ?PrimeQ]] - 1 (* _Jean-François Alcover, Sep 17 2020, as of todate, this only gives 27 terms *)

A285217 Indices of primes in A000712.

Original entry on oeis.org

1, 2, 70, 106, 330, 366, 370, 546, 836, 1370, 1870, 2126, 2616, 4240, 4836, 4956, 9520, 10896, 11446, 14250, 15836, 16170, 18040, 18566, 26516, 28676, 37060, 40546, 40760, 46850, 52060, 57176, 67726, 74776, 78460, 90810, 98216, 108870, 115400, 115990, 123930

Offset: 1

Vaclav Kotesovec, Apr 14 2017

nonn

			70 is in the sequence because A000712(70) = 7592053897 is prime.

Vaclav Kotesovec, Table of n, a(n) for n = 1..69

Cf. A000712, A035359, A046063, A265835.

A355705 Indices k of partition function p where p(k) and p(k) - 2 are twin primes.

Original entry on oeis.org

4, 5, 186, 3542

Offset: 1

Serge Batalov, Jul 15 2022

Because asymptotically size of partitions number function p(n) ~ O(exp(sqrt(n))), and probability of primality of p(n) ~ O(1/sqrt(n)) and combined probability of primality of p(n) and p(n)+-2 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(5) > 10^7.

			4 is a term because A000041(4) = 5, and 3 and 5 are twin primes.
5 is a term because A000041(5) = 7, and 5 and 7 are twin primes.

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

for(n=1, 3600, if(ispseudoprime(p=numbpart(n))&&ispseudoprime(p-2), print1(n, ", ")))

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

A145523 Least integer k > 0 such that A000041(k) is divisible by 2^n.

Original entry on oeis.org

1, 2, 11, 11, 15, 66, 66, 96, 96, 96, 96, 96, 3693, 15005, 18978, 18978, 18978, 43002, 55943, 972190, 1151214, 2799146, 15519397, 15519397, 15519397, 122101417, 210553237, 289585489, 473093534

Offset: 0

M. F. Hasler, Oct 12 2008

The requirement a(n) > 0 is somewhat arbitrary, chosen for agreement with A046641; a(n) >= 0 would have been possible, too, yielding a(0)=0.

a(29) > 10^9.

Cf. A000041, A046641, A000123, A046063.

i=1; c=1; while [ $c -le 21 ]; do echo -n `./A046641 $i`", "; c=`expr $c + 1`; i=`expr $i + $i`; done # M. F. Hasler, Oct 18 2008

a(n) = A046641(2^n).

More terms from M. F. Hasler, Oct 18 2008

a(22)-a(28) from Max Alekseyev, Dec 16 2011

cp's OEIS Frontend

A038601 Prime numbers p such that the number of partitions of p is also a prime.

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A046064 Not a product of partition numbers (A000041).

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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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A285216 Indices of primes in A000219.

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A285217 Indices of primes in A000712.

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A355705 Indices k of partition function p where p(k) and p(k) - 2 are twin primes.

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PARI

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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