A065728 -id:A065728

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A049575

Prime partition numbers.

+10
26

2, 3, 5, 7, 11, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557, 74878248419470886233, 1394313503224447816939

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..100

Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

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, Partition Function P Congruences.

Eric Weisstein's World of Mathematics, Partition Function P.

FORMULA

a(n) = A000041(A046063(n)) = A000040(A051143(n)). - M. F. Hasler, Oct 19 2008

A010051(a(n))*A167392(a(n)) = 1. - Reinhard Zumkeller, Nov 03 2009

MATHEMATICA

lst={}; Do[a=PartitionsP[n]; If[PrimeQ[a], AppendTo[lst, a]], {n, 2*6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jun 14 2009 *)

Select[PartitionsP[Range[1000]], PrimeQ] (* Harvey P. Dale, Mar 11 2013 *)

PROG

(PARI) for(n=0, 10^3, my(p=numbpart(n)); if(isprime(p), print1(p, ", "))); \\ Joerg Arndt, May 09 2013

CROSSREFS

Intersection of A000040 and A000041.

Cf. A046063, A051143.

Cf. A038753, A065728. - Reinhard Zumkeller, Nov 03 2009

KEYWORD

nonn,nice

AUTHOR

G. L. Honaker, Jr.

EXTENSIONS

More terms from James A. Sellers and Christian G. Bower, Oct 15 1999.

STATUS

approved

A065729

n-th partition number (A000041(n)) is a semiprime.

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..100

Hisanori Mishima, Factorization results

Eric Weisstein's World of Mathematics, Semiprime.

EXAMPLE

The 808th partition number 8151756509675604512522473567 = 5963320232189 * 1366982853893003.

MATHEMATICA

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

PROG

(PARI) { n=0; for (m=1, 10^9, p=numbpart(m); if (bigomega(p) == 2, write("b065729.txt", n++, " ", m); if (n==100, return)) ) } \\ Harry J. Smith, Oct 28 2009

CROSSREFS

Intersection of A001358 and A000041.

Cf. A049575, A065728.

KEYWORD

nonn,changed

AUTHOR

Patrick De Geest, Nov 18 2001

EXTENSIONS

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

STATUS

approved

A121061

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

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..59.

FORMULA

A000041(a(n)) in A014612.

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

Cf. A000041, A046063, A065728, A065729.

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Aug 09 2006

EXTENSIONS

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

STATUS

approved

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