A146949 Primes with a prime number of partitions into prime parts.

5, 7, 17, 19, 73, 103, 263, 307, 653, 673, 743, 823, 839, 1109, 1327, 2647, 4391, 4621, 4967, 6389, 6661, 6829, 6871, 12227, 12269, 18839, 19861, 20663, 23497, 23593, 23833, 24499, 25411, 28771, 29717, 36599, 40949, 41617, 46889, 47353, 49033, 50093, 50587, 50599, 51511

Offset: 1

Author

Lekraj Beedassy, Nov 03 2008

Keywords

• nonn

Crossrefs

Cf. A056768.

Programs

Maple

g:=1/(product(1-x^ithprime(j),j=1..500)): gser:= series(g,x=0,3575): a:= proc (n) if isprime(coeff(gser,x,ithprime(n)))=true then ithprime(n) else end if end proc: seq(a(n),n=1..3570); # Emeric Deutsch, Nov 09 2008
##
b:= proc(n, i) local r, m; if n<0 or i<2 then 0 elif n<6 or i<6 then m:= iquo(n, 30, 'r'); (5+15*m+r)*m+ [1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 9, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19][r+1] else b(n, i):= b(n-i, i) +b(n, prevprime(i)) fi end: a:= proc(n) local k; k:= `if`(n=1, 3, nextprime(a(n-1))); while not (isprime(b(k, k))) do k:= nextprime(k) od; a(n):= k end: seq(a(n), n=1..15);  # Alois P. Heinz, Jun 26 2009

Mathematica

jmax = 1000;
pmax = Prime[jmax];
g = 1/Product[1-x^Prime[j], {j, 1, jmax}];
cc = CoefficientList[g + O[x]^pmax, x];
Select[Transpose[{cc, Range[0, Length[cc]-1]}], PrimeQ[#[[1]]] && PrimeQ[#[[2]]]&][[All, 2]] (* Jean-François Alcover, Dec 06 2020, after Emeric Deutsch *)

Formula

Prime number 7 = 5 + 2 = 3 + 2 + 2, with 3 (prime number) partitions into prime parts. So 7 is in the sequence. Similarly with 17 = 13+2+2 = 11+3+3 = 11+2+2+2 = 7+7+3 = 7+5+5 = 7+5+3+2 = 7+3+3+2+2 = 7+2+2+2+2+2 = 5+5+5+2 = 5+5+3+2+2 = 5+3+3+3+3 = 5+3+3+2+2+2 = 5+2+2+2+2+2+2 = 3+3+3+3+3+2 = 3+3+3+2+2+2+2 = 3+2+2+2+2+2+2+2, having 17 (prime number) partitions into prime parts.

Extensions

Edited. - Lekraj Beedassy, Nov 08 2008

More terms from Emeric Deutsch, Nov 09 2008

a(17) - a(28) from Alois P. Heinz, Jun 26 2009

Further terms from Max Alekseyev, May 15 2011