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
Keywords
Crossrefs
Cf. A056768.
Programs
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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
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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