A073901 Number of primes with nonzero digits and digit sum n.

0, 2, 1, 3, 7, 0, 29, 27, 0, 90, 234, 0, 753, 1025, 0, 3876, 9242, 0, 32549, 50112, 0, 180092, 420318, 0, 1525141, 2467286, 0, 9248093, 20668960, 0, 76318859, 130130794, 0, 487397935, 1066434006, 0

Offset: 1

Author

Amarnath Murthy, Aug 18 2002

Keywords

• base
• more
• nonn

Comments

a(3k) = 0 for all k>1.

The number of candidates to consider for a(n) (i.e. the number of integers with nonzero digits and digit sum n) is A104144(n+8). - Robert Israel, Jun 05 2015

Examples

			a(2) = 2: the two primes are 2 and 11. a(5) = 7: the primes are 5, 41, 23, 113, 131, 311 and 2111.
		

Links

• Manfred Scheucher, Sage Script
• Rémy Sigrist, PARI program

Crossrefs

Not the same as A116381.

Cf. A104144.

Programs

Maple

S[1,1]:= [1]:
for x from 2 to 9 do S[1,x]:= [] od:
a[1]:= 0: a[2]:= 2:
for n from 2 to 22 do
  for x from 2 to 9 do S[n,x]:= map(`+`,S[n-1,x-1],1) od:
  S[n,1]:= [seq(op(map(t -> 10*t+1, S[n-1,x])),x=1..9)];
  if n > 3 and n mod 3 = 0 then a[n]:= 0
  else
    if n > 5 then X:= [1,3,7,9] else X:= [$1..9] fi;
    a[n]:= add(numboccur(map(isprime,S[n,x]),true),x=X);
  fi
od:
seq(a[n],n=1..22); # Robert Israel, Jun 05 2015

Mathematica

f[n_] := If[ Mod[n, 3] == 0 && n > 3, 0, Block[{ip = IntegerPartitions@ n, lng = 1 + PartitionsP@ n, cnt = 0, k = 1}, While[k < lng, If[ Max@ ip[[k]] < 10, cnt += Length@ Select[ FromDigits@# & /@ Permutations@ ip[[k]], PrimeQ]]; k++]; cnt]]; Array[f, 30] (* Robert G. Wilson v, Jun 05 2015 *)
DigitSum[n_, b_:10] := Total[IntegerDigits[n, b]];nextodd[c_] := If[ Length[c]==2, Join[ Table[1, {c[[1]]-2}], {c[[2]]+2}], Join[ Table[1, {c[[1]]-1}], {c[[2]]+1}, Drop[c, 2]]]; a[2]=2; a[n_] := If[Mod[n, 3]==0 && n>3, 0, Module[{c, ct}, For[ c = Table[1, {n}]; ct = 0, True, c = nextodd[c], If[ PrimeQ[ FromDigits[c]] && DigitSum[FromDigits[c]]==n, ct++ ]; If[ c[[ -1]] >= n-1, Return[ct]] ] ]]; Table[ a[n], {n, 20}]

PARI

See Links section.

Python

from collections import Counter
from sympy.utilities.iterables import partitions, multiset_permutations
from sympy import isprime
def A073901(n): return sum(1 for p in partitions(n,k=9) for a in multiset_permutations(Counter(p).elements()) if isprime(int(''.join(str(d) for d in a)))) if n==3 or n%3 else 0 # Chai Wah Wu, Feb 21 2024

Extensions

Edited and extended by Robert G. Wilson v, Sep 19 2002

a(20) to a(24) and alternate Mathematica coding from Dean Hickerson, Sep 21 2002

a(25) from Robert G. Wilson v, Sep 26 2002

a(26)-a(31) from Robert G. Wilson v, Nov 14 2005

Corrected and edited by Manfred Scheucher, Jun 01 2015

a(32)-a(33) from Rémy Sigrist, Nov 17 2022

a(34)-a(36) from Michael S. Branicky, Jul 03 2023