A070027 Prime numbers whose initial, all intermediate and final iterated sums of digits are primes.

2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 83, 101, 113, 131, 137, 151, 173, 191, 223, 227, 241, 263, 281, 311, 313, 317, 331, 353, 401, 421, 443, 461, 599, 601, 641, 797, 821, 887, 911, 977, 1013, 1019, 1031, 1033, 1051, 1091, 1103, 1109, 1123, 1163, 1181, 1213

Offset

1,1

Comments

Subsequence of A046704; actually, exactly those numbers for which the orbit under A007953 is a subset of A046704. - M. F. Hasler, Jun 28 2009

Supersequences: A046704 is primes p with digit sum s(p) also prime; A207294 is primes p with s(p) and s(s(p)) also prime.

Disjoint sequences: A104213 is primes p with s(p) not prime; A207293 is primes p with s(p) also prime, but not s(s(p)); A213354 is primes p with s(p) and s(s(p)) also prime, but not s(s(s(p))); A213355 is smallest prime p with k-fold digit sum s(s(..s(p)..)) also prime for all k < n, but not for k = n. - Jonathan Sondow, Jun 13 2012

Links

Alex Costea, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Glyn Harman, Counting Primes whose Sum of Digits is Prime, J. Int. Seq. 15 (2012), Art. 12.2.2.

Jens Lehmann, An Explicit Surjectivity Threshold for Digit Sums of Primes, arXiv:2606.04677 [math.NT], 2026. See pp. 1, 3, 74-77.

Formula

Prime p is a term if and only if p < 10 or A007953(p) is a term. - Michael S. Branicky, May 22 2025

Example

599 is a term because 599, 5+9+9 = 23 and 2+3 = 5 are all prime. 2999 is a term because 2999, 2+9+9+9 = 29, 2+9 = 11 and 1+1 = 2 are all prime. See A062802 and A070026 for related comments.

Mathematica

dspQ[n_] := TrueQ[Union[PrimeQ[NestWhileList[Plus@@IntegerDigits[#] &, n, # > 9 &]]] == {True}]; Select[Prime[Range[200]], dspQ] (* Alonso del Arte, Aug 17 2011 *)
(* Alternative: *)
isdpQ[n_]:=AllTrue[Rest[NestWhileList[Total[IntegerDigits[#]]&, n, #>9&]], PrimeQ]; Select[Prime[Range[300]], isdpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 12 2017 *)

Prog

(PARI) isA070027(n)={ while(isprime(n), n<9 && return(1); n=vector(#n=eval(Vec(Str(n))), i, 1)*n~)} \\ M. F. Hasler, Jun 28 2009
(Python)
from sympy import isprime
def ok(n): return isprime(n) and (n < 10 or ok(sum(map(int, str(n)))))
print([k for k in range(2, 1214) if ok(k)]) # Michael S. Branicky, May 22 2025

Crossrefs

Cf. A070026 (a supersequence), subsequences: A062802, A070028, A070029.

Cf. A007953, A046704, A104213, A207293, A207294, A213354, A213355.

Sequence in context: A386004 A089392 A089695 * A207294 A156658 A118723

Adjacent sequences: A070024 A070025 A070026 * A070028 A070029 A070030

Keyword

nonn,base,easy

Author

Rick L. Shepherd, Apr 14 2002

Status

approved