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

%I A070027 #56 May 22 2025 08:32:04
%S A070027 2,3,5,7,11,23,29,41,43,47,61,83,101,113,131,137,151,173,191,223,227,
%T A070027 241,263,281,311,313,317,331,353,401,421,443,461,599,601,641,797,821,
%U A070027 887,911,977,1013,1019,1031,1033,1051,1091,1103,1109,1123,1163,1181,1213
%N A070027 Prime numbers whose initial, all intermediate and final iterated sums of digits are primes.
%C A070027 Subsequence of A046704; actually, exactly those numbers for which the orbit under A007953 is a subset of A046704. - _M. F. Hasler_, Jun 28 2009
%C A070027 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.
%C A070027 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
%H A070027 Alex Costea, <a href="/A070027/b070027.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
%H A070027 Glyn Harman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Harman/harman2.html">Counting Primes whose Sum of Digits is Prime</a>, J. Integer Seq., 15 (2012), Article 12.2.2.
%F A070027 Prime p is a term if and only if p < 10 or A007953(p) is a term. - _Michael S. Branicky_, May 22 2025
%e A070027 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.
%t A070027 dspQ[n_] := TrueQ[Union[PrimeQ[NestWhileList[Plus@@IntegerDigits[#] &, n, # > 9 &]]] == {True}]; Select[Prime[Range[200]], dspQ] (* _Alonso del Arte_, Aug 17 2011 *)
%t A070027 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 *)
%o A070027 (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
%o A070027 (Python)
%o A070027 from sympy import isprime
%o A070027 def ok(n): return isprime(n) and (n < 10 or ok(sum(map(int, str(n)))))
%o A070027 print([k for k in range(2, 1214) if ok(k)]) # _Michael S. Branicky_, May 22 2025
%Y A070027 Cf. A070026 (a supersequence), subsequences: A062802, A070028, A070029.
%Y A070027 Cf. A007953, A046704, A104213, A207293, A207294, A213354, A213355.
%K A070027 nonn,base,easy
%O A070027 1,1
%A A070027 _Rick L. Shepherd_, Apr 14 2002