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 A070027 Prime numbers whose initial, all intermediate and final iterated sums of digits are primes. 9

%I

%S 2,3,5,7,11,23,29,41,43,47,61,83,101,113,131,137,151,173,191,223,227,

%T 241,263,281,311,313,317,331,353,401,421,443,461,599,601,641,797,821,

%U 887,911,977,1013,1019,1031,1033,1051,1091,1103,1109,1123,1163,1181,1213

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

%C Subsequence of A046704 ; actually, exactly those numbers for which the orbit under A007953 is a subset of A046704. [From _M. F. Hasler_, Jun 28 2009]

%C 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 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

%e 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.

%p P:=proc(i) local a,k,n,w; for n from 1 by 1 to i do a:=ithprime(n); while isprime(a) and floor(a/10)>0 do w:=0; k:=a; while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; a:=w; od; if isprime(a) then print(ithprime(n)); fi; od; end: P(1000); - _Paolo P. Lava_, Jul 30 2008

%t dspQ[n_] := TrueQ[Union[PrimeQ[NestWhileList[Plus@@IntegerDigits[#] &, n, # > 9 &]]] == {True}]; Select[Prime[Range[200]], dspQ] (* From Alonso del Arte, Aug 17 2011 *)

%o (PARI) isA070027(n)={ while(isprime(n), n<9 && return(1); n=vector(#n=eval(Vec(Str(n))),i,1)*n~)} \\ [From _M. F. Hasler_, Jun 28 2009]

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

%Y Cf. also A046704, A104213, A207293, A207294, A213354, A213355.

%K base,easy,nonn

%O 0,1

%A _Rick L. Shepherd_, Apr 14 2002

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