A075177 -id:A075177 - cp's OEIS frontend

A046704 Additive primes: sum of digits is a prime.

2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 113, 131, 137, 139, 151, 157, 173, 179, 191, 193, 197, 199, 223, 227, 229, 241, 263, 269, 281, 283, 311, 313, 317, 331, 337, 353, 359, 373, 379, 397, 401, 409, 421, 443, 449, 461, 463, 467, 487, 557, 571, 577, 593

Offset: 1

Felice Russo

Sum_{a(n) < x} 1/a(n) is asymptotic to (3/2)*log(log(log(x))) as x -> infinity; see Harman (2012). Thus the sequence is infinite. - Jonathan Sondow, Jun 07 2012

Harman 2012 also shows, under a conjecture about primes in short intervals, that there are 3/2 * x/(log x log log x) terms up to x. - Charles R Greathouse IV, Nov 17 2014

			The digit sums of 11 and 13 are 1+1=2 and 1+3=4. Since 2 is prime and 4 is not, 11 is a member and 13 is not. - _Jonathan Sondow_, Jun 07 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Glyn Harman, Counting primes whose sum of digits is prime, J. Integer Seq., 15 (2012), Article 12.2.2.
Glyn Harman, Primes whose sum of digits is prime and metric number theory, Bull. Lond. Math. Soc. 44:5 (2012), pp. 1042-1049.

Indices of additive primes are in A075177.
Cf. A046703, A119450 = Primes with odd digit sum, A081092 = Primes with prime binary digit sum, A104213 = Primes with nonprime digit sum.
Cf. A007953, A010051; intersection of A028834 and A000040.

a046704 n = a046704_list !! (n-1)
a046704_list = filter ((== 1) . a010051 . a007953) a000040_list
-- Reinhard Zumkeller, Nov 13 2011

[ p: p in PrimesUpTo(600) | IsPrime(&+Intseq(p)) ];  // Bruno Berselli, Jul 08 2011

select(n -> isprime(n) and isprime(convert(convert(n,base,10),`+`)), [2,seq(2*i+1,i=1..1000)]); # Robert Israel, Nov 17 2014

Select[Prime[Range[100000]], PrimeQ[Apply[Plus, IntegerDigits[ # ]]]&]

isA046704(n)={local(s,m);s=0;m=n;while(m>0,s=s+m%10;m=floor(m/10));isprime(n) & isprime(s)} \\ Michael B. Porter, Oct 18 2009

is(n)=isprime(n) && isprime(sumdigits(n)) \\ Charles R Greathouse IV, Dec 26 2013

A172216 Smallest k such that sum of digits of prime(n)^k is prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 2, 5, 1, 1, 7, 2, 1, 1, 1, 2, 5, 1, 1, 6, 2, 2, 1, 1, 4, 1, 4, 2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 5, 6, 1, 4, 4, 1, 1, 2, 2, 1, 1, 4, 3, 1, 1, 1, 1, 1, 8, 2, 1, 1, 2, 1, 1, 5, 2, 1, 1, 1, 8, 1, 4, 2, 3, 1, 1, 2, 1, 1, 1, 4, 1, 8, 3, 2, 6, 2, 3, 6, 2, 1, 10, 8, 1

Offset: 1

Klaus Brockhaus, Jan 29 2010

For all n, prime(n)^0 = 1 has nonprime sum of digits 1.

a(n) = 1 iff prime(n) is in A046704, an additive prime. a(n) = 1 iff n is in A075177.

			prime(1) = 2; 2^1 = 2 has prime sum of digits 2. Hence a(1) = 1.
prime(6) = 13; 13^1 = 13 has nonprime sum of digits 4; 13^2 = 169 has nonprime sum of digits 16; 13^3 = 2197 has prime sum of digits 19. Hence a(6) = 3.

Cf. A046704, A075177, A172035.

S:=[]; for n in [1..105] do j:=1; while not IsPrime(&+Intseq(NthPrime(n)^j)) do j+:=1; end while; Append(~S, j); end for; S;

sdp[n_]:=Module[{k=1},While[!PrimeQ[Total[IntegerDigits[Prime[n]^k]]], k++]; k]; Array[sdp,110] (* Harvey P. Dale, Apr 13 2014 *)

A346489 a(n) is the index of the smallest prime whose digits sum to n-th prime.

Original entry on oeis.org

1, 2, 3, 4, 10, 19, 24, 46, 109, 430, 669, 3245, 6057, 7837, 33067, 77777, 476643, 601855, 3556550, 5216954, 15739663, 41146179, 189961757, 882206624, 3325059246, 15581005618, 23007498153, 37607875618

Offset: 1

Ilya Gutkovskiy, Jul 19 2021

			a(10) = 430 because prime(430) = 2999, 2 + 9 + 9 + 9 = 29 = prime(10) and this is the smallest such number.

Cf. A000040, A000720, A007605, A028834, A046704, A046864, A054750, A075177.

Module[{nn=522*10^4,tbl},tbl=Table[{n,Total[IntegerDigits[Prime[n]]]},{n,nn}];Table[SelectFirst[tbl,#[[2]]==Prime[n]&],{n,20}]][[;;,1]] (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, Feb 19 2023 *)

a(n) = A000720(A054750(n)).

a(n) = min {k : A007605(k) = prime(n)}.

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A046704 Additive primes: sum of digits is a prime.

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A172216 Smallest k such that sum of digits of prime(n)^k is prime.

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A346489 a(n) is the index of the smallest prime whose digits sum to n-th prime.

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