A046704 -id:A046704 - cp's OEIS frontend

A249448 Largest n-digit prime whose digit sum is also prime.

7, 89, 991, 9967, 99991, 999983, 9999971, 99999989, 999999937, 9999999943, 99999999821, 999999999989, 9999999999971, 99999999999923, 999999999999883, 9999999999999851, 99999999999999997, 999999999999999967, 9999999999999999919, 99999999999999999989, 999999999999999999829

Offset: 1

Paolo P. Lava, Oct 29 2014

Subsequence of A046704 (primes with digit sum being prime).

Some terms of this sequence are also in A003618, the largest n-digit primes.

			a(1) = 7 because it is the largest prime with just one digit.
a(2) = 89 because it is the largest prime with 2 digits whose sum, 8 + 9 = 17, is a prime.
Again, a(7) = 9999971 because it is the largest prime with 7 digits whose sum is a prime: 9 + 9 + 9 + 9 + 9 + 7 + 1 = 53.

Paolo P. Lava, Table of n, a(n) for n = 1..100

Cf. A046704, A103545, A249447.

P:=proc(q) local a,b,k,n; for k from 0 to q do
for n from 10^(k+1)-1 by -1 to 10^k do if isprime(n) then a:=n; b:=0;
while a>0 do b:=b+(a mod 10); a:=trunc(a/10); od;
if isprime(b) then print(n); break; fi; fi;
od; od; end: P(10^3);

Table[Module[{p=NextPrime[10^n,-1]},While[!PrimeQ[Total[IntegerDigits[p]]],p=NextPrime[p,-1]];p],{n,25}] (* Harvey P. Dale, Jun 20 2023 *)

a(n) = {p = precprime(10^n); while (!isprime(sumdigits(p)), p = precprime(p-1)); p;} \\ Michel Marcus, Oct 29 2014

A330653 The prime numbers whose digit sum, adjacent digit sum concatenation, and adjacent digit difference concatenation are also primes.

Original entry on oeis.org

29, 41, 47, 61, 83, 101, 263, 281, 401, 463, 601, 607, 661, 809, 821, 863, 1129, 1303, 2063, 2267, 3121, 3181, 3301, 3343, 4001, 4603, 5309, 5581, 6007, 6043, 6803, 6863, 7129, 7309, 8009, 8681, 8821, 9721, 9967, 10903, 10909, 14143, 16903, 17209, 18521, 19421, 20063, 20201, 20407, 20807, 21143, 24281, 25147

Offset: 1

Scott R. Shannon, Dec 22 2019

This sequence lists the prime numbers whose digit sum A007953, concatenation of adjacent digit sums A053392, and concatenation of adjacent digit differences A040115, are also primes. Due to the digit sum being prime this is a subsequence of A046704.

For primes up to ten million there are 2268 entries, which is about one prime in every 293. The largest digit sum is 53 for a(1482) = 5986889, the largest adjacent digit sum concatenation is 171818141113 for a(2076) = 8999567, and the largest adjacent digit difference concatenation is 993247 for a(2099) = 9096481.

			a(1) = 29, as 2 + 9 = 11, '2 + 9' = 11, '|2 - 9|' = 7, and 29, 11, 7 are all prime.
a(7) = 263, as 2 + 6 + 3 = 11, '2 + 6' + '6 + 3' = 89, '|2 - 6|' + '|6 - 3|' = 43, and 263, 11, 89, 43 are all prime.
a(25) = 4001, as 4 + 0 + 0 + 1 = 5, '4 + 0' + '0 + 0' + '0 + 1' = 401, '|4 - 0|' + '|0 - 0|' + '|0 - 1|' = 401, and 4001, 5, 401 are all prime.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Chris K. Caldwell, How many primes are there?

Cf. A000040, A007953, A040115, A053392, A046704.

A087340 Primes p such that the sum of the digits of p as well as 1 plus the product of its digits are also primes.

Original entry on oeis.org

2, 11, 23, 29, 41, 43, 47, 61, 67, 89, 223, 227, 229, 263, 269, 281, 463, 643, 661, 821, 827, 883, 887, 1123, 1129, 1163, 1213, 1231, 1237, 1279, 1291, 1297, 1321, 1327, 1361, 1367, 1433, 1439, 1453, 1459, 1471, 1493, 1523, 1543, 1549, 1567, 1613, 1637

Offset: 1

Amarnath Murthy, Sep 06 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A087339.

[p: p in PrimesUpTo(1700) | IsPrime(&+Intseq(p)) and IsPrime(1+&*Intseq(p))]; // Bruno Berselli, Apr 09 2013

Select[Select[Prime[Range[280]], PrimeQ[Plus@@IntegerDigits[ # ]]&], PrimeQ[Times@@IntegerDigits[ # ]+1]&] (from Harvey Dale)

A066725 INTERSECT A046704. - R. J. Mathar, Aug 26 2007

Edited and extended by Robert G. Wilson v, Sep 07 2003

A109982 Primes p such that index of p, the sum of p's digits and the number of p's digits are all primes.

Original entry on oeis.org

11, 41, 67, 83, 157, 179, 191, 241, 283, 331, 353, 401, 461, 599, 739, 773, 797, 919, 991, 10079, 10169, 10433, 10457, 10589, 10631, 10723, 10853, 10909, 11311, 11447, 11867, 11953, 12097, 12143, 12301, 12457, 12479, 12503, 12547, 12763, 13003

Offset: 1

Zak Seidov, Jul 06 2005

			a(414) = 99551 because its index, 9551, the sum, 29 and number, 5, of digits are all primes.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A046704 Additive primes: sum of digits is a prime, A088136 Primes such that sum of first and last digits is prime, A109981 Primes such that the sum of digits and the number of digits are primes.

Select[Prime[Range[200]], PrimeQ[Length[IntegerDigits[ # ]]]&&PrimeQ[Plus@@IntegerDigits[ # ]]&]
Select[Prime[Range[1600]],AllTrue[{PrimePi[#],Total[IntegerDigits[#]], IntegerLength[ #]}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 15 2019 *)

A118571 Sophie Germain primes whose sum of digits is a prime.

Original entry on oeis.org

2, 3, 5, 11, 23, 29, 41, 83, 89, 113, 131, 173, 179, 191, 281, 359, 443, 593, 641, 683, 719, 809, 911, 953, 1013, 1019, 1031, 1103, 1439, 1451, 1499, 1583, 1811, 1901, 2003, 2063, 2069, 2339, 2351, 2393, 2399, 2753, 2939, 3299, 3329, 3389, 3413, 3491, 3761

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 07 2006

			191 is in the sequence because it is a Sophie Germain prime and the sum of its digits 1+9+1 = 11 is a prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A005384 and A028834.

Subsequence of A046704.

Select[Range[4000], PrimeQ[#] && PrimeQ[2*# + 1] && PrimeQ[Plus @@ IntegerDigits[#]] &] (* Amiram Eldar, Feb 08 2021 *)
Select[Prime[Range[600]],AllTrue[{2#+1,Total[IntegerDigits[#]]},PrimeQ]&] (* Harvey P. Dale, Mar 08 2025 *)

A172035 Smallest exponent k > 1 that sum of digits of k-th power of the n-th prime is a prime (n=1,2,...) or 0 if no such k exists.

Original entry on oeis.org

5, 0, 2, 2, 9, 3, 2, 5, 3, 2, 7, 2, 4, 5, 2, 2, 5, 2, 3, 6, 2, 2, 2, 2, 4, 8, 4, 2, 2, 4, 2, 8, 2, 3, 2, 2, 4, 4, 6, 2, 4, 2, 10, 3, 4, 2, 3, 2, 4, 3, 5, 6, 3, 4, 4, 2, 2, 2, 2, 2, 3, 4, 3, 3, 3, 5, 3, 3, 8, 2, 3, 12, 2, 3, 2, 5, 2, 3, 8, 16, 8, 3, 4, 2, 3, 2, 4, 2, 2, 5, 7, 4, 3, 8, 3, 2, 6, 2, 3, 6, 2, 2, 10

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Jan 23 2010

k = 1 is the "trivial" case: sod(prime(n)) = prime(m)

n = 2, prime(2) = 3: 3^k is for k > 1 a multiple of 3^2.

			sod(2^5)=5, sod(5^2)=7, sod(7^2)=13, sod(11^9)=53, sod(13^3)=19, sod(17^2)=19, sod(19^5)=37, sod(23^3)=17, sod(29^2)=13, sod(31^7)=31, sod(37^2)=19, sod(41^4)=31, sod(43^5)=31, sod(47^2)=13, sod(53^2)=19, sod(59^5)=47, sod(61^2)=13, sod(67^3)=19, sod(71^6)=37, sod(73^2)=19, sod(79^2)=13, sod(83^2)=31, sod(89^2)=19, sod(97^4)=43, sod(101^8)=67, sod(103^4)=31, sod(107^2)=19, sod(109^2)=19, sod(113^4)=31, sod(127^2)=19, sod(131^8)=61, sod(137^2)=31, sod(139^3)=37, sod(149^2)=7, sod(151^2)=13, sod(157^4)=31, sod(163^4)=37, sod(167^6)=73, sod(173^2)=31, sod(179^4)=37, sod(181^2)=19, sod(191^10)=97, sod(193^3)=37, sod(197^4)=37, sod(199^2)=19, sod(211^3)=37, sod(223^2)=31, sod(227^4)=43, sod(229^3)=37.

M. Fujiwara, Y. Ogawa: Introduction to truly beautiful Mathematics, Chikuma Shobo, Tokyo 2005.
Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005.
Hans Schubart: Einfuehrung in die klassische und moderne Zahlentheorie, Vieweg, Braunschweig 1974.

Cf. A046704, A007605

Cf. A172216. - Klaus Brockhaus, Jan 29 2010

S:=[ 5, 0 ]; for n in [3..103] do j:=2; while not IsPrime(&+Intseq(NthPrime(n)^j)) do j+:=1; end while; Append(~S, j); end for; S; // Klaus Brockhaus, Jan 29 2010

More terms from Klaus Brockhaus, Jan 29 2010

Edited by Charles R Greathouse IV, Aug 02 2010

A176179 Primes such that the sum of digits, the sum of the squares of digits and the sum of 3rd powers of their digits is also a prime.

Original entry on oeis.org

11, 101, 113, 131, 199, 223, 311, 337, 353, 373, 449, 461, 463, 641, 643, 661, 733, 829, 883, 919, 991, 1013, 1031, 1103, 1301, 1439, 1451, 1471, 1493, 1499, 1697, 1741, 1949, 2089, 2111, 2203, 2333, 2441, 2557, 3011, 3037, 3307, 3323, 3347, 3491, 3583, 3637, 3659, 3673, 3853, 4049, 4111, 4139, 4241, 4337, 4373, 4391, 4409

Offset: 1

Michel Lagneau, Apr 10 2010

See A091365 for the exceptions for the case where the sum of the digits of p is not prime, but the sum of the cubes of the digits of p is prime.

			For the prime number n =5693 we obtain :
5 + 6 + 9 + 3 = 23 ;
5^2 + 6^2 + 9^2 + 3^2 = 151 ;
5^3 + 6^3 + 9^3 + 3^3 = 1097.

Charles W. Trigg, Journal of Recreational Mathematics, Vol. 20(2), 1988.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Mike Mudge, Morph code, Hands On Numbers Count, Personal Computer World, May 1997, p. 290.

Cf. A109181, A046704, A052034, A091366.

with(numtheory):for n from 2 to 10000 do:l:=evalf(floor(ilog10(n))+1):n0:=n:s1:=0:s2:=0:s3:=0:for m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :s1:=s1+u:s2:=s2+u^2:s3:=s3+u^3:od:if type(n,prime)=true and type(s1,prime)=true and type(s2,prime)=true and type(s3,prime)=true then print(n):else fi:od:

okQ[n_]:=Module[{idn=IntegerDigits[n]}, And@@PrimeQ[Total/@{idn,idn^2,idn^3}]]; Select[Prime[Range[600]],okQ]  (* Harvey P. Dale, Jan 18 2011 *)

from sympy import isprime, primerange
def ok(p):
    return all(isprime(sum(int(d)**k for d in str(p))) for k in [1, 2, 3])
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(4409)) # Michael S. Branicky, Nov 23 2021

Corrected and extended by Harvey P. Dale, Jan 18 2011

A239790 The smallest multidigit prime of a sequence of n consecutive primes such that their digit sums are also a sequence of n consecutive primes.

Original entry on oeis.org

11, 41, 41, 191, 402131, 6340271501

Offset: 1

Carlos Rivera, Mar 26 2014

a(7), if it exists, is larger than 2*10^14. - Giovanni Resta, Apr 03 2014
a(7) <= 101100010001001200110001. - Jens Kruse Andersen, Aug 28 2016
a(7) <= 1212030150560200001. - Oscar Volpatti, Aug 25 2025

			a(4)=191 because 191, 193, 197, 199 generates 11, 13, 17, 19.
a(5)=402131 because 402131, 402133, 402137, 402139, 402197 generates 11,13,17,19,23.

Números y Algo Más, Primos cuya suma digital dan primos
Carlos Rivera, Puzzle 1229. Set of consecutive primes such that..., The Prime Puzzles & Problems Connection.
B. Sindelar, Two Sets of Consecutive Primes and their Sum of Digits Connection [Broken link]
Bill Sindelar, Jens Kruse Andersen, Marian Otremba, Two Sets of Consecutive Primes and their Sum of Digits Connection, digest of 12 messages in primenumbers Yahoo group, Aug 26 - Aug 30, 2016.

Cf. A007953, A046704, A240598.

isok(p, n) = if ((p > 10) && isprime(p), my(v=vector(n)); v[1] = p; for (i=2, n, v[i] = nextprime(v[i-1]+1);); my(vs=vector(n, i, sumdigits(v[i]))); if (!isprime(vs[1]), return(0)); for (i=2, n, if (vs[i] != nextprime(vs[i-1]+1), return(0));); return(1););
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Aug 28 2025

10   P=7:KM=0:'puzzle 1290, Meller
   20   P=nxtprm(P):if P>2^32-20 then end
   30   gosub *K:if K<=KM then goto 20
   40   print K,P,Q1:KM=K:goto 20
  100   *K
  110   Z=P:gosub *SODZ
  120   if SODZ<>prmdiv(SODZ) then return
  130   K=1:Q=SODZ:Q1=Q
  140   Z=nxtprm(Z):gosub *SODZ
  150   if SODZ<>nxtprm(Q) then return
  160   K=K+1:Q=nxtprm(Q):goto 140
  200   *SODZ:SODZ=0:L=alen(Z)
  210   for I=1 to L:D=val(mid(str(Z),I+1,1))
  220   SODZ=SODZ+D:next I
  230   return

a(6) from Giovanni Resta, Apr 03 2014

A331031 The prime numbers that are prime-indexed primes and whose digit sum, adjacent digit sum concatenation, and adjacent digit difference concatenation are also primes.

Original entry on oeis.org

41, 83, 401, 2063, 6863, 10909, 20063, 26489, 44621, 105229, 187067, 205507, 233267, 238547, 240047, 243301, 256307, 346763, 367021, 376003, 395581, 555707, 562181, 563467, 600203, 613243, 644843, 675263, 689789, 785801, 787601, 837667, 845381, 954263, 959389, 1070203

Offset: 1

Scott R. Shannon, Jan 07 2020

This sequence lists the prime numbers that are prime-indexed primes, see A006450, and whose digit sum A007953, concatenation of adjacent digit sums A053392, and concatenation of adjacent digit differences A040115, are also primes. This is a subsequence of A006450 and A330653. There are 267 terms for primes up to 20491057.

			a(4) = 2063 as 2063 is the 311th prime, 2+0+6+3 = 11, '2+0'+'0+6'+'6+3' = 269, '|2-0|'+'|0-6|'+'|6-3|' = 263, and 2063, 311, 11, 269, 263 are all prime.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Wikipedia, Super prime.

Cf. A000040, A006450, A330653, A007953, A040115, A053392, A046704.

A068951 Scan the primes, record digit-sum if it is itself prime.

Original entry on oeis.org

2, 3, 5, 7, 2, 5, 11, 5, 7, 11, 7, 13, 11, 17, 2, 5, 5, 11, 13, 7, 13, 11, 17, 11, 13, 17, 19, 7, 11, 13, 7, 11, 17, 11, 13, 5, 7, 11, 7, 13, 11, 17, 13, 19, 19, 5, 13, 7, 11, 17, 11, 13, 17, 19, 17, 13, 19, 17, 23, 7, 13, 11, 13, 17, 13, 17, 17, 13, 19, 13, 19, 17, 23, 17, 11, 13

Offset: 1

Francois Jooste (phukraut(AT)hotmail.com), Mar 10 2002

			a(13)=5 since the 13th prime is 41 and 4+1=5, which is prime.

Cf. A046704.

dig := X->convert((convert(X,base,10)),`+`); a := n->`if`(isprime(dig(ithprime(n)))=true,dig(ithprime(n)),printf(""));

Select[Total[IntegerDigits[#]]&/@Prime[Range[200]],PrimeQ] (* Harvey P. Dale, Oct 17 2020 *)

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A249448 Largest n-digit prime whose digit sum is also prime.

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A330653 The prime numbers whose digit sum, adjacent digit sum concatenation, and adjacent digit difference concatenation are also primes.

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A087340 Primes p such that the sum of the digits of p as well as 1 plus the product of its digits are also primes.

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A109982 Primes p such that index of p, the sum of p's digits and the number of p's digits are all primes.

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A118571 Sophie Germain primes whose sum of digits is a prime.

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A172035 Smallest exponent k > 1 that sum of digits of k-th power of the n-th prime is a prime (n=1,2,...) or 0 if no such k exists.

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A176179 Primes such that the sum of digits, the sum of the squares of digits and the sum of 3rd powers of their digits is also a prime.

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A239790 The smallest multidigit prime of a sequence of n consecutive primes such that their digit sums are also a sequence of n consecutive primes.

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