A062339 -id:A062339 - cp's OEIS frontend

A070247 Palindromic primes with digit sum 5.

5, 131, 10301, 1003001, 100030001, 100111001, 101000010000101, 10000010101000001, 101000000010000000101, 110000000010000000011, 10000000000300000000001, 10000100000100000100001, 100000100000010000001000001, 10000000000000300000000000001, 10000000001000100010000000001

Offset: 1

Amarnath Murthy, May 05 2002

It is conjectured that are just 3 palindromic primes with digit sum 2, namely 2, 11 and 101. If any others exist, they must be of the form 10^(2^k) + 1 with k > 14.
From Jeppe Stig Nielsen, Aug 30 2025: (Start)
It is now known that any additional primes 10^(2^k) + 1 must have k >= 31.
Digit sum 3 yields only one prime, 3, a palindrome in a vacuous way.
Digit sum 4 leads to primes (A062339), but such numbers can never be palindromes. Proof: Let w be any palindrome with digit sum 4. So w = 10^a + 10^b + 10^c + 10^d with a >= b >= c >= d >= 0. But then 10^c + 10^d is a nontrivial divisor of w, showing that w is not prime.
You may have come here searching for the subsequence 5, 131, 10301, 1003001, 100030001, 10000000000300000000001, ... where the largest digit exceeds 1. See A171376 and A100028 for information on them.
(End)

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..386 (all terms below 10^1000; terms n = 1..238 from Chai Wah Wu)
Hans Riesel, Some factors of the numbers Gn = 6^2^n+1 and Hn = 10^2^n+1, Math. Comp. 23 (1969), p. 413-415. With errata reported in Math. Comp. 24 (1970), p. 243.

Cf. A002385, A062341, A070248, A070249.

Do[p = Join[ IntegerDigits[n, 4], Reverse[ Drop[ IntegerDigits[n, 4], -1]]]; q = Plus @@ p; If[q == 5 && PrimeQ[ FromDigits[p]] && q == 5, Print[ FromDigits[p]]], {n, 1, 4 10^8}] (* this coding will not pick up the first entry *)

for(i=0,50,for(j=0,i,p=10^(2*i)+10^(i+j)+10^i+10^(i-j)+1;isprime(p)&&print1(p,", "))) \\ Jeppe Stig Nielsen, Aug 30 2025

Edited by Robert G. Wilson v, May 15 2002

More terms from Chai Wah Wu, Nov 25 2015

A106756 Primes with digit sum = 14.

Original entry on oeis.org

59, 149, 167, 239, 257, 293, 347, 383, 419, 491, 509, 563, 617, 653, 743, 761, 941, 1049, 1193, 1229, 1283, 1319, 1373, 1409, 1427, 1481, 1553, 1571, 1607, 1733, 1823, 1913, 1931, 2039, 2129, 2237, 2273, 2309, 2381, 2417, 2543, 2633, 2741, 2903, 3083

Offset: 1

Zak Seidov, May 16 2005

Or prime numbers in A114527. - Zak Seidov, May 21 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000040 (primes), A007953 (sum of digits), A235225 (digit sum = 14).
Cf. A062339 (same for digit sum s = 4), A106755 (s = 13), A106757 (s = 16), and others listed in A244918 (s = 68).
Cf. A034690, A086793, A114527.

[p: p in PrimesUpTo(4000) | &+Intseq(p) eq 14]; // Vincenzo Librandi, Jul 08 2014

Select[Prime[Range[10000]], Total[IntegerDigits[#]]==14 &] (* Vincenzo Librandi, Jul 08 2014 *)

select( {is_A106756(n)= sumdigits(n)==14 && isprime(n)}, primes([1, 3333])) \\ M. F. Hasler, Mar 09 2022

Intersection of A000040 (primes) and A235225 (digit sum = 14); also equals { p in A000040 | A007953(p) = 14 }. - M. F. Hasler, Mar 09 2022

A106757 Primes with digit sum = 16.

Original entry on oeis.org

79, 97, 277, 349, 367, 439, 457, 547, 619, 673, 691, 709, 727, 853, 907, 1069, 1087, 1249, 1429, 1447, 1483, 1609, 1627, 1663, 1753, 1861, 1933, 1951, 2239, 2293, 2347, 2383, 2437, 2473, 2617, 2671, 2707, 2833, 2851, 3049, 3067, 3229, 3319, 3373, 3391

Offset: 1

Zak Seidov, May 16 2005

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

Cf. A000040 (primes), A007953 (sum of digits), A235227 (digit sum = 16).
Cf. A062339 (same for digit sum s = 4), A106756 (s = 14), A106758 (s = 17), and others listed in A244918 (s = 68).
Subsequence of A062342 (primes whose sum of digits is a multiple of 8) and of A107288 (primes with sum of digits a square).

[p: p in PrimesUpTo(4000) | &+Intseq(p) eq 16]; // Vincenzo Librandi, Jul 08 2014

Reap[Do[If[16==Apply[Plus,IntegerDigits[p=Prime[n]]],Sow[p]],{n,1000}]][[2,1]] (* Zak Seidov, Oct 30 2009 *)
Select[Prime[Range[500]],Total[IntegerDigits[#]]==16&] (* Harvey P. Dale, Nov 14 2011 *)

select( {is_A106757(n)= sumdigits(n)==16 && isprime(n)}, primes([1, 3333])) \\ M. F. Hasler, Mar 09 2022

Intersection of A000040 (primes) and A235227 (digit sum = 16); also equals { p in A000040 | A007953(p) = 16 }. - M. F. Hasler, Mar 09 2022

More terms from Zak Seidov, Oct 30 2009

A107619 Primes with digit sum = 65.

Original entry on oeis.org

29999999, 39899999, 39999899, 48999989, 49898999, 49899989, 49979999, 49997999, 57899999, 57998999, 57999899, 58899989, 58989899, 58998899, 59879999, 59898899, 59898989, 59979989, 59987999, 59988989, 59999879

Offset: 1

Zak Seidov, May 18 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..460

Cf. A062339, A062341, A062343, A106754 - A106787, A107617, A107618.

Cf. Similar sequences listed in A244918.

[p: p in PrimesUpTo(69000000) | &+Intseq(p) eq 65]; // Vincenzo Librandi, Jul 09 2014

Select[Prime[Range[600000]], Total[IntegerDigits[#]]==65 &] (* Vincenzo Librandi, Jul 09 2014 *)

A106755 Primes p with digital sum equal to 13.

Original entry on oeis.org

67, 139, 157, 193, 229, 283, 337, 373, 409, 463, 571, 607, 643, 661, 733, 751, 823, 1039, 1093, 1129, 1237, 1291, 1327, 1381, 1453, 1471, 1543, 1723, 1741, 1831, 2029, 2083, 2137, 2281, 2371, 2551, 2713, 2731, 2803, 3019, 3037, 3109, 3163, 3181, 3217, 3253

Offset: 1

Zak Seidov, May 16 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..4000

Cf. A000040 (primes), A007953 (sum of digits), A143164 (digit sum = 13).

Cf. A062339 (same for digit sum s = 4), ..., A106754 (s = 11), A106756 (s = 14), and others listed in A244918 (s = 68).

[p: p in PrimesUpTo(10000) | &+Intseq(p) eq 13]; // Vincenzo Librandi, Jul 08 2014

Select[Prime[Range[100000]], Total[IntegerDigits[#]]==13 &] (* Vincenzo Librandi, Jul 08 2014 *)

select( {is_A106755(n)=sumdigits(n)==13&&isprime(n)}, primes([1, 3333])) \\ M. F. Hasler, Mar 09 2022

Intersection of A000040 (primes) and A143164 (digit sum = 13); also equals { p in A000040 | A007953(p) = 13 }. - M. F. Hasler, Mar 09 2022

A106807 Primes with digit sum = 67.

Original entry on oeis.org

59899999, 69899899, 69899989, 69979999, 69997999, 69999799, 77899999, 78997999, 78998989, 78999889, 78999979, 79699999, 79879999, 79889899, 79979899, 79979989, 79988899, 79989979, 79996999, 79997899, 79997989

Offset: 1

Zak Seidov, May 18 2005

499999909 is the smallest term that contains 0 as a digit. - Altug Alkan, Mar 25 2018

Robert Israel, Table of n, a(n) for n = 1..10000 (first 128 terms from Vincenzo Librandi)

Cf. A007605, A062339, A062341, A062343, A106754 - A106787, A107617, A107618.

Cf. similar sequences listed in A244018.

[p: p in PrimesUpTo(90000000) | &+Intseq(p) eq 67]; // Vincenzo Librandi, Jul 09 2014

F:= proc(t,d)
  if d = 1 then
     if t<=9 then return [t] else return [] fi
  fi;
  if t > 9*d then return [] fi;
  [seq(op(map(x -> a*10^(d-1)+x, procname(t-a,d-1))), a=0..min(9,t))]
end proc:
select(isprime, F(67,8)); # Robert Israel, Mar 25 2018

Select[Prime[Range[600000]], Total[IntegerDigits[#]]==67 &] (* Vincenzo Librandi, Jul 09 2014 *)

isok(n) = isprime(n) && (sumdigits(n) == 67); \\ Altug Alkan, Mar 25 2018

A133223 Sum of digits of primes (A007605), sorted and with duplicates removed.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103

Offset: 1

Lekraj Beedassy, Dec 19 2007

Presumably this is 3 together with numbers greater than 1 and not divisible by 3 (see A001651). - Charles R Greathouse IV, Jul 17 2013. (This is not a theorem because we do not know if, given s > 3 and not a multiple of 3, there is always a prime with digit-sum s. Cf. A067180, A067523. - N. J. A. Sloane, Nov 02 2018)
From Chai Wah Wu, Nov 04 2018: (Start)
Conjecture: for s > 10 and not a multiple of 3, there exists a prime with digit-sum s consisting only of the digits 2 and 3 (cf. A137269). This conjecture has been verified for s <= 2995.
Conjecture: for s > 18 and not a multiple of 3, there exists a prime with digit-sum s consisting only of the digits 3 and 4. This conjecture has been verified for s <= 1345.
Conjecture: for s > 90 and not a multiple of 3, there exists a prime with digit-sum s consisting only of the digits 8 and 9. This conjecture has been verified for s <= 8995.
Conjecture: for 0 < a < b < 10, gcd(a, b) = 1 and ab not a multiple of 10, if s > 90 and s is not a multiple of 3, then there exists a prime with digit-sum s consisting only of the digits a and b. (End)

Chai Wah Wu, Table of n, a(n) for n = 1..5997

Cf. A007605, A067523, A067180, A106754-A106787, A062339, A062341, A062337, A137269.

Corrected by Jeremy Gardiner, Feb 09 2014

A259866 Primes of the form 3*10^k + 1.

Original entry on oeis.org

31, 3001, 30000001, 30000000001, 30000000000000000000000000001, 3000000000000000000000000000000000001, 30000000000000000000000000000000000000000000000000000000000000000001

Offset: 1

N. J. A. Sloane, Jul 08 2015

nonn

S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]

Cf. A199683, A056807, A062339.

Select[3*10^Range[70]+1,PrimeQ] (* Harvey P. Dale, Jul 05 2020 *)

a(n) = 3*10^A056807(n)+1. - R. J. Mathar, Jul 15 2015

A168586 Smallest n-digit prime with only digits 0 and 1, and having least digit sum (or 0, if no such prime exists).

Original entry on oeis.org

0, 11, 101, 0, 10111, 101111, 1011001, 10010101, 101001001, 1000001011, 10000001101, 100000010101, 1000100000101, 10000000001011, 100000000100101, 1000000000100011, 10000000000001101, 100000000001010001, 1000000000000010011

Offset: 1

Lekraj Beedassy, Nov 30 2009

Robert Israel, Table of n, a(n) for n = 1..800

Cf. A168587, A062339.

f:= proc(n) local s, c, i,Cands;
  for s from 2 to n do
    if s mod 3 = 0 then next fi;
    Cands:= sort(map(t -> 1 + 10^(n-1) + add(10^t[i],i=1..s-2), combinat:-choose([$1..n-2],s-2)));
    for c in Cands do if isprime(c) then return c fi od
  od;
  0
end proc:
map(f, [$1..20]); # Robert Israel, Aug 03 2020

Extended by Ray Chandler, Dec 03 2009

A143863 Primes such that the sum of digits is a perfect power (A001597).

Original entry on oeis.org

13, 17, 31, 53, 71, 79, 97, 103, 107, 211, 233, 251, 277, 349, 367, 431, 439, 457, 503, 521, 547, 619, 673, 691, 701, 709, 727, 853, 907, 997, 1021, 1061, 1069, 1087, 1151, 1201, 1223, 1249, 1429, 1447, 1483, 1511, 1601, 1609, 1627, 1663, 1699, 1753, 1789

Offset: 1

Giovanni Teofilatto, Sep 04 2008

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

Cf. A062338.

Select[Prime[Range[300]],GCD@@FactorInteger[Total[IntegerDigits[#]]][[;;,2]]>1&] (* Harvey P. Dale, Sep 18 2023 *)

Union of A062339, A062343, A106757, A106768, A107618 etc. [From R. J. Mathar, Sep 13 2008]

389, 569, 581, 659, 677 etc. removed by R. J. Mathar, Sep 13 2008

cp's OEIS Frontend

A070247 Palindromic primes with digit sum 5.

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A106756 Primes with digit sum = 14.

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A106757 Primes with digit sum = 16.

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A107619 Primes with digit sum = 65.

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A106755 Primes p with digital sum equal to 13.

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A106807 Primes with digit sum = 67.

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A133223 Sum of digits of primes (A007605), sorted and with duplicates removed.

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A259866 Primes of the form 3*10^k + 1.

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