A067523 -id:A067523 - cp's OEIS frontend

A007605 Sum of digits of n-th prime.

2, 3, 5, 7, 2, 4, 8, 10, 5, 11, 4, 10, 5, 7, 11, 8, 14, 7, 13, 8, 10, 16, 11, 17, 16, 2, 4, 8, 10, 5, 10, 5, 11, 13, 14, 7, 13, 10, 14, 11, 17, 10, 11, 13, 17, 19, 4, 7, 11, 13, 8, 14, 7, 8, 14, 11, 17, 10, 16, 11, 13, 14, 10, 5, 7, 11, 7, 13, 14, 16, 11, 17, 16, 13, 19, 14, 20, 19, 5

Offset: 1

N. J. A. Sloane, Mira Bernstein, and Robert G. Wilson v

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
CNRS press release, Sum of Digits of Prime Numbers Is Evenly Distributed: New Mathematical Proof of Hypothesis
Christian Mauduit and Joël Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers (French) [On a problem posed by Gelfond: the sum of digits of primes] Ann. of Math. (2) 171(2010), no. 3, 1591--1646. MR2680394 (2011j:11137)
Enrique Navarrete, Distributions of Sums of Digits of Primes
Robert G. Wilson v, Letter to C. A. Pickover, Mar. 1993

Cf. A000040, A007953, A038194, A065073, A068395, A133223, A376714, A380192.

For inverse, see A067180 and A067523.

a007605_list = map a007953 a000040_list -- Reinhard Zumkeller, Aug 04 2011

[ &+Intseq(NthPrime(n), 10): n in [1..80] ]; // Klaus Brockhaus, Jun 13 2009

map(t -> convert(convert(t,base,10),`+`), select(isprime, [2,(2*i+1 $ i=1..1000)])); # Robert Israel, Aug 16 2015

Table[Apply[Plus, RealDigits[Prime[n]][[1]]], {n, 1, 100}]
Plus@@ IntegerDigits[Prime[Range[100]]] (* Zak Seidov *)

dsum(n)=my(s);while(n,s+=n%10;n\=10);s
forprime(p=2,1e3,print1(dsum(p)", ")) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = sumdigits(prime(n)); \\ Michel Marcus, Dec 20 2017

from sympy import prime
def a(n): return sum(map(int, str(prime(n))))
print([a(n) for n in range(1, 80)]) # Michael S. Branicky, Feb 03 2021

a(n) = A007953(A000040(n)) = A007953(prime(n)).

A067180 Smallest prime with digit sum n, or 0 if no such prime exists.

Original entry on oeis.org

0, 2, 3, 13, 5, 0, 7, 17, 0, 19, 29, 0, 67, 59, 0, 79, 89, 0, 199, 389, 0, 499, 599, 0, 997, 1889, 0, 1999, 2999, 0, 4999, 6899, 0, 17989, 8999, 0, 29989, 39989, 0, 49999, 59999, 0, 79999, 98999, 0, 199999, 389999, 0, 598999, 599999, 0, 799999, 989999, 0, 2998999, 2999999, 0, 4999999

Offset: 1

Amarnath Murthy, Jan 09 2002

			a(68) = 59999999 because 59999999 is the smallest prime with digit sum = 68;
a(100) = 298999999999 because 298999999999 is the smallest prime with digit sum = 100.

Robert Israel, Table of n, a(n) for n = 1..1000 (first 175 terms from Robert G. Wilson v)

Cf. A054750.

Removal of the 0 terms from this sequence leaves A067523.

g:= proc(s,d) # integers of <=d digits with sum s
  if s > 9*d then return [] fi;
  if d = 1 then return [s] fi;
  [seq(op(map(t -> j*10^(d-1)+ t, g(s-j,d-1))),j=0..9)];
end proc:
f:= proc(n) local d, j,x,y;
  if n mod 3 = 0 then return 0 fi;
  for d from ceil(n/9) do
    if d = 1 then
      if isprime(n) and n < 10 then return n
      else next
      fi
    fi;
    for j from 1 to 9 do
      for y in g(n-j,d-1) do
        x:= 10^(d-1)*j + y;
        if isprime(x) then return x fi;
  od od od;
end proc:
f(1):= 0: f(3):= 3:
map(f, [$1..100]); # Robert Israel, Dec 13 2020

a = Table[0, {100}]; Do[b = Apply[ Plus, IntegerDigits[ Prime[n]]]; If[b < 101 && a[[b]] == 0, a[[b]] = Prime[n]], {n, 1, 10^7} ]; a
f[n_] :=  If[n > 5 && Mod[n, 3] == 0, 0, Block[{k = 1, lmt, lst = {}, ip = IntegerPartitions[n, Round[1 + n/9], {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}]}, lmt = 1 + Length@ ip; While[k < lmt, AppendTo[lst, Select[ FromDigits@# & /@ Permutations@ ip[[k]], PrimeQ[#] &]]; k++]; Min@ Flatten@ lst]]; f[1] = 0; f[4] = 13; Array[f, 70] (* Robert G. Wilson v, Sep 28 2014 *)

A067180(n)={if(n<2, 0, n<4, n, n%3, my(d=divrem(n,9)); forprime(p=d[2]*10^d[1]-1,,sumdigits(p)==n&&return(p)))} \\ M. F. Hasler, Nov 04 2018

a(3k) = 0 for k > 1.

a(3k-2) = A067523(2k-1), a(3k-1) = A067523(2k), for all k > 1. - M. F. Hasler, Nov 04 2018

Edited and extended by Robert G. Wilson v, Mar 01 2002

Edited by Ray Chandler, Apr 24 2007

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

A234429 Numbers which are the digital sum of the square of some prime.

Original entry on oeis.org

4, 7, 9, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172, 175, 178

Offset: 1

Antonio Graciá Llorente, Dec 26 2013

A123157 sorted and duplicates removed.

Cf. A067180, A067523, A123157, A056991.

Cf. A062397, A226803, A226802, A165492, A165459 A165493, A229058, A165502, A165503, A165504.

terms(nn) = {v = []; forprime (p = 1, nn, v = concat(v, sumdigits(p^2));); vecsort(v,,8);} \\ Michel Marcus, Jan 08 2014

From Robert G. Wilson v, Sep 28 2014: (Start)
Except for 3, all primes are congruent to +-1 (mod 3). Therefore, (3n +- 1)^2 = 9n^2 +- 6n + 1 which is congruent to 1 (mod 3).
2, 11, 101, ... (A062397);
5, 149, 1049, ... (A226803);
only 3;
19, 71, 179, 251, 449, 20249, 24499, 100549, ... (A226802);
7, 29, 47, 61, 79, 151, 349, 389, 461, 601, 1051, 1249, 1429, ... (A165492);
13, 23, 31, 41, 59, 103, 131, 139, 211, 229, 239, 347, 401, ... (A165459);
17, 37, 53, 73, 89, 107, 109, 127, 181, 199, 269, 271, 379, ... (A165493);
43, 97, 191, 227, 241, 317, 331, 353, 421, 439, 479, 569, 619, 641, ...;
67, 113, 157, 193, 257, 283, 311, 337, 373, 409, 419, 463, ... (A229058);
163, 197, 233, 307, 359, 397, 431, 467, 487, 523, 541, 577, 593, 631, ...;
83, 137, 173, 223, 263, 277, 281, 367, 443, 457, 547, 587, ... (A165502);
167, 293, 383, 563, 607, 617, 733, 787, 823, 859, 877, 941, 967, 977, ...;
433, 613, 683, 773, 827, 863, 1063, 1117, 1187, 1223, 1567, ... (A165503);
313, 947, 983, 1303, 1483, 1609, 1663, 1933, 1973, 1987, 2063, 2113, ...;
887, 1697, 1723, 1867, 1913, 2083, 2137, 2417, 2543, 2633, ... (A165504);
883, 937, 1367, 1637, 2213, 2447, 2683, 2791, 2917, 3313, 3583, 3833, ...;
1667, 2383, 2437, 2617, 2963, 4219, 4457, 5087, 5281, 6113, 6163, ...;
...  Also see A229058. (End)
Conjectures from Chai Wah Wu, Apr 16 2025: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 5.
G.f.: x*(2*x^4 - x^3 - x^2 - x + 4)/(x - 1)^2. (End)

a(36) from Michel Marcus, Jan 08 2014
a(37)-a(54) from Robert G. Wilson v, Sep 28 2014
More terms from Giovanni Resta, Aug 15 2019

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A007605 Sum of digits of n-th prime.

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A067180 Smallest prime with digit sum n, or 0 if no such prime exists.

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

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A234429 Numbers which are the digital sum of the square of some prime.

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