A067180 -id:A067180 - 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)).

A054750 Smallest prime number whose digits sum to n-th prime.

Original entry on oeis.org

2, 3, 5, 7, 29, 67, 89, 199, 599, 2999, 4999, 29989, 59999, 79999, 389999, 989999, 6999899, 8989999, 59899999, 89999999, 289999999, 799999999, 3999998999, 19999997999, 79999999999, 399999998999, 599999899999, 999998999999

Offset: 1

G. L. Honaker, Jr., Apr 24 2000

a(n) >= A051885(A000040(n)). Indices n for which the equality holds are listed in A055019.

a(n) >= A046864(n). - Michel Marcus, Nov 01 2015

			a(7)=89 because 8+9=17 and 17 is the 7th prime.

Chris Caldwell and G. L. Honaker, Jr., Prime curio for 8999

Cf. A000040, A067180, A046704, A046864, A068951, A007605.

a[n_]:=Module[{k=2}, While[DigitSum[k]!=Prime[n], k=NextPrime[k]]; k]; Array[a,15] (* Stefano Spezia, Mar 27 2025 *)

a(n) = {my(k=2); my(p=prime(n)); while((sumdigits(k) != prime(n)), k=nextprime(k+1)); k;} \\ Michel Marcus, Nov 01 2015

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 31 2000

Edited and extended by Robert G. Wilson v, Feb 26 2002

A067523 The smallest prime with a possible given digit sum.

Original entry on oeis.org

2, 3, 13, 5, 7, 17, 19, 29, 67, 59, 79, 89, 199, 389, 499, 599, 997, 1889, 1999, 2999, 4999, 6899, 17989, 8999, 29989, 39989, 49999, 59999, 79999, 98999, 199999, 389999, 598999, 599999, 799999, 989999, 2998999, 2999999, 4999999, 6999899, 8989999

Offset: 1

Amarnath Murthy, Feb 14 2002

Except for 3 no other prime has a digit sum which is a multiple of 3. Hence the possible digit sums are 2,3,4,5,7,8,10,11,13,14,16,..., etc. Conjecture: For every possible digit sum there exists a prime.

For n > 2, this is (conjecturally) the smallest prime with digit sum A001651(n). - Lekraj Beedassy, Mar 04 2009

Robert Israel, Table of n, a(n) for n = 1..667

Cf. A001651. Equals A067180 with the 0 terms removed.

g:= proc(s, d) # integers of <=d digits with sum s
  local j;
  if s > 9*d then return [] fi;
  if d = 1 then return [s] fi;
  [seq(op(map(t -> j*10^(d-1)+ t, procname(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(3):= 3:
map(f, [2,3,seq(seq(3*i+j,j=1..2),i=1..30)]); # Robert Israel, Jan 18 2024

A067523(n)=if(n<3,n+1,A067180(n*3\/2-1)) \\ M. F. Hasler, Nov 04 2018

a(n) = min(prime(i): A007605(i) = A133223(i)). - R. J. Mathar, Nov 06 2018

More terms from Vladeta Jovovic, Feb 18 2002

Edited by Ray Chandler, Apr 24 2007

A073867 Smallest prime whose digital sum is equal to the n-th composite number, or 0 if no such prime exists.

Original entry on oeis.org

13, 0, 17, 0, 19, 0, 59, 0, 79, 0, 389, 0, 499, 0, 997, 1889, 0, 1999, 0, 6899, 0, 17989, 8999, 0, 39989, 0, 49999, 0, 98999, 0, 199999, 0, 598999, 599999, 0, 799999, 0, 2998999, 2999999, 0, 4999999, 0, 9899999, 0, 19999999, 29999999, 0, 59999999, 0

Offset: 1

Amarnath Murthy, Aug 15 2002

			The first composite number (A002808) is 4 and the least prime whose digital sum is 4 is 13.
The second composite number (A002808) is 6 whose digital sum is == 0 (mod 3) so there is no prime whose fits the definition.

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Equals A067180(A002808(n)). Cf. A111397.

Composite[n_] := FixedPoint[n + PrimePi[ # ] + 1 &, n]; f[n_] := Block[{cn = Composite[n]}, k = 1; While[Plus @@ IntegerDigits@Prime@k != cn, k++ ]; Prime[k]];

a(n)=0 iff that composite number (A002808(n)) is congruent to 0 (modulo 3), otherwise a(n)=A007605(k) for the first k that equals A002808(n).

a(19)-a(32) from Stefan Steinerberger, Nov 09 2005

a(33)-a(56) by Robert G. Wilson v, Nov 10 2005

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

A213354 Primes p with digit sums s(p) and s(s(p)) also prime, but s(s(s(p))) not prime.

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, 79999789, 79999879, 79999987

Offset: 1

Jonathan Sondow, Jun 10 2012

A046704 is primes p with s(p) also prime. A207294 is primes p with s(p) and s(s(p)) also prime. A070027 is primes p with all s(p), s(s(p)), s(s(s(p))), ... also prime. A104213 is primes p with s(p) not prime. A207293 is primes p with s(p) also prime, but not s(s(p)). A213355 is smallest prime p whose k-fold digit sum s(s(..s(p)..)) is also prime for all k < n, but not for k = n.

Contains primes with digit sums 67, 89, 139, 157, 179,...., A207293(.). So A106807 is a subsequence and examples of numbers in this sequence but not in A106807 are A067180(89), A067180(139) etc. - R. J. Mathar, Feb 04 2021

			59899999 and s(59899999) = 5+9+8+9+9+9+9+9 = 67 and s(s(59899999)) = s(67) = 6+7 = 13 are all primes, but s(s(s(59899999))) = s(13) = 1+3 = 4 is not prime. No smaller prime has this property, so a(1) = 59899999 = A213355(3).

Cf. A046704, A070027, A104213, A207293, A207294, A213355, A106807.

Select[Prime[Range[5000000]], PrimeQ[Apply[Plus, IntegerDigits[#]]] && PrimeQ[Apply[Plus, IntegerDigits[Apply[Plus, IntegerDigits[#]]]]] && ! PrimeQ[Apply[Plus, IntegerDigits[Apply[Plus, IntegerDigits[Apply[Plus, IntegerDigits[#]]]]]]] &]

A113625 Irregular triangle in which the n-th row contains all primes having digit sum n (not containing the digit '0') in increasing order.

Original entry on oeis.org

2, 11, 3, 13, 31, 211, 5, 23, 41, 113, 131, 311, 2111, 7, 43, 61, 151, 223, 241, 313, 331, 421, 1123, 1213, 1231, 1321, 2113, 2131, 2221, 2311, 3121, 4111, 11113, 11131, 11311, 12211, 21121, 21211, 22111, 111121, 111211, 112111, 17, 53, 71, 233, 251, 431, 521

Offset: 2

Amarnath Murthy, Nov 10 2005

The number of primes in the n-th row is A073901(n). The smallest prime in the n-th row is A067180(n). The largest prime in the n-th row is A069869(n).

			Starting with row 2, the table is
2, 11
3
13, 31, 211
5, 23, 41, 113, 131, 311, 2111
none
7, 43, 61, 151, 223, 241, 313, 331, 421, 1123,...

Alois P. Heinz, Rows n = 2..17, flattened (Rows n = 2..14 from T. D. Noe)

Cf. A110741 (with contraints on number of digits).

with(combinat):
b:= proc(n, i, l) option remember; `if`(n=0, select(isprime,
      map(x-> parse(cat(x[])), permute(l))), `if`(i<1, [],
      [seq(b(n-i*j, i-1, [l[],i$j])[], j=0..n/i)]))
    end:
T:= n-> sort(b(n, 9, []))[]:
seq(T(n), n=2..8);  # Alois P. Heinz, May 25 2013

Table[If[n > 3 && Mod[n, 3] == 0, {}, p = IntegerPartitions[n]; u = {}; Do[t = Permutations[i]; u = Union[u, Select[FromDigits /@ t, PrimeQ]], {i, p}]; u], {n, 2, 14}]

Edited, corrected and extended by Stefan Steinerberger, Aug 10 2007

Edited by T. D. Noe, Jan 25 2011

A268605 a(1) = 0; a(n+1) is the smallest integer in which the difference between its digits sum and the a(n) digits sum is equal to the n-th prime.

Original entry on oeis.org

0, 2, 5, 19, 89, 1999, 59999, 4999999, 599999999, 199999999999, 399999999999999, 799999999999999999, 8999999999999999999999, 499999999999999999999999999, 29999999999999999999999999999999, 4999999999999999999999999999999999999

Offset: 1

Francesco Di Matteo, Feb 17 2016

First 8 terms are primes (and are also in A061248). Next terms are not always primes.

			a(4) = 19 and 1 + 9 = 10; so a(5) = 89 because 8 + 9 = 17 and 17 - 10 = 7, that is the 4th prime.

Francesco Di Matteo, Table of n, a(n) for n = 1..25

Cf. A269306, A061248, A067180, A051885.

findnext(x, k) = {sx = sumdigits(x); pk = prime(k); y = 1; while (sumdigits(y) - sx != pk, y++); y;}
lista(nn) = {print1(x = 0, ", "); for (k=1, nn, y = findnext(x, k); print1(y, ", "); x = y;);} \\ Michel Marcus, Feb 19 2016

sumprime = 0
isPrime=lambda x: all(x % i != 0 for i in range(int(x**0.5)+1)[2:])
print(0)
for i in range(2,100):
  if isPrime(i):
    alfa = ""
    k = i + sumprime
    sumprime = k
    while k > 9:
      alfa = alfa + "9"
      k = k - 9
    alfa = str(k)+alfa
    print(alfa)

a(n) = A051885( A007504(n-1) ). - R. J. Mathar, Jun 19 2021

NAME adapted to offset by R. J. Mathar, Jun 19 2021

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

A114442 Least prime whose absolute difference between the sum of its even decimal digits and the sum of its odd decimal digits is n.

Original entry on oeis.org

211, 23, 2, 3, 13, 5, 127, 7, 17, 281, 19, 137, 277, 139, 59, 881, 79, 179, 11299, 199, 1559, 2797, 1399, 599, 12799, 997, 1979, 86861, 1999, 13799, 25999, 13999, 49999, 172999, 70999, 19979, 1199929, 39799, 137999, 277999, 139999, 59999, 1299979

Offset: 0

Zak Seidov and Robert G. Wilson v, Feb 04 2006

When a(n) is not A067180(n) and n!=0 (mod 3): 1, 11, 13, 17, 20, 22, 26, 29, 31, 32, 34, 35, 37, 38, 40, 44, 47, 49, 53, 55, 56, 58, 59, 61, 62, 65, 67, 70, 71, 73, 74, 76, 77, 80, 82, 83, ....

			a(0)=211 since 211 is the least prime which meets the criterion; i.e., |2 - (1+1)| = 0.

Cf. A067180, A111309.

f[n_] := Block[{id = IntegerDigits@Prime@n}, Abs[(Plus @@ id) - 2Plus @@ Select[id, OddQ]]]; t = Table[0, {50}]; Do[ a = f[n]; If[ t[[a + 1]] == 0, t[[a + 1]] = n], {n, 100020}]; t

cp's OEIS Frontend

A007605 Sum of digits of n-th prime.

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A054750 Smallest prime number whose digits sum to n-th prime.

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A067523 The smallest prime with a possible given digit sum.

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A073867 Smallest prime whose digital sum is equal to the n-th composite number, 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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A213354 Primes p with digit sums s(p) and s(s(p)) also prime, but s(s(s(p))) not prime.

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A113625 Irregular triangle in which the n-th row contains all primes having digit sum n (not containing the digit '0') in increasing order.

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