A007605 -id:A007605 - cp's OEIS frontend

A090433 Primes p(k) having a smaller sum of digits than k.

11, 13, 23, 61, 101, 103, 107, 109, 151, 163, 211, 223, 227, 241, 251, 271, 311, 313, 317, 331, 337, 347, 401, 421, 431, 433, 443, 461, 503, 509, 521, 523, 701, 911, 1009, 1013, 1021, 1031, 1033, 1051, 1061, 1063, 1103, 1109, 1117, 1123, 1129, 1151

Offset: 1

Reinhard Zumkeller, Dec 01 2003

A090431(a(n)) > 0.

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

Cf. A033548, A090432, A007953, A007605, A090457.

Select[Prime[Range[200]],Total[IntegerDigits[#]]Harvey P. Dale, Mar 05 2017 *)

A239692 Base 6 sum of digits of prime(n).

Original entry on oeis.org

2, 3, 5, 2, 6, 3, 7, 4, 8, 9, 6, 2, 6, 3, 7, 8, 9, 6, 7, 11, 3, 4, 8, 9, 7, 11, 8, 12, 4, 8, 7, 11, 12, 9, 9, 6, 7, 8, 12, 13, 14, 6, 11, 8, 12, 9, 11, 3, 7, 4, 8, 9, 6, 11, 7, 8, 9, 6, 7, 11, 8, 8, 7, 11, 8, 12, 6, 7, 12, 9, 13, 14, 7, 8, 9, 13, 14, 7, 11, 9

Offset: 1

Tom Edgar, Mar 24 2014

a(n) is the rank of prime(n) in the base-6 dominance order on the natural numbers.

			The sixth prime is 13, 13 in base 6 is (2,1) so a(6)=2+1=3.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Tyler Ball and Daniel Juda, Dominance over N, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.

Cf. A007605, A014499, A053827, A239690, A239691.

[&+Intseq(NthPrime(n),6): n in [1..100]]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 6], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

a(n) = sumdigits(prime(n), 6); \\ Michel Marcus, Mar 04 2023

[sum(i.digits(base=6)) for i in primes_first_n(200)]

a(n) = A053827(A000040(n)).

A066538 Sum of the digits of the n-th Mersenne prime (A000668).

Original entry on oeis.org

3, 7, 4, 10, 19, 13, 28, 46, 73, 112, 139, 154, 697, 847, 1675, 3106, 3106, 4258, 5755, 5950, 13216, 13693, 14980, 27202, 28939, 31339, 60337, 116455, 149365, 179488, 291745, 1026544, 1163443, 1704376, 1893388, 4038358, 4092673, 9440671, 18243946, 28445131, 32580433, 35170384, 41201947, 44142151, 50349694, 57766339, 58416637

Offset: 1

Robert G. Wilson v, Jan 06 2002

From Gord Palameta, Jul 21 2018: (Start)
a(38) and a(39) were calculated by Enoch Haga, Sep 07 1999 and Dec 17 2001; a(40) through a(42) were calculated by Andrew Rupinski, Mar 12 2005. (See the Carlos Rivera link.)
It appears that asymptotically a(n)/A000043(n) = 9*log_10(2)/2. (End)

Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Gord Palameta)
Carlos Rivera, Puzzle 74.- SOD(2^6972593-1), The Prime Puzzles & Problems Connection.

Cf. A000043, A000668, A007953, A001348.
Subsequence of: A007953, A007605.
Cf. A001370 (sum of digits of 2^n).

ep = {the exponents from A000043}; a = {}; Do[ a = Append[a, Apply[ Plus, IntegerDigits[ 2^ep[[n]] - 1]]], {n, 1, 47} ]; a
(* Second program: *)
Array[Total@ IntegerDigits[2^MersennePrimeExponent@ # - 1] &, 45] (* Michael De Vlieger, Jul 22 2018 *)

a(n) = A007953(A000668(n)). - Amiram Eldar, Oct 16 2024

Definition corrected by Omar E. Pol, Apr 01 2008

a(38)-a(47) from Gord Palameta, Jul 21 2018

A074462 Average digit (rounded up) in the decimal expansion of prime(n).

Original entry on oeis.org

2, 3, 5, 7, 1, 2, 4, 5, 3, 6, 2, 5, 3, 4, 6, 4, 7, 4, 7, 4, 5, 8, 6, 9, 8, 1, 2, 3, 4, 2, 4, 2, 4, 5, 5, 3, 5, 4, 5, 4, 6, 4, 4, 5, 6, 7, 2, 3, 4, 5, 3, 5, 3, 3, 5, 4, 6, 4, 6, 4, 5, 5, 4, 2, 3, 4, 3, 5, 5, 6, 4, 6, 6, 5, 7, 5, 7, 7, 2, 5, 5, 3, 3, 4, 6, 4, 6, 6, 4, 5, 6, 7, 7, 5, 8, 3, 5, 3, 4, 4, 6, 6, 5, 7, 5, 7, 7, 6, 8, 3, 5, 4, 5, 6, 4, 4, 5, 6, 5, 7

Offset: 1

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 23 2002

			The prime numbers begin with 2,3,5,7,11,13,17,19,23,... so the average digits rounded up are 2, 3, 5, 7, (1+1)/2=1, (1+3)/2=2, (1+7)/2=4, (1+9)/2=5, ceiling((2+3)/2)=3, ...

Cf. A000040, A004427, A074461, A073342.

Cf. A007605, A097944.

Table[Ceiling[Mean[IntegerDigits[p]]],{p,Prime[Range[120]]}] (* Harvey P. Dale, Nov 06 2022 *)

a(n) = my(d=digits(prime(n))); ceil(vecsum(d)/#d); \\ Michel Marcus, Apr 23 2022

a(n) = ceiling(A007605(n)/A097944(n)). - R. J. Mathar, Sep 23 2008

a(n) = A004427(A000040(n)). - Reinhard Zumkeller, May 27 2010

Offset changed to 1, cf. to A073342 added, and extended by R. J. Mathar, Sep 23 2008

A090432 Primes prime(k) having a greater sum of digits than does k.

Original entry on oeis.org

2, 3, 5, 7, 17, 19, 29, 31, 37, 41, 43, 47, 53, 59, 67, 71, 73, 79, 83, 89, 97, 113, 127, 137, 139, 149, 157, 167, 173, 179, 181, 191, 193, 197, 199, 229, 233, 239, 257, 269, 277, 281, 283, 293, 307, 349, 353, 359, 367, 373, 379, 383, 389, 397, 409, 419, 439

Offset: 1

Reinhard Zumkeller, Dec 01 2003

A090431(a(n)) < 0.

Cf. A007605, A007953, A033548, A090431, A090433, A090456.

Prime[#]&/@Select[Range[100],Total[IntegerDigits[Prime[#]]]-Total[IntegerDigits[#]]>0&] (* Michel Lagneau, Nov 07 2015 *)

isok(n) = sumdigits(prime(n)) > sumdigits(n); \\ Michel Marcus, Nov 07 2015

A104247 Primes that are the sum of digits of the first k primes for some k.

Original entry on oeis.org

2, 5, 17, 19, 23, 31, 41, 61, 71, 83, 181, 269, 389, 419, 449, 631, 683, 727, 743, 809, 929, 1039, 1061, 1069, 1091, 1277, 1381, 1481, 1567, 1613, 1747, 1873, 1951, 1993, 2039, 2129, 2281, 2297, 2339, 2381, 2549, 2579

Offset: 1

Zak Seidov, Feb 26 2005

			a(4)=19 because A058049(4)= 5 and sum of digits of the first 5 primes, 2+3+5+7+(1+1)=19 is prime.

M. F. Hasler, Table of n, a(n) for n = 1..1000

Corresponding n's: A058049. Primes: A000040, sum of digits of primes: A007605.

Cf. A051351, A075544.

from sympy import isprime, nextprime
def sd(n): return sum(map(int, str(n)))
def aupto(limit):
    alst, k, p, s = [], 1, 2, 2
    while s <= limit:
        if isprime(s): alst.append(s)
        k += 1; p = nextprime(p); s += sd(p)
    return alst
print(aupto(2579)) # Michael S. Branicky, Jul 18 2021

a(n) = A007605(1) + ... + A007605(A058049(n)).

A104251 Sum of nonprime digits of n-th prime.

Original entry on oeis.org

0, 0, 0, 0, 2, 1, 1, 10, 0, 9, 1, 0, 5, 4, 4, 0, 9, 7, 6, 1, 0, 9, 8, 17, 9, 2, 1, 1, 10, 2, 1, 2, 1, 10, 14, 2, 1, 7, 7, 1, 10, 10, 11, 10, 10, 19, 2, 0, 0, 9, 0, 9, 5, 1, 0, 6, 15, 1, 0, 9, 8, 9, 0, 2, 1, 1, 1, 0, 4, 13, 0, 9, 6, 0, 9, 8, 17, 9, 5, 13, 14, 5, 5, 4, 13, 8, 17, 4, 11, 10, 10, 13, 12

Offset: 1

Zak Seidov, Feb 26 2005

			a(6)=1 because sum of composite (nonprime) digits of prime(6)=13 is 1.

Cf. A000040 (primes).

Cf. A007605 (sum of digits of primes), A104250 (sum of prime digits of n-th prime).

a(n) = A007605(n) - A104250(n).

A178701 An irregular array read by rows. The k-th entry of row r is the number of r-digit primes with digit sum k.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 2, 2, 3, 3, 3, 1, 1, 2, 1, 1, 2, 4, 7, 7, 12, 13, 16, 16, 13, 18, 12, 11, 6, 4, 1, 0, 0, 4, 8, 20, 19, 31, 52, 67, 77, 93, 101, 116, 95, 92, 91, 63, 51, 29, 30, 16, 5, 0, 1, 0, 4, 12, 28, 45, 95, 143, 236, 272, 411, 479, 630, 664, 742, 757, 741, 706, 580, 528, 379, 341, 205, 166, 84, 62, 34, 13, 4, 2, 0, 2, 14, 58, 76, 204, 389, 660, 852, 1448, 1971, 2832, 3101, 4064, 4651, 5393, 5376, 5570, 5785, 5287, 4796

Offset: 1

Robert G. Wilson v, Dec 29 2010

Each row, r, has 6r-1 terms. The first row does not account for the prime 3 and its count of 1.

			To begin the second row, only 11 has digit-sum 2, so the first term is 1; both 13 & 31 have digit-sum 4 so the second term is 2; both 23 & 41 have digit-sum 5, so the third term is 2; etc.
To begin the third row, only 101 -> 2, so its first term is 1, both 103 & 211 -> 4 so its second term is 2; 113, 131, 311 & 401 -> 5, so its third term is 4; etc.
  \k 2, 4,  5,  7,  8,  10,  11,  13,  14,   16,   17,   19,   20,   22,   23,   25,   26, ...
  r\
  1: 1, 0,  1,  1,  0;
  2: 1, 2,  2,  2,  3,   3,   3,   1,   1,    2,    1;
  3: 1, 2,  4,  7,  7,  12,  13,  16,  16,   13,   18,   12,   11,    6,    4,    1,    0;
  4: 0, 4,  8, 20, 19,  31,  52,  67,  77,   93,  101,  116,   95,   92,   91,   63,   51, ...
  5: 0, 4, 12, 28, 45,  95, 143, 236, 272,  411,  479,  630,  664,  742,  757,  741,  706, ...
  6: 0, 2, 14, 58, 76, 204, 389, 660, 852, 1448, 1971, 2832, 3101, 4064, 4651, 5393, 5376, ...
etc.

Robert G. Wilson v, Table of n, a(n) for n = 1..320
Craig Mayhew, Sums of digits of primes

Cf. A000040, A006880, A007605, A177868, A178183, A178447, A178571, A178605, A178876, A178879, A178884.

Row sums (except for the first term) give A006879. The indices k are given by A001651 (beginning with 2).

dir[n_] := Floor[(3 n + 2)/2]; inv[n_] := Floor[(2 n - 1)/3]; f[n_] := Block[{p = NextPrime[10^(n - 1)], t = Table[0, {inv[9 n]}]}, While[p < 10^n, t[[ inv[Plus @@ IntegerDigits@ p]]]++; p = NextPrime@ p]; t]; Array[f, 5] // Flatten

A239693 Base 7 sum of digits of prime(n).

Original entry on oeis.org

2, 3, 5, 1, 5, 7, 5, 7, 5, 5, 7, 7, 11, 7, 11, 5, 5, 7, 7, 5, 7, 7, 11, 11, 13, 5, 7, 5, 7, 5, 7, 11, 11, 13, 5, 7, 7, 7, 11, 11, 11, 13, 11, 13, 5, 7, 7, 13, 11, 13, 11, 11, 13, 11, 11, 11, 11, 13, 13, 11, 13, 17, 13, 11, 13, 11, 13, 13, 5, 7, 5, 5, 7, 7

Offset: 1

Tom Edgar, Mar 24 2014

a(n) is the rank of prime(n) in the base-7 dominance order on the natural numbers.

			The fifth prime is 11, 11 in base 7 is (1,4) so a(5)=1+4=5.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Tyler Ball and Daniel Juda, Dominance over N, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.

Cf. A007605, A014499, A053828, A239690, A239691, A239692.

[&+Intseq(NthPrime(n),7): n in [1..100]]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 7], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

a(n) = sumdigits(prime(n), 7); \\ Michel Marcus, Mar 04 2023

[sum(i.digits(base=7)) for i in primes_first_n(200)]

a(n) = A053828(A000040(n)).

A242368 Primes p such that p + digitsum(p) = q^k for some prime q and k > 1 where digitsum(n) = A007953(n).

Original entry on oeis.org

2, 17, 347, 521, 10601, 28541, 29759, 32027, 39569, 58061, 62969, 100469, 109541, 120401, 130307, 205357, 398129, 426383, 434261, 829883, 896771, 923501, 935063, 1190261, 1216583, 1261109, 1559963, 1697771, 2105381, 2128649, 2505857, 2778851, 2886563, 2920649

Offset: 1

J. M. Bergot, Aug 16 2014

With k>1 the number of entries is greatly reduced compared to simply allowing p+digsum(p) = q. One could allow for k=1 to see how many entries could be found for a variation of this sequence.

			a(4)=521 because 521+5+2+1=529=23^2 and 23 is a prime.

Kevin P. Thompson, Table of n, a(n) for n = 1..1000 (first 181 terms from Michel Marcus)

Cf. A000040, A007953, A007605, A000961.

a242368[n_Integer] := Module[{p, pp}, p = Prime[n]; pp = p + Plus @@ IntegerDigits@p; If[And[Length@FactorInteger[pp] == 1,
    Min[Last@Transpose[FactorInteger[pp]]] > 1], p, 0]]; Rest@Sort@DeleteDuplicates[a242368 /@ Range[10^6]] (* Michael De Vlieger, Aug 16 2014 *)

dsum(n)=n=digits(n); sum(i=1,#n,n[i])
is(p)=isprimepower(p+dsum(p))>1 && isprime(p)
forprime(p=2,1e9,if(is(p), print1(p", "))) \\ Charles R Greathouse IV, Aug 16 2014

More terms from Charles R Greathouse IV, Aug 16 2014

cp's OEIS Frontend

A090433 Primes p(k) having a smaller sum of digits than k.

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A239692 Base 6 sum of digits of prime(n).

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A066538 Sum of the digits of the n-th Mersenne prime (A000668).

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A074462 Average digit (rounded up) in the decimal expansion of prime(n).

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A090432 Primes prime(k) having a greater sum of digits than does k.

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A104247 Primes that are the sum of digits of the first k primes for some k.

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A104251 Sum of nonprime digits of n-th prime.

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A178701 An irregular array read by rows. The k-th entry of row r is the number of r-digit primes with digit sum k.

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