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.

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

Author

Robert G. Wilson v, Dec 29 2010

Keywords

• nonn
• tabf
• base

Comments

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

Examples

			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.
		

Links

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

Crossrefs

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).

Programs

Mathematica

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