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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.

%I A178701 #34 Feb 09 2025 06:34:45
%S A178701 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,
%T A178701 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,
%U A178701 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
%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.
%C A178701 Each row, r, has 6r-1 terms. The first row does not account for the prime 3 and its count of 1.
%H A178701 Robert G. Wilson v, <a href="/A178701/b178701.txt">Table of n, a(n) for n = 1..320</a>
%H A178701 Craig Mayhew, <a href="http://www.bigprimes.net/sum-of-digits/">Sums of digits of primes</a>
%e A178701 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.
%e A178701 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.
%e A178701   \k 2, 4,  5,  7,  8,  10,  11,  13,  14,   16,   17,   19,   20,   22,   23,   25,   26, ...
%e A178701   r\
%e A178701   1: 1, 0,  1,  1,  0;
%e A178701   2: 1, 2,  2,  2,  3,   3,   3,   1,   1,    2,    1;
%e A178701   3: 1, 2,  4,  7,  7,  12,  13,  16,  16,   13,   18,   12,   11,    6,    4,    1,    0;
%e A178701   4: 0, 4,  8, 20, 19,  31,  52,  67,  77,   93,  101,  116,   95,   92,   91,   63,   51, ...
%e A178701   5: 0, 4, 12, 28, 45,  95, 143, 236, 272,  411,  479,  630,  664,  742,  757,  741,  706, ...
%e A178701   6: 0, 2, 14, 58, 76, 204, 389, 660, 852, 1448, 1971, 2832, 3101, 4064, 4651, 5393, 5376, ...
%e A178701 etc.
%t A178701 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
%Y A178701 Cf. A000040, A006880, A007605, A177868, A178183, A178447, A178571, A178605, A178876, A178879, A178884.
%Y A178701 Row sums (except for the first term) give A006879. The indices k are given by A001651 (beginning with 2).
%K A178701 nonn,tabf,base
%O A178701 1,7
%A A178701 _Robert G. Wilson v_, Dec 29 2010