A178701 - OEIS

%I #29 Jan 08 2019 09:39:12

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

%H Robert G. Wilson v, <a href="/A178701/b178701.txt">Table of n, a(n) for n = 1..320</a>

%H Craig Mayhew, <a href="http://www.bigprimes.net/sum-of-digits/">Sums of digits of primes</a>

%e 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 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 \k .2,..4,..5,...7,...8,...10,...11,....13,....14,....16,.....17,.....19,.....20,.....22,......23,......25,......26,...,

%e r\

%e .1: 1,..0,..1,...1,...0}

%e .2: 1,..2,..2,...2,...3,....3,....3,.....1,.....1,.....2,......1}

%e .3: 1,..2,..4,...7,...7,...12,...13,....16,....16,....13,.....18,.....12,.....11,......6,.......4,.......1,.......0}

%e .4: 0,..4,..8,..20,..19,...31,...52,....67,....77,....93,....101,....116,.....95,.....92,......91,......63,......51, ...,

%e .5: 0,..4,.12,..28,..45,...95,..143,...236,...272,...411,....479,....630,....664,....742,.....757,.....741,.....706, ...,

%e .6: 0,..2,.14,..58,..76,..204,..389,...660,...852,..1448,...1971,...2832,...3101,...4064,....4651,....5393,....5376, ...,

%e .7: 0,..5,.21,..95,.138,..420,..773,..1747,..2329,..4616,...6456,..10496,..12743,..18710,...22447,...29209,...32075, ...,

%e .8: 0,..4,.24,.154,.212,..787,.1705,..4214,..5721,.12546,..19040,..34639,..43707,..71879,...92223,..135728,..155461, ...,

%e .9: 0,..2,.26,.226,.372,.1457,.3312,..9159,.13320,.32077,..50752,.102027,.138554,.249053,..331920,..535444,..655423, ...,

%e 10: 0,.11,.42,.278,.547,.2395,.6090,.19204,.27894,.75517,.128909,.284482,.391199,.772365,.1087932,.1919618,.2427462, ...,

%e etc.

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

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

%K nonn,tabf

%O 1,7

%A _Robert G. Wilson v_, Dec 29 2010