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Smallest SanD-d prime p (such that p + d is also prime and digit sum A007953(p(p+d)) = d) with d = 14 + n*18, n >= 0, resp. d = 5 for n = -1.
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%I #10 Apr 09 2019 20:46:34

%S 2,5,149,2543,19961,412253,7601249,39999899,999999893

%N Smallest SanD-d prime p (such that p + d is also prime and digit sum A007953(p(p+d)) = d) with d = 14 + n*18, n >= 0, resp. d = 5 for n = -1.

%C If p is a S(um)anD(difference) prime with d = sum of digits of p(p+d), we call it a SanD-d prime.

%C This sequence lists the smallest SanD-d prime for each possible value of d = {5, 14, 32, 50, 68, ...}, i.e., d = 14 + 18*k with k = -1/2, 0, 1, 2, 3, ...

%C The sequence appears to be increasing. Is there an index where this is no more true?

%H Freeman J. Dyson, Norman E. Frankel, Anthony J. Guttmann: <a href="https://arxiv.org/abs/1904.03573">SanD primes and numbers</a>, arxiv:1904.03573 [math.CA], April 7, 2019

%e a(-1) = 2 is the smallest (and only) SanD-5 prime: 2 and 2 + 5 = 7 both are prime, and the digit sum A007953(2*7) = 1 + 4 = 5.

%e All other SanD primes must have gap and sum (of digits of the product) d = 14 + 18*k, k = 0, 1, 2, ... (corresponding to the index in this sequence):

%e a(0) = 5 is the smallest SanD-14 prime: 5 and 5 + 14 = 19 both are prime and the digit sum A007953(5*19) = 9 + 5 = 14.

%e a(1) = 149 = A307472(1) is the smallest SanD-32 prime: 149 and 149 + 32 = 181 both are prime, and the digit sum A007953(149*181) = 2+6+9+6+9 = 32.

%e a(2) = 2543 = A307473(1) is the smallest SanD-50 prime: 2543 and 2543 + 50 = 2593 both are prime, and the digit sum A007953(2543*2593) = 6+5+9+3+9+9+9 = 50.

%e a(3) = 19961 = A307474(1) is the smallest SanD-68 prime: 19961 and 19961 + 68 = 20029 both are prime, and the digit sum A007953(19961*20029) = 3+9+9+7+9+8+8+6+9 = 68.

%o (PARI) A307480(n,d=max(14+18*n,5))=forprime(p=2,,isprime(p+d)&&sumdigits(p*(p+d))==d&&return(p))

%Y Cf. A307471 - A307478 (d = 14+18k, k=0..7), A307479 (any d: main entry).

%Y Cf. A000040 (primes), A007953 (sum of digits).

%K nonn,base,more

%O -1,1

%A _M. F. Hasler_, Apr 09 2019