A307480 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.

2, 5, 149, 2543, 19961, 412253, 7601249, 39999899, 999999893, 17279999921, 170287917851, 2499999999839, 22293272273609, 344934483634067, 4099999999999811

Offset

-1,1

Comments

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.

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, ...

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

Links

Table of n, a(n) for n=-1..13.

Freeman J. Dyson, Norman E. Frankel, and Anthony J. Guttmann, SanD primes and numbers, arxiv:1904.03573 [math.CA], 2019-2020.

Example

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.
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):
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.
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.
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.
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.

Prog

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

Crossrefs

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

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

Sequence in context: A175525 A069139 A155762 * A062611 A263413 A270585

Adjacent sequences: A307477 A307478 A307479 * A307481 A307482 A307483

Keyword

nonn,base,more

Author

M. F. Hasler, Apr 09 2019

Extensions

a(8)-a(13) from Max Alekseyev, Jun 15 2025

Status

approved