A307479 SanD primes: primes p such that p+d is also prime and digit_sum(p(p+d)) = d, for some d.

2, 5, 17, 23, 29, 53, 59, 83, 113, 149, 167, 179, 239, 281, 383, 389, 431, 443, 491, 509, 569, 659, 1019, 1031, 1061, 1103, 1259, 1289, 1409, 1427, 1439, 1901, 2003, 2081, 2111, 2129, 2207, 2237, 2309, 2357, 2441, 2543, 2657, 2687, 2801, 3137, 3203, 3221, 3359, 3407

Offset

1,1

Comments

Frankel named "S(um)anD(ifference) number" any n such that the sum of digits of n(n+d) equals d, for some d. SanD primes are SanD numbers p such that p and p+d are prime. For given d, we will refer to SanD-d numbers or primes.

The only prime solution with odd d is p = 2, d = 5.

All other SanD primes must have d == 14 (mod 18).

This is the list of all SanD primes, i.e., the union of the SanD-5 prime 2, the SanD-14 primes A307471, SanD-32 primes A307472, SanD-50 primes A307473, etc.

Sequence A307480 lists the smallest SanD-d prime for all possible d = 14 + 18*k, k = -1/2, 0, 1, 2, 3, ...

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Freeman J. Dyson, Norman E. Frankel, Anthony J. Guttmann: SanD primes and numbers, arxiv:1904.03573 [math.CA], April 7, 2019

Example

a(1) = 2 = A307480(-1) 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 a gap and sum d = 14 + 18*k, k = 0, 1, 2, ...
a(2) = 5 = A307471(1) = A307480(0) 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(10) = 149 = A307472(1) = A307480(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(42) = 2543 = A307473(1) = A307480(2) 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(186) = 19961 = A307474(1) = A307480(3) 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.

Maple

filter:= proc(p) local d;
   if not isprime(p) then return false fi;
   for d from 14 by 18 while is(d <= 9*log[10](p*(p+d))+1) do
  if isprime(p+d) and convert(convert(p*(p+d), base, 10), `+`)=d then return true fi
od:
false
end proc:
filter(2):= true:
select(filter, [2, seq(i, i=5..10000, 6)]); # Robert Israel, Apr 10 2019

Prog

(PARI) A307479_upto(LIM=2e4, A=List(2))={ forstep(d=14, oo, 18, my(L=#A); forprime(p=3, LIM, isprime(p+d) && sumdigits(p*(p+d))==d && listput(A, p)); L==#A && break); Set(A)}

Crossrefs

Cf. A307471 - A307478 (d = 14+18k, k=0..7), A307480 (smallest prime for given d).

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

Sequence in context: A042049 A341017 A069689 * A106021 A032605 A099243

Adjacent sequences: A307476 A307477 A307478 * A307480 A307481 A307482

Keyword

nonn,base

Author

M. F. Hasler, Apr 09 2019

Status

approved