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A307479
SanD primes: primes p such that p+d is also prime and digit_sum(p(p+d)) = d, for some d.
10
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
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
KEYWORD
nonn,base
AUTHOR
M. F. Hasler, Apr 09 2019
STATUS
approved