A339805 Numbers k such that the sum of decimal digits of k is the sum of primes dividing k+1 (with repetition).

5, 17, 47, 97, 98, 159, 279, 359, 485, 489, 749, 879, 1679, 1979, 2399, 2499, 3968, 5669, 6749, 7199, 7799, 8099, 8639, 9719, 12799, 19199, 25599, 31999, 37499, 39599, 44799, 68599, 78399, 78749, 79379, 94499, 134999, 143999, 146999, 161999, 172799, 175999, 194399, 199679, 209999, 218699, 259999

Offset: 1

Author

J. M. Bergot and Robert Israel, Dec 18 2020

Keywords

• nonn
• base

Comments

Numbers k such that A007953(k) = A001414(k+1).

If m is not divisible by 10, A007953(10^k*m-1) = A007953(m) - 1 + 9*k while A001414(10^k*m) = A001414(m) + 7*k. Thus if in addition A001414(m) - A007953(m) is odd and positive, 10^k*m-1 is in the sequence where k = (A001414(m) - A007953(m)+1)/2.

Are there infinitely many terms that do not end in 9?

Examples

			a(4) = 97 is in the sequence because the sum of digits of 97 is 9+7 = 16 and the sum of primes dividing 98=2*7*7 is 2+7+7 = 16.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..156

Crossrefs

Cf. A001414, A007953, A063737.

Programs

Maple

sod:= n -> convert(convert(n,base,10),`+`):
spf:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
select(t -> sod(t) = spf(t+1), [$1..10^6]);

PARI

isok(m) = my(f=factor(m+1)); sumdigits(m) == f[, 1]~*f[, 2]; \\ Michel Marcus, Dec 18 2020