A262965 Least number k such that k mod s = prime(n) where s is the sum of the distinct primes dividing k.

12, 10, 14, 15, 26, 57, 38, 85, 87, 62, 111, 129, 86, 603, 159, 177, 122, 201, 219, 146, 237, 927, 267, 545, 309, 206, 327, 218, 1057, 1016, 1359, 411, 278, 1267, 302, 471, 489, 3088, 519, 537, 362, 1561, 386, 597, 398, 1687, 3856, 687, 458, 1897, 717, 482

Offset: 1

Author

Michel Lagneau, Oct 05 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for all n > 0.

Many terms are numbers with two distinct prime divisors, exceptions being a(157) = 15465, a(254) = 25815, a(279) = 28695, a(303) = 31665, ... which have three prime distinct divisors, ...

Examples

			a(5) = 26 because 26 = 2*13 => 26 mod (2+13) = 26 mod 15 = 11 = prime(5).
		

Links

• Michel Lagneau, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A008472.

Programs

Mathematica

Table[k=1;While[Mod[k,Plus@@First[Transpose[FactorInteger[k]]]]!=Prime[n],k++];k,{n,50}]

PARI

spf(k) = my(f = factor(k)); vecsum(f[,1]);
a(n) = {k=2; while (k % spf(k) != prime(n), k++); k;} \\ Michel Marcus, Oct 06 2015

Python

from sympy import prime, primefactors
def a(n):
    k, target = 2, prime(n)
    while k%sum(primefactors(k)) != target: k += 1
    return k
print([a(n) for n in range(1, 53)]) # Michael S. Branicky, Dec 10 2021