A336098 Numbers k such that equation x = k*sopf(x) has no solutions in positive integers.

46, 55, 85, 87, 92, 110, 123, 138, 141, 145, 155, 158, 183, 184, 187, 190, 194, 203, 205, 217, 219, 220, 230, 238, 247, 253, 259, 261, 265, 275, 276, 282, 287, 290, 291, 295, 302, 305, 310, 316, 319, 327, 334, 339, 366, 368, 369, 380, 388, 391, 395, 403, 406, 407, 410, 414, 415, 423, 425, 426, 427, 434

Offset: 1

Author

Vladimir Letsko, Jul 08 2020

Keywords

• nonn

Comments

If k = p^s then p^(s+1) is solution of x = k*sopf(x). Hence powers of primes are not in the sequence.

Let p_1*...*p_t is in the sequence. Then p_1^a_1*...*p_t^a_t is in the sequence for all positive integers a_1, ..., a_t. It means that the sequence is infinite.

Links

• Vladimir Letsko, Mathematical Marathon, Problem 227 (in Russian).

Crossrefs

Cf. A008472, A336099.

Programs

PARI

sopf(n) = vecsum(factor(n)[, 1]); \\ A008472
pp(n) = prod(k=1, n, prime(k)); \\ A002110
sp(n) = sum(k=1, n, prime(k)); \\ A007504
ip(n) = {my(k=1); while (pp(k)/sp(k) <= n, k++); k+1;}
listako(nn) = {my(lim = pp(ip(nn))); my(v = vector(lim, k, k++; k/sopf(k))); my(w = vector(nn-1, k, #select(x->(x==k+1), v))); apply(x->(x+1), Vec(select(x->(x==0), w, 1)));} \\ Michel Marcus, Jul 16 2020