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

1,1

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

Table of n, a(n) for n=1..62.

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

Prog

(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

Crossrefs

Cf. A008472, A336099.

Sequence in context: A043177 A043957 A363762 * A115444 A119385 A330243

Adjacent sequences: A336095 A336096 A336097 * A336099 A336100 A336101

Keyword

nonn

Author

Vladimir Letsko, Jul 08 2020

Status

approved