A336099 -id:A336099 - cp's OEIS frontend

A336296 The least prime p such that equation x = p*sopf(x) (where sopf(x) is the sum of distinct prime factors of x) has exactly n solutions in positive integers.

2, 3, 7, 19, 71, 431, 1259, 4679, 9719, 23399, 7559, 42839, 134399, 181439, 477359, 241919, 262079, 453599

Offset: 1

Vladimir Letsko, Jul 16 2020

It seems that a(n) is the least number for which equation x = p*sopf(x) has exactly n solutions in positive integers not only for prime numbers.

			a(3) = 7 because there are 3 solutions of the equation x = 7*sopf(x), which are {49, 84, 105}, and this is the smallest prime that gives 3 solutions.

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

Cf. A008472, A089352, A336098, A336099, A336297, A157190 (note overlap of values).

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

Original entry on oeis.org

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

Vladimir Letsko, Jul 08 2020

nonn

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.

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

Cf. A008472, A336099.

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

A336297 Prime numbers p such that equation x = p*sopf(x) (where sopf(x) is the sum of distinct prime factors of x) has exactly 1 solution in positive integers.

Original entry on oeis.org

2, 61, 97, 113, 151, 173, 241, 277, 317, 353, 389, 449, 457, 593, 601, 607, 653, 673, 683, 727, 733, 797, 907, 929, 941, 947, 953, 977, 997, 1021, 1051, 1087, 1153, 1181, 1193, 1217, 1249, 1307, 1321, 1361, 1373, 1409, 1433, 1489, 1493, 1523, 1553, 1579, 1597, 1609, 1627

Offset: 1

Vladimir Letsko, Jul 16 2020

nonn

			4 is the unique integer x such that x = 2*sopf(x), a prime, so 2 is a term.

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

Cf. A008472, A336098, A336099, A336296.

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A336296 The least prime p such that equation x = p*sopf(x) (where sopf(x) is the sum of distinct prime factors of x) has exactly n solutions in positive integers.

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

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A336297 Prime numbers p such that equation x = p*sopf(x) (where sopf(x) is the sum of distinct prime factors of x) has exactly 1 solution in positive integers.

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