A336098 -id:A336098 - 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).

A336099 Number of solutions of the equation k = n*sopf(k) in positive integers where sopf(k) is the sum of distinct prime factors of k.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 4, 1, 2, 2, 2, 1, 3, 2, 1, 1, 2, 1, 4, 1, 1, 0, 3, 1, 3, 1, 1, 2, 2, 1, 0, 1, 2, 2, 4, 1, 1, 2, 2, 1, 1, 1, 4, 2, 1, 1, 5, 1, 2, 2, 1, 2, 1, 1, 2, 1

Offset: 2

Vladimir Letsko, Jul 08 2020

nonn

Offset is 2 because a(1) cannot be defined since there are infinitely many solutions for n = 1, the primes.
If n = p^s then p^(s+1) is solution of k = n*sopf(k). Hence a(p^s) > 0. On the other hand there are infinitely many 0's in the sequence. For example a(5^s*11^t) = 0 for all positive integers s, t.
Records appear to occur only at prime n. These are seen in A336296, although note that A336296 is not monotonic, so it includes other terms. - Bill McEachen, Dec 02 2023

			a(3) = 2 because there are exactly 2 solutions of the equation k = 3*sopf(k) in positive integers (9 and 30).

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

Cf. A008472, A336098, A336296.

Cf. A158804 (all possible k's).

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.

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.

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A336099 Number of solutions of the equation k = n*sopf(k) in positive integers where sopf(k) is the sum of distinct prime factors of k.

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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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