A336296 -id:A336296 - cp's OEIS frontend

A157190 Primes which produce records in A157188, at index i=pi(a(n)) (pi=A000720).

2, 3, 59, 71, 1151, 2399, 7559, 42839, 110879, 181439, 241919, 262079, 453599, 665279, 1713599, 2827439, 6425999, 11309759, 12700799, 14137199, 16707599, 37837799, 45239039, 64864799, 82162079, 86486399, 93562559, 260124479, 410810399, 735134399, 950019839

Offset: 1

M. F. Hasler, Mar 11 2009

nonn

Primes that can be written in more ways as p*q-(p+q) (p,q prime) than any smaller prime.

M. F. Hasler, Table of n, a(n), for n=1..40.
Carlos Rivera, Puzzle 482: Two Bergot questions, The Prime Puzzles & Problems Connection.

Cf. A000720, A157187, A157188, A157189, A336296.

A157187(a(n)) = A157188(A000720(a(n))) > A157187(p) for all primes p < a(n).

a(28) corrected by M. F. Hasler, Mar 15 2009

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.

A336445 Integers m such that m/sopf(m) is a prime number where sopf(m) is A008472(m), the sum of the distinct primes dividing m.

Original entry on oeis.org

4, 9, 25, 30, 49, 70, 84, 105, 121, 169, 231, 234, 260, 286, 289, 361, 456, 529, 532, 627, 646, 805, 841, 897, 961, 1116, 1364, 1369, 1581, 1665, 1681, 1798, 1849, 1924, 2064, 2150, 2209, 2632, 2809, 2967, 3055, 3339, 3481, 3526, 3721, 4489, 4543, 4824, 5025, 5041

Offset: 1

Michel Marcus, Jul 22 2020

nonn

All squares of primes (A001248) are terms.

			4 is a term since sopf(4)=2 and 4/2 = 2 is a prime.
30 is a term since sopf(30)=10 and 30/10 = 3 is a prime.

Michel Marcus, Table of n, a(n) for n = 1..1049 (terms up to 10^7)

Cf. A008472 (sopf).
Subsequence of A071139.
A001248 is a subsequence.
Cf. A336296, A336297.

sopf(n)=vecsum(factor(n)[, 1]); \\ A008472
isokp(k) = my(q=k/sopf(k)); (denominator(q)==1) && isprime(q);

A336493 a(n) is the largest integer x such that x/sopf(x) = prime(n) where sopf(x) is the sum of distinct prime factors of x and prime(n) is the n-th prime.

Original entry on oeis.org

4, 30, 70, 105, 286, 260, 646, 627, 897, 1798, 1581, 1924, 3526, 2967, 3055, 3339, 7198, 3721, 6164, 10366, 5840, 9717, 10707, 9256, 9409, 20806, 16377, 23326, 16132, 12769, 21844, 26331, 38086, 29607, 44998, 22801, 33284, 40587, 35905, 29929, 64798, 44164, 73726

Offset: 1

Michel Marcus, Jul 23 2020

nonn

For the primes p=prime(m) that are in A336297, a(m) = prime(m)^2.

			For the 4th prime p=7, the 3 integers (49,84,105) satisfy x/sopf(x)=7, so a(4)=105.

Cf. A008472.
Subsequence of A336445.
Cf. A336296, A336297.

cp's OEIS Frontend

A157190 Primes which produce records in A157188, at index i=pi(a(n)) (pi=A000720).

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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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A336445 Integers m such that m/sopf(m) is a prime number where sopf(m) is A008472(m), the sum of the distinct primes dividing m.

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A336493 a(n) is the largest integer x such that x/sopf(x) = prime(n) where sopf(x) is the sum of distinct prime factors of x and prime(n) is the n-th prime.

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