A157190 -id:A157190 - cp's OEIS frontend

A157191 Indices of records in A157190: prime(a(n)) can be written in more ways as pq-p-q (p,q, prime) than any smaller prime.

1, 2, 17, 20, 190, 357, 959, 4479, 10522, 16447, 21383, 22993, 37977, 53972, 129076, 205244, 440069, 745584, 830983, 918422, 1073653, 2309722, 2731757, 3833005, 4788328, 5025102, 5411333

Offset: 1

M. F. Hasler, Mar 16 2009

nonn

A157191 = A000720 o A157190 ; A157190 = A000040 o A157191.

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.

Original entry on oeis.org

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).

A157188 Number of ways to write prime(n) as p*q-(p+q) with primes p<=q.

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Mar 11 2009

nonn

Records in this sequence are given in A157189, the corresponding primes in A157190.

			a(1)=0 since prime(1) = 2 cannot be written as pq-(p+q) for primes p,q.
a(2)=2 since prime(2) = 3 = 2*5-(2+5) = 3*3-(3+3) are the two possibilities.

C. Rivera (Ed.), Puzzle 482: Two Bergot questions, March 2009.

Cf. A157187-A157190

A157188(n)={ local(c=0,L=sqrtint(n=prime(n)+1)); fordiv( n,d, d>L&break; isprime(d+1) || next; isprime(n/d+1) & c++);c}

A157189 Records in A157188.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 9, 10, 11, 12, 14, 16, 17, 19, 20, 22, 23, 25, 26, 27, 31, 32, 33, 34, 37, 42, 45, 46, 54, 58, 61, 65, 70, 71, 73, 76, 77, 78, 87, 100

Offset: 1

M. F. Hasler, Mar 11 2009

nonn

The corresponding primes are given in A157190.

C. Rivera (Ed.), Puzzle 482: Two Bergot questions, March 2009.

Cf. A157187-A157190

a(n) = A157187(A157190(n)) > A157187(p) for all primes p < A157190(n).

Corrected a(29), deleted spurious term 52 M. F. Hasler, Mar 15 2009

cp's OEIS Frontend

A157191 Indices of records in A157190: prime(a(n)) can be written in more ways as pq-p-q (p,q, prime) than any smaller prime.

Views

Author

Keywords

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A157188 Number of ways to write prime(n) as p*q-(p+q) with primes p<=q.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A157189 Records in A157188.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions