A354515 -id:A354515 - cp's OEIS frontend

A354512 Number of solutions m >= 2 to m - gpf(m) = n, gpf = A006530.

0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 2, 2, 0, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2

Offset: 1

Jianing Song, Aug 16 2022

Number of primes p such that gpf(n+p) = p (such p must be prime factors of n).
Number of distinct prime factors p of n such that n+p is p-smooth.
Clearly we have a(n) <= omega(n) for all n, omega = A001221. The differences are given by A354527.
Is this sequence unbounded? Note that 4 does not appear until a(1660577).

			a(78) = 2 since the prime factors of 78 are 2,3,13, and we have gpf(78+3) = 3 and gpf(78+13) = 13, so the solutions to m - gpf(m) = 78 are m = 78+3 = 81 or m = 78+13 = 91. Note that gpf(78+2) != 2.
a(12) = 0 since the prime factors of 12 are 2,3, and we have gpf(12+2) != 2 and gpf(12+3) != 3.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A006530, A076563, A001221, A354516 (indices of first occurrence of each number), A354527.

Cf. A354514 (0 together with indices of positive terms), A354515 (indices of 0), A354516, A354525 (indices n for which a(n) reaches omega(n)), A354526 (indices n for which a(n) is smaller than omega(n)).

gpf(n) = vecmax(factor(n)[, 1]);
a(n) = my(f=factor(n)[, 1]); sum(i=1, #f, gpf(n+f[i])==f[i])

A354514 Numbers k such that m - gpf(m) = k has solutions m >= 2, gpf = A006530.

Original entry on oeis.org

0, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88

Offset: 1

Jianing Song, Aug 16 2022

Numbers k such that there is a prime p such that gpf(k+p) = p (such p must be a prime factor of n).
Numbers k such that there is a prime factor p of k such that k+p is p-smooth.
A076563 sorted and duplicates removed.

			0 is a term because 0 = p - gpf(p) for every prime p.
if k/gpf(k) <= nextprime(gpf(k)) - 2, where nextprime = A151800, then k is a term since k+gpf(k) <= gpf(k)*(nextprime(gpf(k)) - 1) implies gpf(k+gpf(k)) = gpf(k).

Jianing Song, Table of n, a(n) for n = 1..8650 (all terms <= 10000)

Cf. A006530, A076563, A151800.

0 together with indices of positive terms in A354512. Complement of A354515.

gpf(n) = vecmax(factor(n)[, 1]);
isA354514(n) = if(n, my(f=factor(n)[, 1]); for(i=1, #f, if(gpf(n+f[i])==f[i], return(1))); 0, 1)

cp's OEIS Frontend

A354512 Number of solutions m >= 2 to m - gpf(m) = n, gpf = A006530.

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