A339526 Number of divisors of n that are adjacent to at least one prime.

1, 2, 2, 3, 1, 4, 1, 4, 2, 3, 1, 6, 1, 3, 2, 5, 1, 5, 1, 5, 2, 3, 1, 8, 1, 2, 2, 5, 1, 6, 1, 6, 2, 2, 1, 8, 1, 3, 2, 7, 1, 6, 1, 5, 2, 3, 1, 10, 1, 3, 2, 4, 1, 6, 1, 6, 2, 3, 1, 10, 1, 3, 2, 6, 1, 6, 1, 4, 2, 5, 1, 11, 1, 3, 2, 4, 1, 5, 1, 9, 2, 3, 1, 10, 1, 2, 2, 7, 1, 8, 1

Offset

1,2

Comments

If p > 3 is prime, then a(p) = 1. - Wesley Ivan Hurt, May 21 2021

Links

Table of n, a(n) for n=1..91.

Formula

a(n) = Sum_{d|n} sign(c(d-1) + c(d+1)), where c is the prime characteristic (A010051).

Example

a(6) = 4; All 4 divisors of 6 are adjacent to at least one prime number since 1 ( +1 ) = 2, 2 ( +1 ) = 3, 3 ( -1 ) = 2 and 6 ( +1 ) = 7.

Mathematica

Table[Sum[Sign[PrimePi[n/i + 1] - PrimePi[n/i] + PrimePi[n/i - 1] - PrimePi[n/i - 2]] (1 - Ceiling[n/i] + Floor[n/i]), {i, n}], {n, 100}]
(* Second program: *)
Array[DivisorSum[#, 1 &, AnyTrue[# + {-1, 1}, PrimeQ] &] &, 100] (* Michael De Vlieger, Dec 10 2020 *)

Prog

(PARI) a(n) = sumdiv(n, d, ispseudoprime(d-1)||ispseudoprime(d+1)) \\ Felix Fröhlich, Dec 09 2020

Crossrefs

Cf. A010051.

Sequence in context: A071450 A175457 A322025 * A072078 A322316 A260439

Adjacent sequences: A339523 A339524 A339525 * A339527 A339528 A339529

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Dec 07 2020

Status

approved