A073493 Numbers having exactly one prime gap in their factorization.

10, 14, 20, 21, 22, 26, 28, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 106, 111, 112, 114, 115, 116, 117, 118, 119, 122, 123, 124, 126, 129

Offset: 1

Author

Reinhard Zumkeller, Aug 03 2002

Keywords

• nonn

Examples

			200 is a term, as 200 = 2*2*2*5*5 with one gap between 2 and 5.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005117, A073490, A073492, A073494, A073495, A073487.

Programs

Haskell

a073493 n = a073493_list !! (n-1)
a073493_list = filter ((== 1) . a073490) [1..]
-- Reinhard Zumkeller, Dec 20 2013

Mathematica

pa[n_, k_] := If[k == NextPrime[n], 0, 1]; Select[Range[130], Total[pa @@@ Partition[First /@ FactorInteger[#], 2, 1]] == 1 &] (* Jayanta Basu, Jul 01 2013 *)

Python

from sympy import primefactors, nextprime
def ok(n):
    pf = primefactors(n)
    return sum(p2 != nextprime(p1) for p1, p2 in zip(pf[:-1], pf[1:])) == 1
print(list(filter(ok, range(1, 130)))) # Michael S. Branicky, Oct 14 2021

Formula

A073490(a(n)) = 1.