A260985 Numbers k such that A001222(k) - A001221(k) is an odd prime.

16, 48, 64, 72, 80, 81, 108, 112, 162, 176, 192, 200, 208, 240, 256, 272, 288, 304, 320, 336, 360, 368, 392, 405, 432, 448, 464, 496, 500, 504, 528, 540, 560, 567, 592, 600, 624, 625, 648, 656, 675, 688, 704, 729, 752, 756, 768, 792, 800, 810, 816, 832, 848

Offset: 1

Author

Carlos Eduardo Olivieri, Aug 06 2015

Keywords

• nonn
• easy

Comments

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0626525..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

Examples

			16 is in the sequence because A001222(16) - A001221(16) = 3.
80 is in the sequence because A001222(80) - A001221(80) = 3.
192 is in the sequence because A001222(192) - A001221(192) = 5.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..3000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A001221, A001222, A046660, A059956, A112526.

Subsequence of A013929.

Subsequences: A195087, A195089, A195091.

Programs

Mathematica

Select[Range[10^3], !PrimeQ[#] && PrimeQ[p = PrimeOmega[#] - PrimeNu[#]] && OddQ[p] &]

PARI

isok(n) = (d=bigomega(n)-omega(n)) && (d != 2) && isprime(d); \\ Michel Marcus, Aug 07 2015

Python

from sympy import isprime, primefactors
def omega(n): return 0 if n==1 else len(primefactors(n))
def bigomega(n): return 0 if n==1 else bigomega(n//min(primefactors(n))) + 1
def ok(n):
    d = bigomega(n) - omega(n)
    return d%2 and isprime(d)
print([n for n in range(1, 1001) if ok(n)]) # Indranil Ghosh, Apr 25 2017