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Numbers k such that A001222(k) - A001221(k) is an odd prime.
1

%I #31 Sep 25 2024 10:28:08

%S 16,48,64,72,80,81,108,112,162,176,192,200,208,240,256,272,288,304,

%T 320,336,360,368,392,405,432,448,464,496,500,504,528,540,560,567,592,

%U 600,624,625,648,656,675,688,704,729,752,756,768,792,800,810,816,832,848

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

%C 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

%H G. C. Greubel, <a href="/A260985/b260985.txt">Table of n, a(n) for n = 1..3000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

%e 16 is in the sequence because A001222(16) - A001221(16) = 3.

%e 80 is in the sequence because A001222(80) - A001221(80) = 3.

%e 192 is in the sequence because A001222(192) - A001221(192) = 5.

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

%o (PARI) isok(n) = (d=bigomega(n)-omega(n)) && (d != 2) && isprime(d); \\ _Michel Marcus_, Aug 07 2015

%o (Python)

%o from sympy import isprime, primefactors

%o def omega(n): return 0 if n==1 else len(primefactors(n))

%o def bigomega(n): return 0 if n==1 else bigomega(n//min(primefactors(n))) + 1

%o def ok(n):

%o d = bigomega(n) - omega(n)

%o return d%2 and isprime(d)

%o print([n for n in range(1, 1001) if ok(n)]) # _Indranil Ghosh_, Apr 25 2017

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

%Y Subsequence of A013929.

%Y Subsequences: A195087, A195089, A195091.

%K nonn,easy

%O 1,1

%A _Carlos Eduardo Olivieri_, Aug 06 2015