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

%I A260985 #31 Sep 25 2024 10:28:08
%S A260985 16,48,64,72,80,81,108,112,162,176,192,200,208,240,256,272,288,304,
%T A260985 320,336,360,368,392,405,432,448,464,496,500,504,528,540,560,567,592,
%U A260985 600,624,625,648,656,675,688,704,729,752,756,768,792,800,810,816,832,848
%N A260985 Numbers k such that A001222(k) - A001221(k) is an odd prime.
%C A260985 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 A260985 G. C. Greubel, <a href="/A260985/b260985.txt">Table of n, a(n) for n = 1..3000</a>
%H A260985 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%e A260985 16 is in the sequence because A001222(16) - A001221(16) = 3.
%e A260985 80 is in the sequence because A001222(80) - A001221(80) = 3.
%e A260985 192 is in the sequence because A001222(192) - A001221(192) = 5.
%t A260985 Select[Range[10^3], !PrimeQ[#] && PrimeQ[p = PrimeOmega[#] - PrimeNu[#]] && OddQ[p] &]
%o A260985 (PARI) isok(n) = (d=bigomega(n)-omega(n)) && (d != 2) && isprime(d); \\ _Michel Marcus_, Aug 07 2015
%o A260985 (Python)
%o A260985 from sympy import isprime, primefactors
%o A260985 def omega(n): return 0 if n==1 else len(primefactors(n))
%o A260985 def bigomega(n): return 0 if n==1 else bigomega(n//min(primefactors(n))) + 1
%o A260985 def ok(n):
%o A260985     d = bigomega(n) - omega(n)
%o A260985     return d%2 and isprime(d)
%o A260985 print([n for n in range(1, 1001) if ok(n)]) # _Indranil Ghosh_, Apr 25 2017
%Y A260985 Cf. A001221, A001222, A046660, A059956, A112526.
%Y A260985 Subsequence of A013929.
%Y A260985 Subsequences: A195087, A195089, A195091.
%K A260985 nonn,easy
%O A260985 1,1
%A A260985 _Carlos Eduardo Olivieri_, Aug 06 2015