A375143 - OEIS

%I #9 Aug 02 2024 02:51:07

%S 72,108,200,392,432,500,648,675,968,1125,1323,1352,1372,1800,2000,

%T 2312,2592,2700,2888,3087,3267,3528,3888,4232,4500,4563,5000,5292,

%U 5324,5400,5488,6125,6728,7688,7803,8575,8712,8788,9000,9747,9800,10125,10584,10952,11979

%N Numbers whose prime factorization has a minimum exponent that is larger than 1 and is 1 less than the maximum exponent.

%C Numbers k such that 2 <= A051904(k) = A051903(k) - 1.

%C Numbers that are product of two coprime nonsquarefree powers of squarefree numbers (A072777) with consecutive exponents.

%H Amiram Eldar, <a href="/A375143/b375143.txt">Table of n, a(n) for n = 1..10000</a>

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

%F Sum_{n>=1} 1/a(n) = Sum_{k>=2} f(k) = 0.053695635500385312854..., where f(k) = Product_{p prime} (1 + 1/p^k + 1/p^(k+1)) - zeta(k)/zeta(2*k) - zeta(k+1)/zeta(2*k+2) + 1 is the sum of reciprocals of the subset of numbers m with A051904(m) = k.

%e 72 = 2^3 * 3^2 is a term since A051904(72) = 2 is larger than 1 and is 1 less than A051903(72) = 3.

%t q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 2 <= Min[e] == Max[e] - 1]; Select[Range[12000], q]

%o (PARI) is(k) = {my(e = factor(k)[,2]); k > 1 && 2 <= vecmin(e) && vecmin(e) + 1 == vecmax(e);}

%Y Cf. A051903, A051904, A072777.

%Y Subsequence of A001694.

%Y Subsequences: A143610, A167747 \ {1, 2, 12}, A093136 \ {1, 2, 20}, A179666, A179702, A190472, A375073.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Aug 01 2024