A375142 - OEIS

%I #10 Aug 02 2024 08:58:12

%S 4,8,9,12,16,18,20,24,25,27,28,32,36,40,44,45,48,49,50,52,54,56,60,63,

%T 64,68,75,76,80,81,84,88,90,92,96,98,99,100,104,112,116,117,120,121,

%U 124,125,126,128,132,135,136,140,147,148,150,152,153,156,160,162,164

%N Numbers whose powerful part (A057521) is a power of a squarefree number that is larger than 1 (A072777).

%C Subsequence of A013929 and first differs from it at n = 27: A013929(27) = 72 = 2^3 * 3^2 is not a term of this sequence.

%C Numbers whose prime factorization has one distinct exponent that does not equal 1.

%C Numbers that are a product of a squarefree number (A005117) and a power of a different squarefree number that is not squarefree.

%C The asymptotic density of this sequence is Sum_{k>=2} (d(k)-1)/zeta(2) = 0.36113984820338109927..., where d(k) = zeta(k) * Product_{p prime} (1 + Sum_{i=k+1..2*k-1} (-1)^i/p^i), if k is even, and d(k) = (zeta(2*k)/zeta(k)) * Product_{p prime} (1 + 2/p^k + Sum_{i=k+1..2*k-1} (-1)^(i+1)/p^i) if k is odd > 1.

%H Amiram Eldar, <a href="/A375142/b375142.txt">Table of n, a(n) for n = 1..10000</a>

%H Eckford Cohen, <a href="https://projecteuclid.org/euclid.pjm/1103036708">Arithmetical notes. III. Certain equally distributed sets of integers</a>, Pacific Journal of Mathematics, Vol. 12, No. 1 (1962), pp. 77-84.

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

%e 12 = 2^2 * 3 is a term because its powerful part, 4 = 2^2, is a power of a squarefree number, 2, that is larger than 1.

%t q[n_] := Count[Union[FactorInteger[n][[;; , 2]]], _?(# > 1 &)] == 1; Select[Range[200], q]

%o (PARI) is(k) = {my(e = select(x -> (x > 1), Set(factor(k)[,2]))); #e == 1;}

%Y Cf. A005117, A057521.

%Y Subsequence of A013929.

%Y Subsequences: A067259, A072777, A190641, A336591.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Aug 01 2024