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Cubefull exponentially odd numbers: numbers whose prime factorization contains only odd exponents that are larger than 1.
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%I #30 Mar 24 2024 03:59:30

%S 1,8,27,32,125,128,216,243,343,512,864,1000,1331,1944,2048,2187,2197,

%T 2744,3125,3375,3456,4000,4913,6859,7776,8192,9261,10648,10976,12167,

%U 13824,16000,16807,17496,17576,19683,24389,25000,27000,29791,30375,31104,32768,35937

%N Cubefull exponentially odd numbers: numbers whose prime factorization contains only odd exponents that are larger than 1.

%C This sequence is a permutation of A355038.

%C This sequence is also a permutation of the exponentially odd numbers (A268335) multiplied by the square of their squarefree kernel (A007947).

%C a(n)/rad(a(n)) is a permutation of the squares.

%C a(n)/rad(a(n))^2 is a permutation of the exponentially odd numbers.

%H Amiram Eldar, <a href="/A335988/b335988.txt">Table of n, a(n) for n = 1..10000</a>

%F Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p*(p^2-1))) = 1.2312911... (A065487).

%e 8 = 2^3 is a term since the exponent of its prime factor 2 is 3 which is odd and larger than 1.

%t Join[{1}, Select[Range[10^5], AllTrue[Last /@ FactorInteger[#], #1 > 1 && OddQ[#1] &] &]]

%o (Python)

%o from math import isqrt, prod

%o from sympy import factorint

%o def afind(N): # all terms up to limit N

%o cands = (n**2*prod(factorint(n**2)) for n in range(1, isqrt(N//2)+2))

%o return sorted(c for c in cands if c <= N)

%o print(afind(4*10**4)) # _Michael S. Branicky_, Jun 16 2022

%Y Intersection of A001694 and A268335.

%Y Intersection of A036966 and A268335.

%Y A355038 in ascending order.

%Y A030078, A050997, A092759, A179665, A079395 and A138031 are subsequences.

%Y Cf. A007947, A065487.

%K nonn

%O 1,2

%A _Amiram Eldar_, Jul 03 2020