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A268388 "Fermi-Dirac composites": numbers k for which A064547(k) > 1.

%I A268388 #29 Jul 09 2025 04:40:46
%S A268388 6,8,10,12,14,15,18,20,21,22,24,26,27,28,30,32,33,34,35,36,38,39,40,
%T A268388 42,44,45,46,48,50,51,52,54,55,56,57,58,60,62,63,64,65,66,68,69,70,72,
%U A268388 74,75,76,77,78,80,82,84,85,86,87,88,90,91,92,93,94,95,96,98,99,100,102,104,105,106,108,110,111,112,114,115,116,117,118,119,120
%N A268388 "Fermi-Dirac composites": numbers k for which A064547(k) > 1.
%H A268388 Antti Karttunen, <a href="/A268388/b268388.txt">Table of n, a(n) for n = 1..10000</a>
%e A268388 6 = 2^1 * 3^1 is present, as there are altogether two 1-bits in the exponents (1 and 1 also in binary), which is more than one.
%e A268388 64 = 2^6 is present, as the binary representation of 6 is "110", which contains more than one 1-bit. This is also the first term not present in A139118.
%t A268388 Select[Range[120], Plus @@ DigitCount[Last /@ FactorInteger[#], 2, 1] > 1 &] (* _Amiram Eldar_, Nov 27 2020 *)
%o A268388 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A268388 (define A268388 (MATCHING-POS 1 1 (lambda (n) (> (A064547 n) 1))))
%o A268388 (PARI) isok(n) = my(f = factor(n)[,2]); sum(k=1, #f, hammingweight(f[k])) > 1; \\ _Michel Marcus_, Feb 10 2016
%o A268388 (Python)
%o A268388 from sympy import primepi, integer_nthroot
%o A268388 def A268388(n):
%o A268388     def f(x): return int(n+1+sum(primepi(integer_nthroot(x,1<<i)[0]) for i in range(x.bit_length().bit_length())))
%o A268388     m, k = n, f(n)
%o A268388     while m != k: m, k = k, f(k)
%o A268388     return m # _Chai Wah Wu_, Feb 22 2025
%Y A268388 Subsequence of A002808.
%Y A268388 Cf. A050376 (complement without 1).
%Y A268388 Cf. A064547.
%Y A268388 Cf. A176699 (subsequence), A000379 (also subsequence, without the initial 1).
%Y A268388 Different from A139118.
%K A268388 nonn
%O A268388 1,1
%A A268388 _Antti Karttunen_, Feb 09 2016, after _Vladimir Shevelev_'s Apr 2010 comment in A176699