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A342081 Numbers without an inferior odd divisor > 1.

%I A342081 #18 Nov 01 2024 05:14:00
%S A342081 1,2,3,4,5,6,7,8,10,11,13,14,16,17,19,20,22,23,26,28,29,31,32,34,37,
%T A342081 38,41,43,44,46,47,52,53,58,59,61,62,64,67,68,71,73,74,76,79,82,83,86,
%U A342081 88,89,92,94,97,101,103,104,106,107,109,113,116,118,122,124
%N A342081 Numbers without an inferior odd divisor > 1.
%C A342081 We define a divisor d|n to be inferior if d <= n/d. Inferior divisors are counted by A038548 and listed by A161906.
%C A342081 Numbers n such that n is either a power of 2 or has a single odd prime factor > sqrt(n). Equivalently, numbers n such that all odd prime factors are > sqrt(n). - _Chai Wah Wu_, Mar 08 2021
%H A342081 Amiram Eldar, <a href="/A342081/b342081.txt">Table of n, a(n) for n = 1..10000</a>
%e A342081 The divisors > 1 of 72 are {2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}, of which {3, 9} are odd and {2, 3, 4, 6, 8} are inferior, with intersection {3}, so 72 is not in the sequence.
%t A342081 Select[Range[100],Function[n,Select[Divisors[n]//Rest,OddQ[#]&&#<=n/#&]=={}]]
%o A342081 (Python)
%o A342081 from sympy import primefactors
%o A342081 A342081_list = [n for n in range(1,10**3) if len([p for p in primefactors(n) if p > 2 and p*p <= n]) == 0] # _Chai Wah Wu_, Mar 08 2021
%o A342081 (PARI) is(n) = #select(x -> x > 2 && x^2 <= n, factor(n)[, 1]) == 0; \\ _Amiram Eldar_, Nov 01 2024
%Y A342081 The strictly inferior version is the same with A001248 added.
%Y A342081 Positions of 1's in A069288.
%Y A342081 The superior version is A116882, with complement A116883.
%Y A342081 The complement is A342082.
%Y A342081 A006530 selects the greatest prime factor.
%Y A342081 A020639 selects the smallest prime factor.
%Y A342081 A038548 counts superior (or inferior) divisors, with strict case A056924.
%Y A342081 - Odd -
%Y A342081 A000009 counts partitions into odd parts, ranked by A066208.
%Y A342081 A001227 counts odd divisors.
%Y A342081 A026424 lists numbers with odd Omega.
%Y A342081 A027193 counts odd-length partitions.
%Y A342081 A058695 counts partitions of odd numbers.
%Y A342081 A067659 counts strict partitions of odd length, ranked by A030059.
%Y A342081 A340101 counts factorizations into odd factors; A340102 also has odd length.
%Y A342081 A340854/A340855 cannot/can be factored with odd minimum factor.
%Y A342081 A341594 counts strictly superior odd divisors
%Y A342081 A341675 counts superior odd divisors.
%Y A342081 - Inferior: A033676, A066839, A161906.
%Y A342081 - Superior: A033677, A051283, A059172, A063538/A063539.
%Y A342081 - Strictly Inferior A333805, A341674.
%Y A342081 - Strictly Superior: A064052/A048098, A341645/A341646.
%Y A342081 Cf. A000005, A000203, A001055, A001221, A001222, A001414, A207375, A244991, A300272, A340832, A340931.
%K A342081 nonn
%O A342081 1,2
%A A342081 _Gus Wiseman_, Mar 06 2021