cp's OEIS Frontend

A360013 Numbers whose exponent of 2 in their canonical prime factorization is larger than all the other exponents.

%I A360013 #47 Jul 10 2025 04:23:39
%S A360013 2,4,8,12,16,20,24,28,32,40,44,48,52,56,60,64,68,72,76,80,84,88,92,96,
%T A360013 104,112,116,120,124,128,132,136,140,144,148,152,156,160,164,168,172,
%U A360013 176,184,188,192,200,204,208,212,220,224,228,232,236,240,244,248,256
%N A360013 Numbers whose exponent of 2 in their canonical prime factorization is larger than all the other exponents.
%C A360013 Numbers k such that A007814(k) > A051903(A000265(k)).
%C A360013 The powers of 2 (A000079), except for 1, are all terms.
%C A360013 The product of any two terms (not necessarily distinct) is also a term.
%C A360013 This sequence is a disjoint union of {2} and the subsequences of numbers m of the form 2^k*o where o = A000265(m), the odd part of m, is a k-free number, for k >= 2. These subsequences include, for k = 2, numbers of the form 4*o where o is an odd squarefree number (A056911); for k = 3, numbers of the form 8*o where o is an odd cubefree number; etc.
%C A360013 The asymptotic density of this sequence is Sum_{k>=2} 1/(zeta(k)*2*(2^k-1)) = 0.222707226888193809... .
%C A360013 The asymptotic mean of the exponent of 2 in the prime factorization of the terms of this sequence is Sum_{k>=2} k/(zeta(k)*2*(2^k-1)) / Sum_{k>=2} 1/(zeta(k)*2*(2^k-1)) = 3.10346728882748723133... . [corrected by _Amiram Eldar_, Jul 10 2025]
%C A360013 This sequence is a subsequence of A360015 and the asymptotic density of this sequence within A360015 is exactly 1/2.
%C A360013 Also even numbers whose multiset of prime factors has unique mode 2. - _Gus Wiseman_, Jul 10 2023
%H A360013 Amiram Eldar, <a href="/A360013/b360013.txt">Table of n, a(n) for n = 1..10000</a>
%F A360013 a(n) = 2*A360015(n). - _Gus Wiseman_, Jul 10 2023
%e A360013 From _Gus Wiseman_, Jul 09 2023: (Start)
%e A360013 108 = 2*2*3*3*3 is missing because its mode is not 2.
%e A360013 180 = 2*2*3*3*5 is missing because 2 is not the unique mode.
%e A360013 120 = 2*2*2*3*5 is present because its unique mode is 2.
%e A360013 The terms together with their prime factorizations begin:
%e A360013    2 = 2
%e A360013    4 = 2*2
%e A360013    8 = 2*2*2
%e A360013   12 = 2*2*3
%e A360013   16 = 2*2*2*2
%e A360013   20 = 2*2*5
%e A360013   24 = 2*2*2*3
%e A360013   28 = 2*2*7
%e A360013   32 = 2*2*2*2*2
%e A360013   40 = 2*2*2*5
%e A360013   44 = 2*2*11
%e A360013   48 = 2*2*2*2*3
%e A360013   52 = 2*2*13
%e A360013   56 = 2*2*2*7
%e A360013   60 = 2*2*3*5
%e A360013   64 = 2*2*2*2*2*2
%e A360013 (End)
%t A360013 q[n_] := Module[{e = IntegerExponent[n, 2], m}, m = n/2^e; (m == 1 && e > 0) || AllTrue[FactorInteger[m][[;; , 2]], # < e &]]; Select[Range[256], q]
%o A360013 (PARI) is(n) = {my(e = valuation(n, 2), m = n >> e); (m == 1 && e > 0) || (m > 1 && vecmax(factor(m)[,2]) < e)};
%Y A360013 Cf. A000079, A000265, A007814, A051903, A056911.
%Y A360013 Equals A360015 \ A360014.
%Y A360013 Partitions of this type are counted by A241131.
%Y A360013 Allowing any unique mode gives A356862, complement A362605.
%Y A360013 Allowing any unique co-mode gives A359178, complement A362606.
%Y A360013 Not requiring the mode to be unique gives A360015.
%Y A360013 The opposite version is A362616, counted by A362612.
%Y A360013 For co-mode instead of mode we have A364061, counted by A364062.
%Y A360013 With least prime factor instead of 2, we have A364160, counted by A364193.
%Y A360013 With a different factorization, we have the subsequence A335738.
%Y A360013 A124010 gives prime signature, ordered A118914.
%Y A360013 A362611 counts modes in prime factorization, triangle A362614.
%Y A360013 A362613 counts co-modes in prime factorization, triangle A362615.
%Y A360013 A363486 gives least mode in prime indices, A363487 greatest.
%Y A360013 Cf. A001222, A002865, A327473, A327476, A362608, A362610, A363723, A363727.
%K A360013 nonn,easy
%O A360013 1,1
%A A360013 _Amiram Eldar_, Jan 21 2023