cp's OEIS Frontend

A360015 Numbers whose exponent of 2 in their canonical prime factorization is equal to the maximal exponent.

%I A360015 #30 Jul 10 2025 04:27:19
%S A360015 1,2,4,6,8,10,12,14,16,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,
%T A360015 52,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,92,94,96,100,
%U A360015 102,104,106,110,112,114,116,118,120,122,124,128,130,132,134,136,138
%N A360015 Numbers whose exponent of 2 in their canonical prime factorization is equal to the maximal exponent.
%C A360015 Numbers k such that A007814(k) = A051903(k).
%C A360015 The powers of 2 (A000079) are all terms.
%C A360015 The product of any two terms (not necessarily distinct) is also a term.
%C A360015 This sequence is a disjoint union of {1} and the subsequences of numbers m of the form 2^(k-1)*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 2*o where o is an odd squarefree number (A056911); for k = 3, numbers of the form 4*o where o is an odd cubefree number; etc.
%C A360015 The asymptotic density of this sequence is Sum_{k>=2} 1/(zeta(k)*(2^k-1)) = 0.44541445377638761933... .
%C A360015 The asymptotic mean of the exponent of 2 in the prime factorization of the terms of this sequence is Sum_{k>=2} (k-1)/(zeta(k)*(2^k-1)) / Sum_{k>=2} 1/(zeta(k)*(2^k-1)) = 2.10346728882748723133... . [corrected by _Amiram Eldar_, Jul 10 2025]
%C A360015 Also numbers whose multiset of prime factors has low (i.e. least) mode 2. Here, a mode in a multiset is an element that appears at least as many times as each of the others; for example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}. - _Gus Wiseman_, Jul 14 2023
%H A360015 Amiram Eldar, <a href="/A360015/b360015.txt">Table of n, a(n) for n = 1..10000</a>
%F A360015 Disjoint union of A360013 and A360014.
%F A360015 a(n) = A360013(n)/2. - _Gus Wiseman_, Jul 14 2023
%e A360015 From _Gus Wiseman_, Jul 14 2023: (Start)
%e A360015 108 = 2*2*3*3*3 is missing because its mode is not 2.
%e A360015 180 = 2*2*3*3*5 is present because it has low mode 2.
%e A360015 The terms together with their prime factorizations begin:
%e A360015    1 =
%e A360015    2 = 2
%e A360015    4 = 2*2
%e A360015    6 = 2*3
%e A360015    8 = 2*2*2
%e A360015   10 = 2*5
%e A360015   12 = 2*2*3
%e A360015   14 = 2*7
%e A360015   16 = 2*2*2*2
%e A360015   20 = 2*2*5
%e A360015   22 = 2*11
%e A360015   24 = 2*2*2*3
%e A360015   26 = 2*13
%e A360015   28 = 2*2*7
%e A360015   30 = 2*3*5
%e A360015   32 = 2*2*2*2*2
%e A360015   34 = 2*17
%e A360015   36 = 2*2*3*3
%e A360015 (End)
%t A360015 q[n_] := IntegerExponent[n, 2] == Max[FactorInteger[n][[;; , 2]]]; q[1] = True; Select[Range[150], q]
%o A360015 (PARI) is(n) = n == 1 || vecmax(factor(n)[,2]) == valuation(n, 2);
%Y A360015 Cf. A000079, A000265, A007814, A051903, A056911.
%Y A360015 Partitions of this type are counted by A241131.
%Y A360015 The case of unique mode is A360013, complement here A360014.
%Y A360015 For unique minimal prime exponent we have A364061, counted by A364062.
%Y A360015 For minimal prime exponent we have A364158, counted by A364159.
%Y A360015 A027746 lists prime factors (with multiplicity).
%Y A360015 A112798 lists prime indices, length A001222, sum A056239.
%Y A360015 A362611 counts modes in prime indices, triangle A362614.
%Y A360015 Cf. A002865, A327473, A327476, A356862, A359178, A362605, A362613, A363486.
%K A360015 nonn,easy
%O A360015 1,2
%A A360015 _Amiram Eldar_, Jan 21 2023