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A381439 Numbers whose exponent of 2 in their canonical prime factorization is not larger than all the other exponents.

%I A381439 #6 Mar 02 2025 16:04:19
%S A381439 3,5,6,7,9,10,11,13,14,15,17,18,19,21,22,23,25,26,27,29,30,31,33,34,
%T A381439 35,36,37,38,39,41,42,43,45,46,47,49,50,51,53,54,55,57,58,59,61,62,63,
%U A381439 65,66,67,69,70,71,73,74,75,77,78,79,81,82,83,85,86,87,89
%N A381439 Numbers whose exponent of 2 in their canonical prime factorization is not larger than all the other exponents.
%C A381439 First differs from A335740 in lacking 72, which has prime indices {1,1,1,2,2} and section-sum partition (3,3,1).
%C A381439 Also numbers whose section-sum partition of prime indices (A381436) ends with a number > 1.
%C A381439 Includes all squarefree numbers (A005117) except 2.
%F A381439 Positive integers n such that A007814(n) <= A375669(n).
%e A381439 The terms together with their prime indices begin:
%e A381439      3: {2}        25: {3,3}        45: {2,2,3}
%e A381439      5: {3}        26: {1,6}        46: {1,9}
%e A381439      6: {1,2}      27: {2,2,2}      47: {15}
%e A381439      7: {4}        29: {10}         49: {4,4}
%e A381439      9: {2,2}      30: {1,2,3}      50: {1,3,3}
%e A381439     10: {1,3}      31: {11}         51: {2,7}
%e A381439     11: {5}        33: {2,5}        53: {16}
%e A381439     13: {6}        34: {1,7}        54: {1,2,2,2}
%e A381439     14: {1,4}      35: {3,4}        55: {3,5}
%e A381439     15: {2,3}      36: {1,1,2,2}    57: {2,8}
%e A381439     17: {7}        37: {12}         58: {1,10}
%e A381439     18: {1,2,2}    38: {1,8}        59: {17}
%e A381439     19: {8}        39: {2,6}        61: {18}
%e A381439     21: {2,4}      41: {13}         62: {1,11}
%e A381439     22: {1,5}      42: {1,2,4}      63: {2,2,4}
%e A381439     23: {9}        43: {14}         65: {3,6}
%t A381439 Select[Range[100],FactorInteger[2*#][[1,2]]-1<=Max@@Last/@Rest[FactorInteger[2*#]]&]
%Y A381439 The LHS (exponent of 2) is A007814.
%Y A381439 The complement is A360013 = 2*A360015 (if we prepend 1), counted by A241131 (shifted right and starting with 1 instead of 0).
%Y A381439 The case of equality is A360014, inclusive A360015.
%Y A381439 The RHS (greatest exponent of an odd prime factor) is A375669.
%Y A381439 These are positions of terms > 1 in A381437.
%Y A381439 Partitions of this type are counted by A381544.
%Y A381439 A000040 lists the primes, differences A001223.
%Y A381439 A051903 gives greatest prime exponent, least A051904.
%Y A381439 A055396 gives least prime index, greatest A061395.
%Y A381439 A056239 adds up prime indices, row sums of A112798.
%Y A381439 A122111 represents conjugation in terms of Heinz numbers.
%Y A381439 A239455 counts Look-and-Say partitions, complement A351293.
%Y A381439 A381436 gives section-sum partition of prime indices, Heinz number A381431.
%Y A381439 A381438 counts partitions by last part part of section-sum partition.
%Y A381439 Cf. A000720, A001221, A001222, A001694, A003557, A005117, A066328, A130091, A181819, A212166, A380955.
%K A381439 nonn
%O A381439 1,1
%A A381439 _Gus Wiseman_, Mar 02 2025