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

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89

Offset: 1

Author

Gus Wiseman, Mar 02 2025

Keywords

• nonn

Comments

First differs from A335740 in lacking 72, which has prime indices {1,1,1,2,2} and section-sum partition (3,3,1).

Also numbers whose section-sum partition of prime indices (A381436) ends with a number > 1.

Includes all squarefree numbers (A005117) except 2.

Examples

			The terms together with their prime indices begin:
     3: {2}        25: {3,3}        45: {2,2,3}
     5: {3}        26: {1,6}        46: {1,9}
     6: {1,2}      27: {2,2,2}      47: {15}
     7: {4}        29: {10}         49: {4,4}
     9: {2,2}      30: {1,2,3}      50: {1,3,3}
    10: {1,3}      31: {11}         51: {2,7}
    11: {5}        33: {2,5}        53: {16}
    13: {6}        34: {1,7}        54: {1,2,2,2}
    14: {1,4}      35: {3,4}        55: {3,5}
    15: {2,3}      36: {1,1,2,2}    57: {2,8}
    17: {7}        37: {12}         58: {1,10}
    18: {1,2,2}    38: {1,8}        59: {17}
    19: {8}        39: {2,6}        61: {18}
    21: {2,4}      41: {13}         62: {1,11}
    22: {1,5}      42: {1,2,4}      63: {2,2,4}
    23: {9}        43: {14}         65: {3,6}
		

Crossrefs

The LHS (exponent of 2) is A007814.

The complement is A360013 = 2*A360015 (if we prepend 1), counted by A241131 (shifted right and starting with 1 instead of 0).

The case of equality is A360014, inclusive A360015.

The RHS (greatest exponent of an odd prime factor) is A375669.

These are positions of terms > 1 in A381437.

Partitions of this type are counted by A381544.

A000040 lists the primes, differences A001223.

A051903 gives greatest prime exponent, least A051904.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A122111 represents conjugation in terms of Heinz numbers.

A239455 counts Look-and-Say partitions, complement A351293.

A381436 gives section-sum partition of prime indices, Heinz number A381431.

A381438 counts partitions by last part part of section-sum partition.

Cf. A000720, A001221, A001222, A001694, A003557, A005117, A066328, A130091, A181819, A212166, A380955.

Programs

Mathematica

Select[Range[100],FactorInteger[2*#][[1,2]]-1<=Max@@Last/@Rest[FactorInteger[2*#]]&]

Formula

Positive integers n such that A007814(n) <= A375669(n).