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

1,1

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.

Links

Table of n, a(n) for n=1..66.

Formula

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

Example

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}

Mathematica

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

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.

Sequence in context: A186145 A364058 A335740 * A352873 A047984 A288513

Adjacent sequences: A381436 A381437 A381438 * A381440 A381441 A381442

Keyword

nonn,new

Author

Gus Wiseman, Mar 02 2025

Status

approved