A371167 Positive integers with more divisors (A000005) than distinct divisors of prime indices (A370820).

1, 2, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 34, 36, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 60, 62, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 82, 84, 85, 88, 90, 92, 93, 96, 98, 99, 100, 102, 104, 105, 108, 110

Offset

1,2

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Links

Table of n, a(n) for n=1..65.

Formula

A000005(a(n)) > A370820(a(n)).

Example

The prime indices of 814 are {1,5,12}, and there are 8 divisors (1,2,11,22,37,74,407,814) and 7 distinct divisors of prime indices (1,2,3,4,5,6,12), so 814 is in the sequence.
The prime indices of 1859 are {5,6,6}, and there are 6 divisors (1,11,13,143,169,1859) and 5 distinct divisors of prime indices (1,2,3,5,6), so 1859 is in the sequence.
The terms together with their prime indices begin:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    12: {1,1,2}
    14: {1,4}
    15: {2,3}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    21: {2,4}
    22: {1,5}
    24: {1,1,1,2}
    25: {3,3}
    27: {2,2,2}
    28: {1,1,4}
    30: {1,2,3}

Mathematica

Select[Range[100], Length[Divisors[#]]>Length[Union @@ Divisors/@PrimePi/@First/@If[#==1, {}, FactorInteger[#]]]&]

Crossrefs

For prime factors on the LHS we have A370348, counted by A371171.

The RHS is A370820, for prime factors instead of divisors A303975.

For (equal to) instead of (greater than) we get A371165, counted by A371172.

For (less than) instead of (greater than) we get A371166.

Other equalities: A319899, A370802 (A371130), A371128, A371177 (A371178).

Other inequalities: A371168 (A371173), A371169, A371170.

A000005 counts divisors.

A001221 counts distinct prime factors.

A027746 lists prime factors, A112798 indices, length A001222.

A239312 counts divisor-choosable partitions, ranks A368110.

A355731 counts choices of a divisor of each prime index, firsts A355732.

A370320 counts non-divisor-choosable partitions, ranks A355740.

A370814 counts divisor-choosable factorizations, complement A370813.

Cf. A000792, A003963, A355529, A355739, A355741, A368100, A370808, A371127.

Sequence in context: A085492 A338924 A065090 * A324560 A355822 A367226

Adjacent sequences: A371164 A371165 A371166 * A371168 A371169 A371170

Keyword

nonn

Author

Gus Wiseman, Mar 14 2024

Status

approved