A371177 Positive integers whose prime indices include all distinct divisors of all prime indices.

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 30, 32, 34, 36, 40, 42, 44, 48, 50, 54, 60, 62, 64, 66, 68, 72, 80, 82, 84, 88, 90, 96, 100, 102, 108, 110, 118, 120, 124, 126, 128, 132, 134, 136, 144, 150, 160, 162, 164, 166, 168, 170, 176, 180, 186, 192, 198, 200

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.

Also positive integers with as many distinct prime factors (A001221) as distinct divisors of prime indices (A370820).

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

A001221(a(n)) = A370820(a(n)).

Example

The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}

Maple

pindices:= proc(n) uses NumberTheory; map(pi, PrimeFactors(n)) end proc:
filter:= proc(n) local F;
  F:= pindices(n);
  andmap(t -> NumberTheory:-Divisors(t) subset F, F)
end proc:
select(filter, [1, seq(i, i=2..200, 2)]); # Robert Israel, Dec 26 2025

Mathematica

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

Crossrefs

The LHS is A001221, distinct case of A001222.

The RHS is A370820, for prime factors A303975.

For bigomega on the LHS we have A370802, counted by A371130.

For divisors on the LHS we have A371165, counted by A371172.

Partitions of this type are counted by A371178, strict A371128.

The complement is A371179, counted by A371132.

A000005 counts divisors.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length.

A305148 counts partitions without divisors, strict A303362, ranks A316476.

Cf. A000837, A003963, A239312, A285573, A355529, A370813, A371168.

Sequence in context: A049225 A378508 A351910 * A076450 A097379 A371288

Adjacent sequences: A371174 A371175 A371176 * A371178 A371179 A371180

Keyword

nonn

Author

Gus Wiseman, Mar 18 2024

Status

approved