A355386 Position of first appearance of n in A355382, where A355382(m) = number of divisors d of m such that bigomega(d) = omega(m); or a(n) = -1 if n does not appear in A355382.

1, 12, 36, 120, 180, 360, 840, 1260, 5400, 27000, 2520, 5040, 6300, 7560, 15120, 12600, 25200

Offset

1,2

Comments

The first position of -1 appears to be 18, pointed out by Amiram Eldar.

The terms are not always increasing.

The statistic omega = A001221 counts distinct prime factors (without multiplicity).

The statistic bigomega = A001222 counts prime factors with multiplicity.

Links

Table of n, a(n) for n=1..17.

Example

The terms together with their prime indices begin:
      1: {}
     12: {1,1,2}
     36: {1,1,2,2}
    120: {1,1,1,2,3}
    180: {1,1,2,2,3}
    360: {1,1,1,2,2,3}
    840: {1,1,1,2,3,4}
   1260: {1,1,2,2,3,4}
   5400: {1,1,1,2,2,2,3,3}
  27000: {1,1,1,2,2,2,3,3,3}
   2520: {1,1,1,2,2,3,4}
   5040: {1,1,1,1,2,2,3,4}
   6300: {1,1,2,2,3,3,4}
   7560: {1,1,1,2,2,2,3,4}
  15120: {1,1,1,1,2,2,2,3,4}
The terms together with their divisors satisfying the condition begin:
      1:   1
     12:   4,   6
     36:   4,   6,   9
    120:   8,  12,  20,  30
    180:  12,  18,  20,  30,  45
    360:   8,  12,  18,  20,  30,  45
    840:  24,  40,  56,  60,  84, 140, 210
   1260:  36,  60,  84,  90, 126, 140, 210, 315
   5400:   8,  12,  18,  20,  27,  30,  45,  50,  75
  27000:   8,  12,  18,  20,  27,  30,  45,  50,  75, 125
   2520:  24,  36,  40,  56,  60,  84,  90, 126, 140, 210, 315
   5040:  16,  24,  36,  40,  56,  60,  84,  90, 126, 140, 210, 315
   6300:  36,  60,  84,  90, 100, 126, 140, 150, 210, 225, 315, 350, 525

Mathematica

tf=Table[Length[Select[Divisors[n], PrimeOmega[#]==PrimeNu[n]&]], {n, 1000}];
Table[Position[tf, n][[1, 1]], {n, Select[Union[tf], SubsetQ[tf, Range[#]]&]}]

Crossrefs

These are the positions of first appearances in A355382, which is the version of A181591 without multiplicity.

A000005 counts divisors.

A001221 counts prime indices without multiplicity.

A001222 counts prime indices with multiplicity.

A070175 gives representatives for bigomega and omega, triangle A303555.

A355383 counts cmpsbl. pairs of partitions with containment, comps. A355384.

Cf. A000712, A022811, A056239, A071625, A181819, A319910, A339006.

Sequence in context: A058880 A282097 A055551 * A073403 A191817 A270840

Adjacent sequences: A355383 A355384 A355385 * A355387 A355388 A355389

Keyword

nonn,more

Author

Gus Wiseman, Jul 02 2022

Status

approved