A355391 Position of first appearance of n in A181591 = binomial(bigomega(n), omega(n)).

1, 4, 8, 16, 32, 24, 128, 256, 512, 48, 2048, 4096, 8192, 16384, 96, 65536, 131072, 262144, 524288, 240, 192, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 384, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 480, 768, 137438953472

Offset

1,2

Comments

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

The statistic bigomega = A001222 counts prime factors with multiplicity.

We have A181591(2^k) = k, so the sequence is fully defined. Positions meeting this maximum are A185024, complement A006987.

Links

Amiram Eldar, Table of n, a(n) for n = 1..168

Formula

binomial(bigomega(a(n)), omega(a(n))) = n.

Example

The terms together with their prime indices begin:
       1: {}
       4: {1,1}
       8: {1,1,1}
      16: {1,1,1,1}
      32: {1,1,1,1,1}
      24: {1,1,1,2}
     128: {1,1,1,1,1,1,1}
     256: {1,1,1,1,1,1,1,1}
     512: {1,1,1,1,1,1,1,1,1}
      48: {1,1,1,1,2}
    2048: {1,1,1,1,1,1,1,1,1,1,1}
    4096: {1,1,1,1,1,1,1,1,1,1,1,1}
    8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
   16384: {1,1,1,1,1,1,1,1,1,1,1,1,1,1}
      96: {1,1,1,1,1,2}
   65536: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  131072: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  262144: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  524288: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
     240: {1,1,1,1,2,3}
     192: {1,1,1,1,1,1,2}

Mathematica

s=Table[Binomial[PrimeOmega[n], PrimeNu[n]], {n, 1000}];
Table[Position[s, k][[1, 1]], {k, Select[Union[s], SubsetQ[s, Range[#]]&]}]

Prog

(PARI) f(n) = binomial(bigomega(n), omega(n)); \\ A181591
a(n) = my(k=1); while (f(k) != n, k++); k; \\ Michel Marcus, Jul 10 2022

Crossrefs

Positions of powers of 2 are A185024, complement A006987.

Counting multiplicity gives A355386.

The sorted version is A355392.

A000005 counts divisors.

A001221 counts prime factors without multiplicity.

A001222 count prime factors with multiplicity.

A070175 gives representatives for bigomega and omega, triangle A303555.

Cf. A022811, A056239 , A071625, A118914, A181819, A323014, A323023, A355383 (with multiplicity A339006), A355384.

Sequence in context: A298803 A320947 A329778 * A053163 A125626 A141031

Adjacent sequences: A355388 A355389 A355390 * A355392 A355393 A355394

Keyword

nonn

Author

Gus Wiseman, Jul 04 2022

Extensions

a(22)-a(28) from Michel Marcus, Jul 10 2022

a(29)-a(37) from Amiram Eldar, Jul 10 2022

Status

approved