A304660 A run-length describing inverse to A181819. The multiplicity of prime(k) in a(n) is the k-th smallest prime index of n, which is A112798(n,k).

1, 2, 4, 6, 8, 18, 16, 30, 36, 54, 32, 150, 64, 162, 108, 210, 128, 450, 256, 750, 324, 486, 512, 1470, 216, 1458, 900, 3750, 1024, 2250, 2048, 2310, 972, 4374, 648, 7350, 4096, 13122, 2916, 10290, 8192, 11250, 16384, 18750, 4500, 39366, 32768, 25410, 1296

Offset

1,2

Comments

A permutation of A133808. a(n) is the smallest member m of A133808 such that A181819(m) = n.

Least integer whose prime signature is given by the partition with Heinz number n. - Andrew Howroyd, Oct 06 2025

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Formula

a(n) = Product_{i = 1..Omega(n)} prime(i)^A112798(n,i).

Example

Sequence of normalized prime multisets together with the normalized prime multisets of their images begins:
   1:        {} -> {}
   2:       {1} -> {1}
   3:       {2} -> {1,1}
   4:     {1,1} -> {1,2}
   5:       {3} -> {1,1,1}
   6:     {1,2} -> {1,2,2}
   7:       {4} -> {1,1,1,1}
   8:   {1,1,1} -> {1,2,3}
   9:     {2,2} -> {1,1,2,2}
  10:     {1,3} -> {1,2,2,2}
  11:       {5} -> {1,1,1,1,1}
  12:   {1,1,2} -> {1,2,3,3}
  13:       {6} -> {1,1,1,1,1,1}
  14:     {1,4} -> {1,2,2,2,2}
  15:     {2,3} -> {1,1,2,2,2}
  16: {1,1,1,1} -> {1,2,3,4}
  17:       {7} -> {1,1,1,1,1,1,1}
  18:   {1,2,2} -> {1,2,2,3,3}

Mathematica

Table[With[{y=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]}, Times@@Power[Array[Prime, Length[y]], y]], {n, 100}]

Prog

(PARI)
C(sig)={my(v=vecsort(sig, , 4)); prod(k=1, #sig, prime(k)^sig[k])}
a(n)={my(f=factor(n)); C(if(n==1, [], concat(vector(#f~, i, primepi(f[i, 1]) * vector(f[i, 2], j, 1)))))} \\ Andrew Howroyd, Oct 06 2025

Crossrefs

Cf. A055932, A056239, A112798, A130091, A133808, A181819, A181821, A182850, A182857, A275870, A304455.

Sequence in context: A100055 A393277 A334031 * A344902 A104001 A048784

Adjacent sequences: A304657 A304658 A304659 * A304661 A304662 A304663

Keyword

nonn

Author

Gus Wiseman, May 16 2018

Status

approved