A064839 -id:A064839 - cp's OEIS frontend

A326438 Number of subsets of {1..n} such that no two elements have the same sorted prime signature.

1, 2, 4, 6, 12, 16, 32, 40, 80, 120, 180, 216, 432, 504, 672, 840, 1680, 1920, 2880, 3240, 4320, 5184, 6048, 6720, 13440, 17920, 20480, 30720, 38400, 42240, 84480, 92160, 184320, 207360, 230400, 253440, 506880, 549120, 599040, 648960, 973440

Offset: 0

Gus Wiseman, Jul 06 2019

nonn

The sorted prime signature (A118914) of a positive integer is the multiset of exponents in its standard factorization into prime numbers.

			The a(0) = 1 through a(5) = 16 subsets:
  {}  {}   {}     {}     {}       {}
      {1}  {1}    {1}    {1}      {1}
           {2}    {2}    {2}      {2}
           {1,2}  {3}    {3}      {3}
                  {1,2}  {4}      {4}
                  {1,3}  {1,2}    {5}
                         {1,3}    {1,2}
                         {1,4}    {1,3}
                         {2,4}    {1,4}
                         {3,4}    {1,5}
                         {1,2,4}  {2,4}
                         {1,3,4}  {3,4}
                                  {4,5}
                                  {1,2,4}
                                  {1,3,4}
                                  {1,4,5}

Cf. A001221, A001222, A025487, A064839, A085089, A112798, A118914, A124010, A181819, A325263, A325365, A326439, A326441.

prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
Table[Length[Select[Subsets[Range[n]],UnsameQ@@prisig/@#&]],{n,0,10}]

A335286 n is the a(n)-th positive integer having its sequence of exponents in canonical prime factorization.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 2, 2, 5, 1, 6, 3, 4, 1, 7, 1, 8, 2, 5, 6, 9, 1, 3, 7, 2, 3, 10, 1, 11, 1, 8, 9, 10, 1, 12, 11, 12, 2, 13, 2, 14, 4, 5, 13, 15, 1, 4, 2, 14, 6, 16, 1, 15, 3, 16, 17, 17, 1, 18, 18, 7, 1, 19, 3, 19, 8, 20, 4, 20, 1, 21, 21, 3, 9, 22, 5, 22, 2

Offset: 1

David A. Corneth, May 30 2020

			a(14) = 3 as 14 has prime signature [1, 1] and it's the third positive integer having that prime signature, after 6 and 10.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A046523, A064839, A071364, A120426, A254524.

p:= proc() 0 end:
a:= proc(n) option remember; local t; a(n-1); t:=
      (l-> mul(ithprime(i)^l[i][2], i=1..nops(l)
       ))(sort(ifactors(n)[2])); p(t):= p(t)+1
    end: a(0):=0:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 01 2020

A071364[n_] := If[n == 1, 1, With[{f = FactorInteger[n]}, Times @@ (Prime[Range[Length[f]]]^f[[All, 2]])]];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A071364[n]}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 10 2022 *)

first(n) = { my(m = Map(), res = vector(n)); for(i = 1, n, c = factor(i)[,2]; if(mapisdefined(m, c), res[i] = mapget(m, c) + 1; mapput(m, c, res[i]) , res[i] = 1; mapput(m, c, 1) ) ); res }

Ordinal transform of A071364. - Alois P. Heinz, Jun 01 2020

A077559 Final terms of rows of A077558.

Original entry on oeis.org

1, 3, 7, 49, 17, 22, 29, 6859, 841, 38, 47, 76, 61, 57, 62, 7890481, 83, 147, 101, 164, 86, 91, 127, 472, 10609, 111, 1225043, 244, 163, 285, 179, 38579489651, 134, 142, 145, 14884, 223, 159, 166, 872, 241, 399, 263, 388, 412, 202, 283, 2512, 57121, 475, 214

Offset: 1

Amarnath Murthy, Nov 10 2002

nonn

Cf. A064839, A077558.

More terms from Ray Chandler, Jul 17 2003

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A326438 Number of subsets of {1..n} such that no two elements have the same sorted prime signature.

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A335286 n is the a(n)-th positive integer having its sequence of exponents in canonical prime factorization.

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A077559 Final terms of rows of A077558.

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