A325263 -id:A325263 - cp's OEIS frontend

A064839 List the natural numbers starting a new row only with each new least prime signature (A025487). a(n) is the column position associated with n.

1, 1, 2, 1, 3, 1, 4, 1, 2, 2, 5, 1, 6, 3, 4, 1, 7, 2, 8, 3, 5, 6, 9, 1, 3, 7, 2, 4, 10, 1, 11, 1, 8, 9, 10, 1, 12, 11, 12, 2, 13, 2, 14, 5, 6, 13, 15, 1, 4, 7, 14, 8, 16, 3, 15, 4, 16, 17, 17, 1, 18, 18, 9, 1, 19, 3, 19, 10, 20, 4, 20, 1, 21, 21, 11, 12, 22, 5, 22, 2, 2, 23, 23, 2, 24, 25, 26

Offset: 1

Alford Arnold, Oct 24 2001

Row 2 records the primes (A000040). Rows 3 and 4 record the semiprimes (A001358). Rows 5, 6 and 9 record the 3-almost primes (A014612) etc. A058933 is a similar sequence based on k-almost primes.
The graph of this sequence is interesting for large n because it shows multiple curves, one for each prime signature. For example, the six highest curves on the graph of a(n) for n up to 10^4 are for the (1,1), (1,1,1), (1), (2,1,1), (2,1), and (1,1,1,1) prime signatures. The (1) curve dominates until n=58; the (1,1) curve dominates until n=1279786, when the (1,1,1) curve intersects the (1,1) curve. Each (1,1,...,1) curve dominates for a finite number of n.
Ordinal transform of A101296. - Antti Karttunen, May 15 2017
a(n) is the number of positive integers up to n with the same prime signature as n. For example, the a(20) = 3 numbers are {12, 18, 20}. - Gus Wiseman, Jul 08 2019
Ordinal transform of A046523. - Alois P. Heinz, May 31 2020

			The list begins as follows:
1
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 ...
4 9 25 49 ...
6 10 14 15 21 22 26 33 34 35 38 39 46 51 ...
8 27 ...
12 18 20 28 44 45 50 52 ...
16 ...
Note: the above array, without the initial 1, is given by A095904 (and its transpose A179216). - _Antti Karttunen_, May 15 2017

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A014612, A025487, A046523, A058933, A101296.
Cf. also arrays A095904, A179216.
Cf. A001222, A085089, A118914, A124010, A325263, A325365, A326438, A326439.

p:= proc() 0 end:
a:= proc(n) option remember; local t; a(n-1);
      t:= (l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
           sort(map(i-> i[2], ifactors(n)[2]), `>`));
      p(t):= p(t)+1
    end: a(0):=0:
seq(a(n), n=1..100);  # Alois P. Heinz, May 31 2020

prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
Table[Count[Array[prisig,n],prisig[n]],{n,30}] (* Gus Wiseman, Jul 08 2019 *)

More terms from Naohiro Nomoto, Oct 31 2001

A325365 Number of maximal subsets of {1..n} containing n such that no two elements have the same sorted prime signature.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 4, 8, 4, 20, 4, 12, 12, 48, 8, 56, 16, 64, 48, 48, 36, 324, 162, 81, 567, 378, 168, 1680, 168, 1848, 264, 264, 264, 2640, 240, 288, 288, 3456, 576, 7488, 1152, 4032, 4032, 2016, 1872, 28080, 9360, 6240, 3360, 6720, 3584, 28672, 6144

Offset: 1

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(1) = 1 through a(12) = 20 subsets (A = 10, B = 11, C = 12) are the following. The common cardinality of sets in column n is A085089(n).
  1  12  13  124  145  1246  1467  12468  12689  1248A  1468B  12468C
             134       1346        13468  13689  1289A  148AB  1248AC
                       1456        14568  15689  1348A  1689B  12689C
                                   14678  16789  1389A  189AB  1289AC
                                                 1458A         13468C
                                                 1478A         1348AC
                                                 1589A         13689C
                                                 1789A         1389AC
                                                               14568C
                                                               1458AC
                                                               14678C
                                                               1468BC
                                                               1478AC
                                                               148ABC
                                                               15689C
                                                               1589AC
                                                               16789C
                                                               1689BC
                                                               1789AC
                                                               189ABC

Cf. A001221, A001222, A025487, A064839, A085089, A112798, A118914, A124010, A181819, A324762, A325263, A325365, A326438, A326441.

prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
Table[Times@@Length/@Split[Sort[Array[prisig,n]]]/Count[Array[prisig,n],prisig[n]],{n,30}]

a(n) = A326439(n)/A064839(n).

A326439 Number of maximal subsets of {1..n} such that no two elements have the same sorted prime signature.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 4, 8, 16, 20, 20, 24, 36, 48, 48, 56, 112, 128, 192, 240, 288, 324, 324, 486, 567, 1134, 1512, 1680, 1680, 1848, 1848, 2112, 2376, 2640, 2640, 2880, 3168, 3456, 6912, 7488, 14976, 16128, 20160, 24192, 26208, 28080, 28080, 37440, 43680

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(9) = 8 subsets:
  {}  {1}  {12}  {12}  {124}  {124}  {1246}  {1246}  {12468}  {12468}
                 {13}  {134}  {134}  {1346}  {1346}  {13468}  {12689}
                              {145}  {1456}  {1456}  {14568}  {13468}
                                             {1467}  {14678}  {13689}
                                                              {14568}
                                                              {14678}
                                                              {15689}
                                                              {16789}

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A001221, A001222, A025487, A064839, A085089, A112798, A118914, A124010, A181819, A324762, A325263, A325365, A326438, A326441.

prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
Table[Times@@(Length/@Split[Sort[Array[prisig,n]]]),{n,0,30}]

a(n)={if(n==0, 1, my(M=Map()); for(i=1, n, my(f=factor(i)[,2], s=sum(k=1, #f, x^f[k]), z); mapput(M, s, if(mapisdefined(M, s, &z), z + 1, 1))); vecprod(Mat(M)[,2]))} \\ Andrew Howroyd, Aug 30 2019

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

Original entry on oeis.org

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}]

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A064839 List the natural numbers starting a new row only with each new least prime signature (A025487). a(n) is the column position associated with n.

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A325365 Number of maximal subsets of {1..n} containing n such that no two elements have the same sorted prime signature.

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A326439 Number of maximal subsets of {1..n} such that no two elements have the same sorted prime signature.

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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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