A325365 -id:A325365 - cp's OEIS frontend

A325404 Number of reversed integer partitions y of n such that the k-th differences of y are distinct for all k >= 0 and are disjoint from the i-th differences for i != k.

1, 1, 1, 1, 2, 3, 2, 4, 4, 4, 5, 7, 5, 11, 12, 11, 12, 20, 15, 24, 22, 27, 28, 37, 28, 45, 43, 48, 50, 66, 58, 79, 72, 84, 87, 112, 106, 135, 128, 158, 147, 186, 180, 218, 220, 265, 246, 304, 303, 354, 340, 412, 418, 471, 463, 538, 543, 642, 600, 711, 755

Offset: 0

Gus Wiseman, May 02 2019

nonn

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The Heinz numbers of these partitions are given by A325405.

			The a(1) = 1 through a(12) = 5 reversed partitions (A = 10, B = 11, C = 12):
  (1)  (2)  (3)  (4)   (5)   (6)   (7)   (8)   (9)   (A)   (B)    (C)
                 (13)  (14)  (15)  (16)  (17)  (18)  (19)  (29)   (39)
                       (23)        (25)  (26)  (27)  (28)  (38)   (57)
                                   (34)  (35)  (45)  (37)  (47)   (1B)
                                                     (46)  (56)   (2A)
                                                           (1A)
                                                           (146)

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A279945, A325325, A325349, A325353, A325354, A325365, A325368, A325391, A325393, A325405, A325406, A325468.

Table[Length[Select[Reverse/@IntegerPartitions[n],UnsameQ@@Join@@Table[Differences[#,k],{k,0,Length[#]}]&]],{n,0,30}]

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.

Original entry on oeis.org

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

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

A325263 Number of subsets of {1..n} containing n such that no two elements have the same sorted prime signature.

Original entry on oeis.org

1, 2, 2, 6, 4, 16, 8, 40, 40, 60, 36, 216, 72, 168, 168, 840, 240, 960, 360, 1080, 864, 864, 672, 6720, 4480, 2560, 10240, 7680, 3840, 42240, 7680, 92160, 23040, 23040, 23040, 253440, 42240, 49920, 49920, 324480, 74880, 524160, 112320, 336960, 336960, 181440

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(5) = 16 subsets:
  {1}  {2}    {3}    {4}      {5}      {6}        {7}
       {1,2}  {1,3}  {1,4}    {1,5}    {1,6}      {1,7}
                     {2,4}    {4,5}    {2,6}      {4,7}
                     {3,4}    {1,4,5}  {3,6}      {6,7}
                     {1,2,4}           {4,6}      {1,4,7}
                     {1,3,4}           {5,6}      {1,6,7}
                                       {1,2,6}    {4,6,7}
                                       {1,3,6}    {1,4,6,7}
                                       {1,4,6}
                                       {1,5,6}
                                       {2,4,6}
                                       {3,4,6}
                                       {4,5,6}
                                       {1,2,4,6}
                                       {1,3,4,6}
                                       {1,4,5,6}

Cf. A001221, A001222, A025487, A064839, A085089, A112798, A118914, A124010, A181819, A325365, A326438, A326439, A326441.

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

a(n) = A326438(n)/(1 + A064839(n)).

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}]

cp's OEIS Frontend

A325404 Number of reversed integer partitions y of n such that the k-th differences of y are distinct for all k >= 0 and are disjoint from the i-th differences for i != k.

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

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A325263 Number of subsets of {1..n} containing 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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