A325337 -id:A325337 - cp's OEIS frontend

A130091 Numbers having in their canonical prime factorization mutually distinct exponents.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116

Offset: 1

Reinhard Zumkeller, May 06 2007

nonn

This sequence does not contain any number of the form 36n-6 or 36n+6, as such numbers are divisible by 6 but not by 4 or 9. Consequently, this sequence does not contain 24 consecutive integers. The quest for the greatest number of consecutive integers in this sequence has ties to the ABC conjecture (see the MathOverflow link). - Danny Rorabaugh, Sep 23 2015
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions with distinct multiplicities. The enumeration of these partitions by sum is given by A098859. - Gus Wiseman, May 04 2019
Aktaş and Ram Murty (2017) called these terms "special numbers" ("for lack of a better word"). They prove that the number of terms below x is ~ c*x/log(x), where c > 1 is a constant. - Amiram Eldar, Feb 25 2021
Sequence A005940(1+A328592(n)), n >= 1, sorted into ascending order. - Antti Karttunen, Apr 03 2022

			From _Gus Wiseman_, May 04 2019: (Start)
The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  23: {9}
  24: {1,1,1,2}
  25: {3,3}
  27: {2,2,2}
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Kevser Aktaş and M. Ram Murty, On the number of special numbers, Proceedings - Mathematical Sciences, Vol. 127, No. 3 (2017), pp. 423-430; alternative link.
MathOverflow, Consecutive numbers with mutually distinct exponents in their canonical prime factorization
Carlo Sanna, On the number of distinct exponents in the prime factorization of an integer, arXiv:1902.09224 [math.NT], 2019.
Eric Weisstein's World of Mathematics, Prime Factorization

Complement of A130092. A351564 is the characteristic function.
Subsequence of A351294.
Cf. A000961, A006939, A181818, A304686, A319161, A342028, A342029, A342030, A342031, A342032 (subsequences).
Cf. A005940, A048767, A048768, A056239, A098859, A112798, A118914, A181796, A217605, A325326, A325337, A325368, A327498, A327523, A328592, A336423, A336424, A336569, A336570, A336571, A343012, A343013.

filter:= proc(t) local f;
f:= map2(op,2,ifactors(t)[2]);
nops(f) = nops(convert(f,set));
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 30 2015

t[n_] := FactorInteger[n][[All, 2]]; Select[Range[400],  Union[t[#]] == Sort[t[#]] &]  (* Clark Kimberling, Mar 12 2015 *)

isok(n) = {nbf = omega(n); f = factor(n); for (i = 1, nbf, for (j = i+1, nbf, if (f[i, 2] == f[j, 2], return (0)););); return (1);} \\ Michel Marcus, Aug 18 2013

isA130091(n) = issquarefree(factorback(apply(e->prime(e), (factor(n)[, 2])))); \\ Antti Karttunen, Apr 03 2022

a(n) < A130092(n) for n<=150, a(n) > A130092(n) for n>150.

A048767 If n = Product (p_j^k_j) then a(n) = Product ( prime(k_j)^pi(p_j) ) where pi is A000720.

Original entry on oeis.org

1, 2, 4, 3, 8, 8, 16, 5, 9, 16, 32, 12, 64, 32, 32, 7, 128, 18, 256, 24, 64, 64, 512, 20, 27, 128, 25, 48, 1024, 64, 2048, 11, 128, 256, 128, 27, 4096, 512, 256, 40, 8192, 128, 16384, 96, 72, 1024, 32768, 28, 81, 54, 512, 192, 65536, 50, 256, 80, 1024, 2048

Offset: 1

Naohiro Nomoto

If the prime power factors p^e of n are replaced by prime(e)^pi(p), then the prime terms q in the sequence pertain to 2^m with m > 1, since pi(2) = 1. - Michael De Vlieger, Apr 25 2017

Also the Heinz number of the integer partition obtained by applying the map described in A217605 (which interchanges the parts with their multiplicities) to the integer partition with Heinz number n, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The image of this map (which is the union of this sequence) is A130091. - Gus Wiseman, May 04 2019

			For n=6, 6 = (2^1)*(3^1), a(6) = ([first prime]^pi(2))*([first prime]^pi(3)) = (2^1)*(2^2) = 8.
From _Gus Wiseman_, May 04 2019: (Start)
For n = 1..20, the prime indices of n together with the prime indices of a(n) are the following:
   1: {} {}
   2: {1} {1}
   3: {2} {1,1}
   4: {1,1} {2}
   5: {3} {1,1,1}
   6: {1,2} {1,1,1}
   7: {4} {1,1,1,1}
   8: {1,1,1} {3}
   9: {2,2} {2,2}
  10: {1,3} {1,1,1,1}
  11: {5} {1,1,1,1,1}
  12: {1,1,2} {1,1,2}
  13: {6} {1,1,1,1,1,1}
  14: {1,4} {1,1,1,1,1}
  15: {2,3} {1,1,1,1,1}
  16: {1,1,1,1} {4}
  17: {7} {1,1,1,1,1,1,1}
  18: {1,2,2} {1,2,2}
  19: {8} {1,1,1,1,1,1,1,1}
  20: {1,1,3} {1,1,1,2}
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Stephan Wagner, The Number of Fixed Points of Wilf's Partition Involution, The Electronic Journal of Combinatorics, 20(4) (2013), #P13.

Cf. A008477, A048768, A048769, A064988.

Cf. A056239, A098859, A109298, A112798, A118914, A130091, A217605, A320348, A325326, A325337.

A048767 := proc(n)
    local a,p,e,f;
    a := 1 ;
    for f in ifactors(n)[2] do
        p := op(1,f) ;
        e := op(2,f) ;
        a := a*ithprime(e)^numtheory[pi](p) ;
    end do:
    a ;
end proc: # R. J. Mathar, Nov 08 2012

Table[{p, k} = Transpose@ FactorInteger[n]; Times @@ (Prime[k]^PrimePi[p]), {n, 58}] (* Ivan Neretin, Jun 02 2016 *)
Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; e >= 0 :> Prime[e]^PrimePi[p]] &, 65] (* Michael De Vlieger, Apr 25 2017 *)

a(1)=1 prepended by Alois P. Heinz, Jul 26 2015

A320348 Number of partition into distinct parts (a_1, a_2, ... , a_m) (a_1 > a_2 > ... > a_m and Sum_{k=1..m} a_k = n) such that a1 - a2, a2 - a_3, ... , a_{m-1} - a_m, a_m are different.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 4, 4, 6, 9, 7, 13, 12, 13, 16, 22, 17, 28, 28, 31, 36, 50, 45, 63, 62, 74, 78, 102, 92, 123, 123, 146, 148, 191, 181, 228, 233, 280, 283, 348, 350, 420, 437, 518, 523, 616, 641, 727, 774, 884, 911, 1038, 1102, 1240, 1292, 1463, 1530, 1715, 1861, 2002

Offset: 1

Seiichi Manyama, Oct 11 2018

nonn

Also the number of integer partitions of n whose parts cover an initial interval of positive integers with distinct multiplicities. Also the number of integer partitions of n whose multiplicities cover an initial interval of positive integers and are distinct (see A048767 for a bijection). - Gus Wiseman, May 04 2019

			n = 9
[9]        *********  a_1 = 9.
           ooooooooo
------------------------------------
[8, 1]             *        a_2 = 1.
            *******o  a_1 - a_2 = 7.
            oooooooo
------------------------------------
[7, 2]            **        a_2 = 2.
             *****oo  a_1 - a_2 = 5.
             ooooooo
------------------------------------
[5, 4]          ****        a_2 = 4.
               *oooo  a_1 - a_2 = 1.
               ooooo
------------------------------------
a(9) = 4.
From _Gus Wiseman_, May 04 2019: (Start)
The a(1) = 1 through a(11) = 9 strict partitions with distinct differences (where the last part is taken to be 0) are the following (A = 10, B = 11). The Heinz numbers of these partitions are given by A325388.
  (1)  (2)  (3)  (4)   (5)   (6)   (7)   (8)   (9)   (A)    (B)
                 (31)  (32)  (51)  (43)  (53)  (54)  (64)   (65)
                       (41)        (52)  (62)  (72)  (73)   (74)
                                   (61)  (71)  (81)  (82)   (83)
                                                     (91)   (92)
                                                     (631)  (A1)
                                                            (632)
                                                            (641)
                                                            (731)
The a(1) = 1 through a(10) = 6 partitions covering an initial interval of positive integers with distinct multiplicities are the following. The Heinz numbers of these partitions are given by A325326.
  1  11  111  211   221    21111   2221     22211     22221      222211
              1111  2111   111111  22111    221111    2211111    322111
                    11111          211111   2111111   21111111   2221111
                                   1111111  11111111  111111111  22111111
                                                                 211111111
                                                                 1111111111
The a(1) = 1 through a(10) = 6 partitions whose multiplicities cover an initial interval of positive integers and are distinct are the following (A = 10). The Heinz numbers of these partitions are given by A325337.
  (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)    (9)    (A)
                 (211)  (221)  (411)  (322)  (332)  (441)  (433)
                        (311)         (331)  (422)  (522)  (442)
                                      (511)  (611)  (711)  (622)
                                                           (811)
                                                           (322111)
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..500 (terms 1..100 from Seiichi Manyama)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000009, A320347.

Cf. A007294, A007862, A048767, A098859, A179269, A320509, A320510, A325324, A325325, A325349, A325367, A325404, A325468.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Differences[Append[#,0]]&]],{n,30}] (* Gus Wiseman, May 04 2019 *)

A317829 Number of set partitions of multiset {1, 2, 2, 3, 3, 3, ..., n X n}.

Original entry on oeis.org

1, 1, 4, 52, 2776, 695541, 927908528, 7303437156115, 371421772559819369, 132348505150329265211927, 355539706668772869353964510735, 7698296698535929906799439134946965681, 1428662247641961794158621629098030994429958386, 2405509035205023556420199819453960482395657232596725626

Offset: 0

Antti Karttunen, Aug 10 2018

nonn

Number of factorizations of the superprimorial A006939(n) into factors > 1. - Gus Wiseman, Aug 21 2020

			For n = 2 we have a multiset {1, 2, 2} which can be partitioned as {{1}, {2}, {2}} or {{1, 2}, {2}} or {{1}, {2, 2}} or {{1, 2, 2}}, thus a(2) = 4.

Index entries for sequences related to partitions

Cf. A033312, A317826.
Subsequence of A317828.
A000142 counts submultisets of the same multiset.
A022915 counts permutations of the same multiset.
A337069 is the strict case.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A076716 counts factorizations of factorials.
A076954 can be used instead of A006939 (cf. A307895, A325337).
A181818 lists products of superprimorials, with complement A336426.
Cf. A000178, A002110, A022559, A027423, A124010, A303279, A318284, A322583, A336417, A336496.

g:= proc(n, k) option remember; uses numtheory; `if`(n>k, 0, 1)+
     `if`(isprime(n), 0, add(`if`(d>k or max(factorset(n/d))>d, 0,
        g(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> g(mul(ithprime(i)^i, i=1..n)$2):
seq(a(n), n=0..5);  # Alois P. Heinz, Jul 26 2020

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[facs[chern[n]]],{n,3}] (* Gus Wiseman, Aug 21 2020 *)

\\ See A318284 for count.
a(n) = {if(n==0, 1, count(vector(n,i,i)))} \\ Andrew Howroyd, Aug 31 2020

a(n) = A317826(A033312(n+1)) = A317826((n+1)!-1) = A001055(A076954(n)).
a(n) = A001055(A006939(n)). - Gus Wiseman, Aug 21 2020
a(n) = A318284(A002110(n)). - Andrew Howroyd, Aug 31 2020

a(0)=1 prepended and a(7) added by Alois P. Heinz, Jul 26 2020

a(8)-a(13) from Andrew Howroyd, Aug 31 2020

A325326 Heinz numbers of integer partitions covering an initial interval of positive integers with distinct multiplicities.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 18, 24, 32, 48, 54, 64, 72, 96, 108, 128, 144, 162, 192, 256, 288, 324, 360, 384, 432, 486, 512, 540, 576, 600, 648, 720, 768, 864, 972, 1024, 1152, 1200, 1350, 1440, 1458, 1500, 1536, 1620, 1728, 1944, 2048, 2160, 2250, 2304, 2400, 2592

Offset: 1

Gus Wiseman, May 01 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A320348.

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     8: {1,1,1}
    12: {1,1,2}
    16: {1,1,1,1}
    18: {1,2,2}
    24: {1,1,1,2}
    32: {1,1,1,1,1}
    48: {1,1,1,1,2}
    54: {1,2,2,2}
    64: {1,1,1,1,1,1}
    72: {1,1,1,2,2}
    96: {1,1,1,1,1,2}
   108: {1,1,2,2,2}
   128: {1,1,1,1,1,1,1}
   144: {1,1,1,1,2,2}
   162: {1,2,2,2,2}
   192: {1,1,1,1,1,1,2}
   256: {1,1,1,1,1,1,1,1}
   288: {1,1,1,1,1,2,2}
   324: {1,1,2,2,2,2}
   360: {1,1,1,2,2,3}
   384: {1,1,1,1,1,1,1,2}

Cf. A000837, A047966, A055932, A056239, A098859, A112798, A130091, A317081, A317089, A320348, A325329, A325337, A325369, A325372.

normQ[n_Integer]:=n==1||PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]];
Select[Range[100],normQ[#]&&UnsameQ@@Last/@FactorInteger[#]&]

Intersection of normal numbers (A055932) and numbers with distinct prime exponents (A130091).

A337070 Number of strict chains of divisors starting with the superprimorial A006939(n).

Original entry on oeis.org

1, 2, 16, 1208, 1383936, 32718467072, 20166949856488576, 391322675415566237681536

Offset: 0

Gus Wiseman, Aug 15 2020

The n-th superprimorial is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

			The a(0) = 1 through a(2) = 16 chains:
  1  2    12
     2/1  12/1
          12/2
          12/3
          12/4
          12/6
          12/2/1
          12/3/1
          12/4/1
          12/4/2
          12/6/1
          12/6/2
          12/6/3
          12/4/2/1
          12/6/2/1
          12/6/3/1

A022915 is the maximal case.
A076954 can be used instead of A006939 (cf. A307895, A325337).
A336571 is the case with distinct prime multiplicities.
A336941 is the case ending with 1.
A337071 is the version for factorials.
A000005 counts divisors.
A000142 counts divisors of superprimorials.
A006939 lists superprimorials or Chernoff numbers.
A067824 counts chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A253249 counts chains of divisors.
A317829 counts factorizations of superprimorials.
Cf. A001055, A002033, A167865, A181818, A336420, A336423, A336942, A337074.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
chnsc[n_]:=If[n==1,{{1}},Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,Most[Divisors[n]]}],{n}]];
Table[Length[chnsc[chern[n]]],{n,0,3}]

a(n) = 2*A336941(n) for n > 0.

a(n) = A067824(A006939(n)).

A307895 Numbers whose prime exponents, starting from the largest prime factor through to the smallest, form an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 19, 20, 23, 28, 29, 31, 37, 41, 43, 44, 45, 47, 52, 53, 59, 61, 63, 67, 68, 71, 73, 76, 79, 83, 89, 92, 97, 99, 101, 103, 107, 109, 113, 116, 117, 124, 127, 131, 137, 139, 148, 149, 151, 153, 157, 163, 164, 167, 171, 172, 173

Offset: 1

Gus Wiseman, May 04 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose multiplicities, starting from the largest part through to the smallest, form an initial interval of positive integers. The enumeration of these partitions by sum is given by A179269.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   11: {5}
   12: {1,1,2}
   13: {6}
   17: {7}
   19: {8}
   20: {1,1,3}
   23: {9}
   28: {1,1,4}
   29: {10}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   44: {1,1,5}
   45: {2,2,3}

Cf. A055932, A056239, A098859, A109298, A112798, A130091, A179269, A317090, A325326, A325337, A325460.

Select[Range[100],Last/@If[#==1,{},FactorInteger[#]]==Range[PrimeNu[#],1,-1]&]

A325370 Numbers whose prime signature has multiplicities covering an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 101, 103, 104

Offset: 1

Gus Wiseman, May 02 2019

nonn

First differs from A319161 in lacking 420.
The prime signature (A118914) is the multiset of exponents appearing in a number's prime factorization.
Numbers whose prime signature covers an initial interval are given by A317090.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose multiplicities have multiplicities covering an initial interval of positive integers. The enumeration of these partitions by sum is given by A325330.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   12: {1,1,2}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
   23: {9}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
For example, the prime indices of 1890 are {1,2,2,2,3,4}, whose multiplicities give the prime signature {1,1,1,3}, and since this does not cover an initial interval (2 is missing), 1890 is not in the sequence.

Cf. A000009, A055932, A056239, A112798, A118914, A317081, A317089, A317090, A319161, A325326, A325330, A325337, A325369, A325371.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[100],normQ[Length/@Split[Sort[Last/@FactorInteger[#]]]]&]

A336942 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities) starting with the superprimorial A006939(n) and ending with 1.

Original entry on oeis.org

1, 1, 5, 95, 8823, 4952323, 20285515801, 714092378624317

Offset: 0

Gus Wiseman, Aug 14 2020

			The a(0) = 1 through a(2) = 5 chains:
  {1}  {2,1}  {12,1}
              {12,2,1}
              {12,3,1}
              {12,4,1}
              {12,4,2,1}

A076954 can be used instead of A006939 (cf. A307895, A325337).
A336423 and A336571 are not restricted to A006939.
A336941 is the version not restricted by A130091.
A337075 is the version for factorials.
A074206 counts chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
Cf. A000005, A001055, A002033, A032741, A067824, A124010, A167865, A336419, A336420, A336500, A336568.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
chnstr[n_]:=If[n==1,1,Sum[chnstr[d],{d,Select[Most[Divisors[n]],UnsameQ@@Last/@FactorInteger[#]&]}]];
Table[chnstr[chern[n]],{n,0,3}]

a(n) = A336423(A006939(n)) = A336571(A006939(n)).

A325329 Number of integer partitions of n whose multiplicities appear with distinct multiplicities.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 7, 13, 18, 25, 30, 52, 57, 81, 109, 140, 167, 230, 267, 354, 428, 532, 630, 815, 942, 1166, 1385, 1695, 1966, 2440, 2810, 3422, 4008, 4828, 5630, 6847, 7905, 9527, 11135, 13340, 15498, 18636, 21591, 25769, 30086, 35630, 41379, 49150, 56880

Offset: 0

Gus Wiseman, May 01 2019

nonn

The Heinz numbers of these partitions are given by A325369.

Partitions whose parts appear with distinct multiplicities are counted by A098859, with Heinz numbers A130091.

			The a(0) = 1 through a(8) = 13 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
           (11)  (21)   (22)    (32)     (33)      (43)       (44)
                 (111)  (31)    (41)     (42)      (52)       (53)
                        (1111)  (11111)  (51)      (61)       (62)
                                         (222)     (421)      (71)
                                         (321)     (3211)     (431)
                                         (2211)    (1111111)  (521)
                                         (111111)             (2222)
                                                              (3221)
                                                              (3311)
                                                              (4211)
                                                              (32111)
                                                              (11111111)
For example, in (4,2,1,1), the multiplicities are 1 and 2, and 2 appears 1 time while 1 appears 2 times, so (4,2,1,1) is counted under a(8).

Cf. A098859, A130091, A317081, A320348, A325326, A325330, A325331, A325333, A325337, A325369.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[Sort[Length/@Split[#]]]&]],{n,0,30}]

cp's OEIS Frontend

A130091 Numbers having in their canonical prime factorization mutually distinct exponents.

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A048767 If n = Product (p_j^k_j) then a(n) = Product ( prime(k_j)^pi(p_j) ) where pi is A000720.

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A320348 Number of partition into distinct parts (a_1, a_2, ... , a_m) (a_1 > a_2 > ... > a_m and Sum_{k=1..m} a_k = n) such that a1 - a2, a2 - a_3, ... , a_{m-1} - a_m, a_m are different.

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A317829 Number of set partitions of multiset {1, 2, 2, 3, 3, 3, ..., n X n}.

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A325326 Heinz numbers of integer partitions covering an initial interval of positive integers with distinct multiplicities.

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A337070 Number of strict chains of divisors starting with the superprimorial A006939(n).

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A307895 Numbers whose prime exponents, starting from the largest prime factor through to the smallest, form an initial interval of positive integers.

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