A001462 -id:A001462 - cp's OEIS frontend

A117144 Partitions of n in which each part k occurs at least k times.

1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 9, 10, 12, 15, 16, 19, 21, 25, 28, 32, 34, 41, 46, 51, 55, 64, 70, 79, 86, 97, 106, 119, 129, 146, 159, 175, 190, 214, 232, 256, 277, 306, 334, 367, 394, 434, 472, 515, 556, 607, 654, 714, 770, 836, 901, 978, 1048, 1140, 1226, 1322

Offset: 0

Emeric Deutsch, Mar 06 2006

nonn

The Heinz numbers of these integer partitions are given by A324525. - Gus Wiseman, Mar 09 2019

			a(9)=5 because we have [3,3,3], [2,2,2,2,1], [2,2,2,1,1,1], [2,2,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1].
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(1) = 1 through a(9) = 5 integer partitions:
  1  11  111  22    221    222     2221     2222      333
              1111  11111  2211    22111    22211     22221
                           111111  1111111  221111    222111
                                            11111111  2211111
                                                      111111111
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A001462, A003114, A006141, A033461, A039900, A047993, A052335, A064174, A090858, A114638, A115584, A276078, A280204.

Cf. A324518, A324520, A324525, A324572.

g:=product((1-x^k+x^(k^2))/(1-x^k),k=1..100): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..66);
# second Maple program:
b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +add(b(n-i*j, i-1), j=i..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Dec 28 2016

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-1], {j, i, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]>=i,{i,Union[#]}]&]],{n,0,30}] (* Gus Wiseman, Mar 09 2019 *)
nmax = 100; CoefficientList[Series[Product[(1-x^k+x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 28 2024 *)

G.f.: Product_{k>=1} (1-x^k+x^(k^2))/(1-x^k).

A324736 Number of subsets of {1...n} containing all prime indices of the elements.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 15, 22, 43, 79, 127, 175, 343, 511, 851, 1571, 3141, 4397, 8765, 13147, 25243, 46843, 76795, 115171, 230299, 454939, 758203, 1516363, 2916079, 4356079, 8676079, 12132079, 24264157, 45000157, 73800253, 145685053, 291369853, 437054653, 728424421

Offset: 0

Gus Wiseman, Mar 13 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Also the number of subsets of {1...n} containing no prime indices of the non-elements up to n.

			The a(0) = 1 through a(6) = 15 subsets:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {1}    {1}      {1}        {1}          {1}
           {1,2}  {1,2}    {1,2}      {1,2}        {1,2}
                  {1,2,3}  {1,4}      {1,4}        {1,4}
                           {1,2,3}    {1,2,3}      {1,2,3}
                           {1,2,4}    {1,2,4}      {1,2,4}
                           {1,2,3,4}  {1,2,3,4}    {1,2,6}
                                      {1,2,3,5}    {1,2,3,4}
                                      {1,2,3,4,5}  {1,2,3,5}
                                                   {1,2,3,6}
                                                   {1,2,4,6}
                                                   {1,2,3,4,5}
                                                   {1,2,3,4,6}
                                                   {1,2,3,5,6}
                                                   {1,2,3,4,5,6}
An example for n = 18 is {1,2,4,7,8,9,12,16,17,18}, whose elements have the following prime indices:
   1: {}
   2: {1}
   4: {1,1}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  12: {1,1,2}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
All of these prime indices {1,2,4,7} belong to the subset, as required.

Andrew Howroyd, Table of n, a(n) for n = 0..100

The strict integer partition version is A324748. The integer partition version is A324753. The Heinz number version is A290822. An infinite version is A324698.
Cf. A000720, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A279861, A290689, A304360, A320426.
Cf. A324697, A324737, A324741, A324743.

Table[Length[Select[Subsets[Range[n]],SubsetQ[#,PrimePi/@First/@Join@@FactorInteger/@DeleteCases[#,1]]&]],{n,0,10}]

pset(n)={my(b=0, f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n,k,pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(!bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 15 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 15 2019

A324572 Number of integer partitions of n whose multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are equal to the distinct parts in decreasing order.

Original entry on oeis.org

1, 1, 0, 0, 2, 0, 1, 0, 1, 1, 2, 0, 3, 0, 2, 0, 4, 1, 2, 1, 4, 1, 3, 1, 5, 3, 5, 1, 6, 2, 6, 1, 7, 2, 7, 2, 11, 4, 8, 3, 11, 5, 10, 4, 13, 5, 11, 5, 16, 8, 14, 5, 19, 8, 18, 6, 22, 8, 22, 7, 26, 10, 25, 8, 33, 12, 29, 11, 36, 13, 34, 12, 40, 16, 41, 14, 47, 17, 45, 16, 55

Offset: 0

Gus Wiseman, Mar 08 2019

nonn

These are a kind of self-describing partitions (cf. A001462, A304679).
The Heinz numbers of these partitions are given by A324571.
The case where the distinct parts are taken in increasing order is counted by A033461, with Heinz numbers given by A109298.

			The first 19 terms count the following integer partitions:
   1: (1)
   4: (22)
   4: (211)
   6: (3111)
   8: (41111)
   9: (333)
  10: (511111)
  10: (322111)
  12: (6111111)
  12: (4221111)
  12: (33222)
  14: (71111111)
  14: (52211111)
  16: (811111111)
  16: (622111111)
  16: (4444)
  16: (442222)
  17: (43331111)
  18: (9111111111)
  18: (7221111111)
  19: (533311111)

Cf. A001156, A033461, A109298, A117144, A276078, A324524, A324571.

Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

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

More terms from Alois P. Heinz, Mar 08 2019

A324704 Lexicographically earliest sequence containing 1 and all numbers > 2 divisible by prime(m) for some m already in the sequence.

Original entry on oeis.org

1, 4, 6, 7, 8, 10, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 26, 28, 29, 30, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 84

Offset: 1

Gus Wiseman, Mar 11 2019

nonn

A self-describing sequence, similar to A304360.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
   1: {}
   4: {1,1}
   6: {1,2}
   7: {4}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}

Complement of A324698.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324699, A324700, A324701, A324702, A324703, A324705.

aQ[n_]:=Switch[n,1,True,2,False,,!And@@Cases[FactorInteger[n],{p,k_}:>!aQ[PrimePi[p]]]];
Select[Range[100],aQ]

A332833 Number of compositions of n whose run-lengths are neither weakly increasing nor weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 8, 27, 75, 185, 441, 1025, 2276, 4985, 10753, 22863, 48142, 100583, 208663, 430563, 884407, 1809546, 3690632, 7506774, 15233198, 30851271, 62377004, 125934437, 253936064, 511491634, 1029318958, 2069728850, 4158873540, 8351730223, 16762945432

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(6) = 3 and a(7) = 8 compositions:
  (1221)   (2113)
  (2112)   (3112)
  (11211)  (11311)
           (12112)
           (21112)
           (21121)
           (111211)
           (112111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Unimodal Sequence.

The case of partitions is A332641.
The version for unsorted prime signature is A332831.
The version for the compositions themselves (not run-lengths) is A332834.
The complement is counted by A332835.
Unimodal compositions are A001523.
Partitions with weakly increasing run-lengths are A100883.
Compositions that are not unimodal are A115981.
Compositions with equal run-lengths are A329738.
Compositions whose run-lengths are unimodal are A332726.
Compositions whose run-lengths are not unimodal are A332727.
Partitions with weakly increasing or weakly decreasing run-lengths: A332745.
Compositions with weakly increasing run-lengths are A332836.
Compositions that are neither unimodal nor is their negation are A332870.
Cf. A001462, A072704, A072706, A107429, A181819, A329398, A329744, A329746, A329766, A332273, A332640, A332746.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,10}]

a(n) = 2^(n - 1) - 2 * A332836(n) + A329738(n).

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

A332835 Number of compositions of n whose run-lengths are either weakly increasing or weakly decreasing.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 29, 56, 101, 181, 327, 583, 1023, 1820, 3207, 5631, 9905, 17394, 30489, 53481, 93725, 164169, 287606, 503672, 881834, 1544018, 2703161, 4731860, 8283291, 14499392, 25379278, 44422866, 77754798, 136093756, 238204369, 416923752, 729728031

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(6) = 29 compositions:
  (6)    (141)  (213)   (1113)  (21111)
  (51)   (114)  (132)   (222)   (12111)
  (15)   (33)   (123)   (2211)  (11121)
  (42)   (321)  (3111)  (2121)  (11112)
  (24)   (312)  (1311)  (1212)  (111111)
  (411)  (231)  (1131)  (1122)
Missing are: (2112), (1221), (11211).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Unimodal Sequence

The version for the compositions themselves (not run-lengths) is A329398.
Compositions with equal run-lengths are A329738.
The case of partitions is A332745.
The version for unsorted prime signature is the complement of A332831.
The complement is counted by A332833.
Unimodal compositions are A001523.
Partitions with weakly decreasing run-lengths are A100882.
Partitions with weakly increasing run-lengths are A100883.
Compositions that are not unimodal are A115981.
Compositions whose negation is unimodal are A332578.
Compositions whose run-lengths are unimodal are A332726.
Neither weakly increasing nor weakly decreasing compositions are A332834.
Compositions with weakly increasing run-lengths are A332836.
Compositions that are neither unimodal nor is their negation are A332870.
Cf. A001462, A181819, A329744, A329766, A332273, A332640, A332641, A332746.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,20}]

a(n) = 2 * A332836(n) - A329738(n).

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

A316774 a(n) = n for n < 2, a(n) = freq(a(n-1),n) + freq(a(n-2),n) for n >= 2, where freq(i,j) is the number of times i appears in [a(0),a(1),...,a(j-1)].

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 2, 4, 5, 3, 3, 6, 4, 4, 8, 5, 3, 6, 6, 6, 8, 6, 7, 6, 7, 8, 5, 6, 10, 8, 5, 8, 9, 6, 9, 10, 4, 7, 8, 9, 9, 8, 11, 8, 9, 13, 6, 10, 12, 4, 7, 10, 8, 13, 11, 4, 9, 13, 9, 10, 12, 7, 7, 12, 9, 11, 11, 8, 14, 11, 6, 15, 11, 7, 13, 11, 11, 16, 9, 10

Offset: 0

Peter Illig, Jul 12 2018

In other words, a(n) = (number of times a(n-1) has appeared) plus (number of times a(n-2) has appeared). - N. J. A. Sloane, Dec 13 2019
What is the asymptotic behavior of this sequence?
Does it contain every positive integer at least once?
Does it contain every positive integer at most finitely many times?
Additional comments from Peter Illig's "Puzzles" link below (Start):
Sometimes referred to as "The Devil's Sequence" (by me), due to the early presence of three consecutive 6's (and my inability to understand it). The next time a number occurs three times in a row isn't until a(355677).
If each n does appear only finitely many times, approximately how many times does it appear? (It seems to be close to 2n.)
What are the best possible upper/lower bounds on a(n)?
Let r(k) be the smallest n such that {0,1,2,...,k} is contained in {a(0),...,a(n)}. What is the asymptotic behavior of r(k)? (It seems to be close to k^2/2.)
(End)

			For n=4, a(n-1) = a(n-2) = 2, and 2 appears twice in the first 4 terms. So a(4) = 2 + 2 = 4.

Alois P. Heinz, Table of n, a(n) for n = 0..65536
"Horseshoe_Crab" Reddit User, Properties of a Strange, Rather Meta Sequence. [In case this link breaks, the main point of the discussion is to propose the sequence and suggest other initial values. - _N. J. A. Sloane_, Dec 13 2019]
Peter Illig, Problems. [No date, probably 2018]
Samuel B. Reid, Density plot of one billion terms
Rémy Sigrist, Density plot of the first 10000000 terms

Cf. A001462, A316973 (freq(n)), A316905 (when n appears), A316984 (when n last appears), A330439 (total number of times a(n) has appeared so far).
For records see A330330, A330331.
See A306246 and A329934 for similar sequences with different initial conditions.
A330332 considers the frequencies of the three previous terms.

b:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= `if`(n<2, n, b(a(n-1))+b(a(n-2)));
      b(t):= b(t)+1; t
    end:
seq(a(n), n=0..200);  # Alois P. Heinz, Jul 12 2018

a = prev = {0, 1};
Do[
AppendTo[prev, Count[a, prev[[1]]] + Count[a, prev[[2]]]];
AppendTo[a, prev[[3]]];
prev = prev[[2 ;;]] , {78}]
a (* Peter Illig, Jul 12 2018 *)

from itertools import islice
from collections import Counter
def agen():
    a = [0, 1]; c = Counter(a); yield from a
    while True:
        a = [a[-1], c[a[-1]] + c[a[-2]]]; c[a[-1]] += 1; yield a[-1]
print(list(islice(agen(), 80))) # Michael S. Branicky, Oct 13 2022

Definition clarified by N. J. A. Sloane, Dec 13 2019

A324698 Lexicographically earliest sequence containing 2 and all numbers > 1 whose prime indices already belong to the sequence.

Original entry on oeis.org

2, 3, 5, 9, 11, 15, 23, 25, 27, 31, 33, 45, 47, 55, 69, 75, 81, 83, 93, 97, 99, 103, 115, 121, 125, 127, 135, 137, 141, 155, 165, 197, 207, 211, 225, 235, 243, 249, 253, 257, 275, 279, 291, 297, 309, 341, 345, 347, 363, 375, 379, 381, 405, 411, 415, 419, 423

Offset: 1

Gus Wiseman, Mar 10 2019

nonn

A self-describing sequence, similar to A304360.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
   2: {1}
   3: {2}
   5: {3}
   9: {2,2}
  11: {5}
  15: {2,3}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  31: {11}
  33: {2,5}
  45: {2,2,3}
  47: {15}
  55: {3,5}
  69: {2,9}
  75: {2,3,3}
  81: {2,2,2,2}
  83: {23}
  93: {2,11}
  97: {25}
  99: {2,2,5}

Complement of A324704.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324699, A324700, A324701, A324702, A324703, A324705.

aQ[n_]:=Switch[n,1,False,2,True,,And@@Cases[FactorInteger[n],{p,k_}:>aQ[PrimePi[p]]]];
Select[Range[100],aQ]

A324741 Number of subsets of {1...n} containing no prime indices of the elements.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 19, 30, 54, 96, 156, 248, 440, 688, 1120, 1864, 3664, 5856, 11232, 16896, 31296, 53952, 91008, 137472, 270528, 516720, 863088, 1710816, 3173856, 4836672, 9329472, 14897376, 29788128, 52256448, 88429248, 166037184, 331648704, 497685888, 829449600

Offset: 0

Gus Wiseman, Mar 15 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(0) = 1 through a(6) = 19 subsets:
  {}  {}   {}   {}     {}     {}       {}
      {1}  {1}  {1}    {1}    {1}      {1}
           {2}  {2}    {2}    {2}      {2}
                {3}    {3}    {3}      {3}
                {1,3}  {4}    {4}      {4}
                       {1,3}  {5}      {5}
                       {2,4}  {1,3}    {6}
                       {3,4}  {1,5}    {1,3}
                              {2,4}    {1,5}
                              {2,5}    {2,4}
                              {3,4}    {2,5}
                              {4,5}    {3,4}
                              {2,4,5}  {3,6}
                                       {4,5}
                                       {4,6}
                                       {5,6}
                                       {2,4,5}
                                       {3,4,6}
                                       {4,5,6}
An example for n = 20 is {5,6,7,9,10,12,14,15,16,19,20}, with prime indices:
   5: {3}
   6: {1,2}
   7: {4}
   9: {2,2}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  19: {8}
  20: {1,1,3}
None of these prime indices {1,2,3,4,8} belong to the subset, as required.

Andrew Howroyd, Table of n, a(n) for n = 0..100

The maximal case is A324743. The strict integer partition version is A324751. The integer partition version is A324756. The Heinz number version is A324758. An infinite version is A304360.
Cf. A000720, A001462, A007097, A076078, A084422, A112798, A276625, A279861, A290689, A290822, A304360, A306844.
Cf. A324695, A324736, A324742.

Table[Length[Select[Subsets[Range[n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,0,10}]

pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n,k,pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(!bitand(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 16 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 16 2019

A324525 Numbers divisible by prime(k)^k for each prime index k.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 27, 32, 36, 54, 64, 72, 81, 108, 125, 128, 144, 162, 216, 243, 250, 256, 288, 324, 432, 486, 500, 512, 576, 625, 648, 729, 864, 972, 1000, 1024, 1125, 1152, 1250, 1296, 1458, 1728, 1944, 2000, 2048, 2187, 2250, 2304, 2401, 2500, 2592

Offset: 1

Gus Wiseman, Mar 08 2019

nonn

These are a kind of self-describing numbers (cf. A001462, A304679).
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The prime signature of a number is the multiset of multiplicities (or exponents) in its prime factorization.
Also Heinz numbers of integer partitions where the multiplicity of k is at least k (A117144). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins as follows. For example, 36 = prime(1) * prime(1) * prime(2) * prime(2) is a term because the prime multiplicities are {2,2}, which are greater than or equal to the prime indices {1,2}.
    1: {}
    2: {1}
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   18: {1,2,2}
   27: {2,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   54: {1,2,2,2}
   64: {1,1,1,1,1,1}
   72: {1,1,1,2,2}
   81: {2,2,2,2}
  108: {1,1,2,2,2}
  125: {3,3,3}
  128: {1,1,1,1,1,1,1}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001156, A033461, A056239, A062457, A112798, A117144, A118914 (prime signature), A124010, A181819, A276078, A304679.
Cf. A109298, A324524, A324571, A324572, A324587, A324588.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

q:= n-> andmap(i-> i[2]>=numtheory[pi](i[1]), ifactors(n)[2]):
select(q, [$1..3000])[];  # Alois P. Heinz, Mar 08 2019

Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k>=PrimePi[p]]&]
seq[max_] := Module[{ps = {2}, p, s = {1}, s1, s2, emax}, While[ps[[-1]]^Length[ps] < max, AppendTo[ps, NextPrime[ps[[-1]]]]]; Do[p = ps[[k]]; emax = Floor[Log[p, max]]; s1 = Join[{1}, p^Range[k, emax]]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= max &]; s = Union[s, s2], {k, 1, Length[ps]}]; s]; seq[3000] (* Amiram Eldar, Nov 23 2020 *)

Closed under multiplication.

Sum_{n>=1} 1/a(n) = Product_{k>=1} 1 + 1/(prime(k)^(k-1) * (prime(k)-1)) = 2.35782843100111139159... - Amiram Eldar, Nov 23 2020

cp's OEIS Frontend

A117144 Partitions of n in which each part k occurs at least k times.

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A324736 Number of subsets of {1...n} containing all prime indices of the elements.

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A324572 Number of integer partitions of n whose multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are equal to the distinct parts in decreasing order.

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A324704 Lexicographically earliest sequence containing 1 and all numbers > 2 divisible by prime(m) for some m already in the sequence.

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A332833 Number of compositions of n whose run-lengths are neither weakly increasing nor weakly decreasing.

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A332835 Number of compositions of n whose run-lengths are either weakly increasing or weakly decreasing.

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A316774 a(n) = n for n < 2, a(n) = freq(a(n-1),n) + freq(a(n-2),n) for n >= 2, where freq(i,j) is the number of times i appears in [a(0),a(1),...,a(j-1)].

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