A296150 -id:A296150 - cp's OEIS frontend

A316888 Heinz numbers of aperiodic integer partitions into relatively prime parts whose reciprocal sum is 1.

2, 195, 3185, 6475, 10527, 16401, 20445, 20535, 21045, 25365, 46155, 164255, 171941, 218855, 228085, 267883, 312785, 333925, 333935, 335405, 343735, 355355, 414295, 442975, 474513, 527425, 549575, 607475, 633777, 691041, 711321, 722425, 753865, 804837, 822783

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A partition is aperiodic if its multiplicities are relatively prime.
Does not contain 29888089, which belongs to A316890 and is the Heinz number of a periodic partition.

			The partition (6,4,4,3) with Heinz number 3185 is aperiodic, has relatively prime parts, and 1/6 + 1/4 + 1/4 + 1/3 = 1, so 3185 belongs to the sequence.
The sequence of partitions whose Heinz numbers belong to the sequence begins: (1), (6,3,2), (6,4,4,3), (12,4,3,3), (10,5,5,2), (20,5,4,2), (15,10,3,2), (12,12,3,2), (18,9,3,2), (24,8,3,2), (42,7,3,2).

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A007916, A051908, A100953, A289509, A296150, A316855, A316856, A316857, A316888-A316904.

Select[Range[2,100000],And[GCD@@FactorInteger[#][[All,2]]==1,GCD@@PrimePi/@FactorInteger[#][[All,1]]==1,Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]==1]&]

A316904 Heinz numbers of aperiodic integer partitions into relatively prime parts whose reciprocal sum is an integer.

Original entry on oeis.org

2, 18, 72, 162, 195, 250, 288, 294, 390, 500, 588, 648, 780, 1125, 1152, 1176, 1458, 1560, 1755, 2000, 2250, 2352, 2592, 2646, 3120, 3185, 3510, 4000, 4500, 4608, 4704, 4802, 5292, 6240, 6370, 6475, 7020, 8450, 9000, 9408, 10125, 10368, 10527, 10584, 12480

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A partition is aperiodic if its multiplicities are relatively prime.

			The sequence of partitions whose Heinz numbers belong to this sequence begins: (1), (221), (22111), (22221), (632), (3331), (2211111), (4421), (6321), (33311), (44211), (2222111).

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A296150, A316854, A316855, A316856, A316857, A316888-A316904.

Select[Range[2,20000],And[GCD@@FactorInteger[#][[All,2]]==1,GCD@@PrimePi/@FactorInteger[#][[All,1]]==1,IntegerQ[Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]]]&]

A352491 n minus the Heinz number of the conjugate of the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, -1, 1, -3, 0, -9, 3, 0, -2, -21, 2, -51, -10, -3, 9, -111, 3, -237, 0, -15, -26, -489, 10, -2, -70, 2, -12, -995, 0, -2017, 21, -39, -158, -19, 15, -4059, -346, -105, 12, -8151, -18, -16341, -36, -5, -722, -32721, 26, -32, 5, -237, -108, -65483, 19, -53

Offset: 1

Gus Wiseman, Mar 20 2022

sign

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Problem: What is the image? In the nonnegative case it appears to start: 0, 1, 2, 3, 5, 7, 9, ...

			The partition (4,4,1,1) has Heinz number 196 and its conjugate (4,2,2,2) has Heinz number 189, so a(196) = 196 - 189 = 7.

Positions of zeros are A088902, counted by A000700.
A similar sequence is A175508.
Positions of nonzero terms are A352486, counted by A330644.
Positions of negative terms are A352487, counted by A000701.
Positions of nonnegative terms are A352488, counted by A046682.
Positions of nonpositive terms are A352489, counted by A046682.
Positions of positive terms are A352490, counted by A000701.
A000041 counts integer partitions, strict A000009.
A003963 is product of prime indices, conjugate A329382.
A008480 counts permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 is partition conjugation using Heinz numbers, parts A321649/A321650.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A173018 counts permutations by excedances, weak A123125.
A238744 is partition conjugate of prime signature, ranked by A238745.
Cf. A000720, A114324, A301987, A324850, A325037, A325038, A325040, A347450.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[n-Times@@Prime/@conj[primeMS[n]],{n,30}]

a(n) = n - A122111(n).

A371781 Numbers with biquanimous prime signature.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159

Offset: 1

Gus Wiseman, Apr 09 2024

nonn

First differs from A320911 in lacking 900.
First differs from A325259 in having 1 and lacking 120.
A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 (aerated) and ranked by A357976.
Also numbers n with a unitary divisor d|n having exactly half as many prime factors as n, counting multiplicity.

			The prime signature of 120 is (3,1,1), which is not biquanimous, so 120 is not in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

A number's prime signature is given by A124010.
For prime indices we have A357976, counted by A002219 aerated.
The complement for prime indices is A371731, counted by A371795, A006827.
The complement is A371782, counted by A371840.
Partitions of this type are counted by A371839.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A237258 aerated counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371783 counts k-quanimous partitions.
A371791 counts biquanimous sets, complement A371792.
Cf. A035470, A064914, A232466, A299702, A336137, A371737, A371796.
Subsequence of A028260.

biquanimous:= proc(L) local s,x,i,P; option remember;
  s:= convert(L,`+`); if s::odd then return false fi;
  P:= mul(1+x^i,i=L);
  coeff(P,x,s/2) > 0
end proc:
select(n -> biquanimous(ifactors(n)[2][..,2]), [$1..200]); # Robert Israel, Apr 22 2024

g[n_]:=Select[Divisors[n],GCD[#,n/#]==1&&PrimeOmega[#]==PrimeOmega[n/#]&];
Select[Range[100],g[#]!={}&]
(* second program: *)
q[n_] := Module[{e = FactorInteger[n][[;; , 2]], sum, x}, sum = Plus @@ e; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, e}], x][[1 + sum/2]] > 0]; q[1] = True; Select[Range[200], q] (* Amiram Eldar, Jul 24 2024 *)

A372442 (Greatest binary index of n) minus (greatest prime index of n).

Original entry on oeis.org

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

Offset: 2

Gus Wiseman, May 07 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

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.

For sum instead of maximum we have A372428, zeros A372427.
Positions of zeros are A372436.
For minimum instead of maximum we have A372437, zeros {}.
For length instead of maximum we have A372441, zeros A071814.
Positions of odd terms are A372588, even A372589.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A003963, A014499, A174090, A243055, A355536, A359495, A372429-A372432, A372589-A372591.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Max[bix[n]]-Max[prix[n]],{n,2,100}]

a(n) = A070939(n) - A061395(n) = A029837(n) - A061395(n) for n > 1.

A277905 Irregular table: Each row n (n >= 0) lists in ascending order all A018819(n) numbers k for which A048675(k) = n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 9, 12, 16, 10, 18, 24, 32, 15, 20, 27, 36, 48, 64, 30, 40, 54, 72, 96, 128, 7, 25, 45, 60, 80, 81, 108, 144, 192, 256, 14, 50, 90, 120, 160, 162, 216, 288, 384, 512, 21, 28, 75, 100, 135, 180, 240, 243, 320, 324, 432, 576, 768, 1024, 42, 56, 150, 200, 270, 360, 480, 486, 640, 648, 864, 1152, 1536, 2048, 35, 63, 84, 112, 125, 225, 300, 400

Offset: 1

Antti Karttunen, Nov 14 2016

Each row beginning with an odd number (rows with even index) is followed by a row of the same length, with the same terms, but multiplied by 2. See also comments in the Formula section of A018819.
Note that although the indexing of rows start from zero, the indexing of this sequence starts from 1, with a(1) = 1.
Also Heinz numbers of integer partitions whose binary rank is n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1). For example, row n = 6 is 15, 20, 27, 36, 48, 64, corresponding to the partitions (3,2), (3,1,1), (2,2,2), (2,2,1,1), (2,1,1,1,1), (1,1,1,1,1,1). - Gus Wiseman, May 25 2024
Also, row n lists in ascending order all A018819(n) numbers k for which A097248(k) = A019565(n). - Flávio V. Fernandes, Jul 19 2025

			The irregular table begins as:
  row terms
   0   1;
   1   2;
   2   3,  4;
   3   6,  8;
   4   5,  9,  12,  16;
   5  10, 18,  24,  32;
   6  15, 20,  27,  36,  48,  64;
   7  30, 40,  54,  72,  96, 128;
   8   7, 25,  45,  60,  80,  81, 108, 144, 192, 256;
   9  14, 50,  90, 120, 160, 162, 216, 288, 384, 512;
  10  21, 28,  75, 100, 135, 180, 240, 243, 320, 324, 432,  576,  768, 1024;
  11  42, 56, 150, 200, 270, 360, 480, 486, 640, 648, 864, 1152, 1536, 2048;
...

Antti Karttunen, Table of n, a(n) for n = 1..166
Index entries for sequences that are permutations of the natural numbers

Cf. A019565 (the left edge, the only terms that are squarefree).
Cf. A000079 (the trailing edge).
Cf. A048675, A260443, A277886, A277896, A277903, A277904.
Row lengths are A018819 (number of partitions of binary rank n).
A000009 counts strict partitions, ranks A005117.
A029837 stc_sum or A070939 bin_len, opposite A070940 binexp_lastpos_1.
A048675 gives binary rank of prime indices, distinct A087207.
A048793 lists binary indices, product A096111, reverse A272020.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, cf. A001222, A003963, A056239, A296150.
A372890 adds up binary ranks of partitions, strict A372888.
Cf. A000040, A000041, A001511, A005940, A118462, A215366, A225620, A246868.
Cf. A097248, A048675.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Select[Range[0,2^k],Total[2^(prix[#]-1)]==k&],{k,0,10}] (* Gus Wiseman, May 25 2024 *)

(definec (A277905 n) (A277905bi (A277903 n) (A277904 n)))
(define (A277905bi row col) (let outloop ((k (A019565 row)) (col col)) (if (zero? col) k (let inloop ((j (+ 1 k))) (if (= (A048675 j) row) (outloop j (- col 1)) (inloop (+ 1 j))))))) ;; Very slow implementation.
;; Implementation based on a naive recurrence:
(definec (A277905 n) (if (= 1 n) n (let ((maybe_next (A277896 (A277905 (- n 1))))) (if (not (zero? maybe_next)) maybe_next (A019565 (A277903 n))))))

a(1) = 1; for n > 1, if A277896(a(n-1)) > 0, then a(n) = A277896(a(n-1)), otherwise a(n) = A019565(A277903(n)). [A naive recurrence for a one-dimensional version.]
Other identities. For all n >= 1:
A048675(a(n)) = A277903(n).

A299925 Number of chains in Young's lattice from () to the partition with Heinz number n.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 8, 4, 12, 16, 16, 16, 32, 40, 44, 8, 64, 44, 128, 52, 136, 96, 256, 40, 88, 224, 88, 152, 512, 204, 1024, 16, 384, 512, 360, 136, 2048, 1152, 1024, 152, 4096, 744, 8192, 416, 496, 2560, 16384, 96, 720, 496, 2624, 1088, 32768, 360, 1216, 504

Offset: 1

Gus Wiseman, Feb 21 2018

nonn

a(n) is the number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions skew-partitions. A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(9) = 12 tableaux:
1 3   1 2
2 4   3 4
.
1 3   1 2   1 2   1 2   1 1
2 3   3 3   2 3   1 3   2 3
.
1 2   1 2   1 1   1 1
2 2   1 2   2 2   1 2
.
1 1
1 1
The a(9) = 12 chains of Heinz numbers:
1<9,
1<2<9, 1<3<9, 1<4<9, 1<6<9,
1<2<3<9, 1<2<4<9, 1<2<6<9, 1<3<6<9, 1<4<6<9,
1<2<3<6<9, 1<2<4<6<9.

Cf. A000085, A001222, A056239, A063834, A112798, A122111, A138178, A153452, A238690, A296150, A296188, A296561, A297388, A299202.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
hncQ[a_,b_]:=And@@GreaterEqual@@@Transpose[PadRight[{Reverse[primeMS[b]],Reverse[primeMS[a]]}]];
chns[x_,y_]:=chns[x,y]=Join[{{x,y}},Join@@Function[c,Append[#,y]&/@chns[x,c]]/@Select[Range[x+1,y-1],hncQ[x,#]&&hncQ[#,y]&]];
Table[Length[chns[1,n]],{n,30}]

A304464 Start with the normalized multiset of prime factors of n > 1. Given a multiset, take the multiset of its multiplicities. Repeat this until a multiset of size 1 is obtained. a(n) is the unique element of this multiset.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 3, 2, 2, 5, 2, 6, 2, 2, 4, 7, 2, 8, 2, 2, 2, 9, 2, 2, 2, 3, 2, 10, 3, 11, 5, 2, 2, 2, 2, 12, 2, 2, 2, 13, 3, 14, 2, 2, 2, 15, 2, 2, 2, 2, 2, 16, 2, 2, 2, 2, 2, 17, 2, 18, 2, 2, 6, 2, 3, 19, 2, 2, 3, 20, 2, 21, 2, 2, 2, 2, 3, 22, 2, 4, 2, 23

Offset: 1

Gus Wiseman, May 13 2018

nonn

a(1) = 0 by convention.

			Starting with the normalized multiset of prime factors of 360, we obtain {1,1,1,2,2,3} -> {1,2,3} -> {1,1,1} -> {3}, so a(360) = 3.

Cf. A000005, A001222, A001597, A005117, A007916, A055932, A056239, A112798, A181819, A182850, A182857, A275870, A296150, A303945, A304465.

Table[If[n===1,0,NestWhile[Sort[Length/@Split[#]]&,If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]],Length[#]>1&]//First],{n,100}]

a(prime(n)) = n.
a(p^n) = n where p is any prime number and n > 1.
a(product of n > 1 distinct primes) = n.

A306438 Number of non-crossing set partitions whose block sizes are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 2, 4, 1, 6, 1, 5, 5, 1, 1, 10, 1, 10, 6, 6, 1, 10, 3, 7, 5, 15, 1, 30, 1, 1, 7, 8, 7, 30, 1, 9, 8, 20, 1, 42, 1, 21, 21, 10, 1, 15, 4, 21, 9, 28, 1, 35, 8, 35, 10, 11, 1, 105, 1, 12, 28, 1, 9, 56, 1, 36, 11, 56, 1, 70, 1, 13, 28, 45, 9

Offset: 1

Gus Wiseman, Feb 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(18) = 10 non-crossing set partitions of type (2, 2, 1) are:
  {{1},{2,3},{4,5}}
  {{1},{2,5},{3,4}}
  {{1,2},{3},{4,5}}
  {{1,2},{3,4},{5}}
  {{1,2},{3,5},{4}}
  {{1,3},{2},{4,5}}
  {{1,4},{2,3},{5}}
  {{1,5},{2},{3,4}}
  {{1,5},{2,3},{4}}
  {{1,5},{2,4},{3}}
Missing from this list are the following crossing set partitions:
  {{1},{2,4},{3,5}}
  {{1,3},{2,4},{5}}
  {{1,3},{2,5},{4}}
  {{1,4},{2},{3,5}}
  {{1,4},{2,5},{3}}

Germain Kreweras, Sur les partitions non croisées d'un cycle, Discrete Math. 1 333-350 (1972).
Wikipedia, Noncrossing partition.

Other orderings are A125181 and A134264.

Cf. A000041, A000108, A000110, A000670, A001263, A001764, A008480, A056239, A112798, A124794, A306437.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,1,With[{y=primeMS[n]},Binomial[Total[y],Length[y]-1]*(Length[y]-1)!/Product[Count[y,i]!,{i,Max@@y}]]],{n,80}]

a(n) = falling(m, k - 1)/Product_i (y)_i! where m is the sum of parts (A056239(n)), k is the number of parts (A001222(n)), y is the integer partition with Heinz number n (row n of A296150), (y)_i is the number of i's in y, and falling(x, y) is the falling factorial x(x - 1)(x - 2) ... (x - y + 1) [Kreweras].

Equivalently, a(n) = falling(A056239(n), A001222(n) - 1)/A112624(n).

A316655 Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 5, 4, 12, 9, 12, 17, 33, 29, 44, 26, 90, 90, 261, 68, 168, 93, 766, 144, 197, 307, 575, 269, 2312, 428, 7068, 236, 625, 1017, 863, 954, 21965, 3409, 2342, 712

Offset: 1

Gus Wiseman, Jul 09 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches.

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

			Sequence of sets of trees begins:
1:
2: 1
3: (11)
4: (12)
5: (1(11)), (111)
6: (1(12)), (2(11)), (112)
7: (1(1(11))), (1(111)), ((11)(11)), (11(11)), (1111)
8: (1(23)), (2(13)), (3(12)), (123)
9: (1(1(22))), (1(2(12))), (1(122)), (2(1(12))), (2(2(11))), (2(112)), ((11)(22)), ((12)(12)), (11(22)), (12(12)), (22(11)), (1122)

Cf. A000081, A000311, A000669, A001678, A005804, A056239, A141268, A181821, A292504, A296150, A300660, A304660.

Cf. A316651, A316652, A316653, A316654, A316656.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
gro[m_]:=If[Length[m]==1,m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
Table[Length[gro[Flatten[MapIndexed[Table[#2,{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]],{n,20}]

a(prime(n)) = A000669(n).

a(2^n) = A000311(n).

a(37)-a(40) from Robert Price, Sep 13 2018

cp's OEIS Frontend

A316888 Heinz numbers of aperiodic integer partitions into relatively prime parts whose reciprocal sum is 1.

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A316904 Heinz numbers of aperiodic integer partitions into relatively prime parts whose reciprocal sum is an integer.

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A352491 n minus the Heinz number of the conjugate of the integer partition with Heinz number n.

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A371781 Numbers with biquanimous prime signature.

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A372442 (Greatest binary index of n) minus (greatest prime index of n).

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A277905 Irregular table: Each row n (n >= 0) lists in ascending order all A018819(n) numbers k for which A048675(k) = n.

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A299925 Number of chains in Young's lattice from () to the partition with Heinz number n.

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A304464 Start with the normalized multiset of prime factors of n > 1. Given a multiset, take the multiset of its multiplicities. Repeat this until a multiset of size 1 is obtained. a(n) is the unique element of this multiset.

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