A182857 -id:A182857 - cp's OEIS frontend

A182856 a(0) = 1; for n > 0, a(n) = smallest positive integer whose prime signature contains, for k = 1 to n, exactly one positive number appearing exactly k times.

1, 2, 60, 1801800, 11657093261814000, 7167827541370578634694420017740000, 291943326350524088652207164949980988754136887856059678357800000

Offset: 0

Matthew Vandermast, Jan 05 2011

a(n) = smallest number m such that A181819(m) = A006939(n).  a(n) belongs to A182855 iff n > 1.
Next term has 105 digits.
Smallest number k with A323022(k) = n, where A323022(m) is the number of distinct multiplicities in the prime signature of m. - Gus Wiseman, Jan 03 2019

			The canonical prime factorization of a(3) = 1801800 is 2^3*3^2*5^2*7*11*13. The prime signature of 1801800 is therefore (3,2,2,1,1,1). Note that (3,2,2,1,1,1) contains exactly one number that appears once (3), one number that appears twice (2), and one number that appears three times (1).

David A. Corneth, Table of n, a(n) for n = 0..12
Dario Alpern, Factorization using the Elliptic Curve Method

Cf. A006939, A181819, A182855.

Cf. A056239, A059404, A062770, A071625, A118914, A181821, A182857, A323022.

Table[Product[Times@@Prime[i*(i-1)/2+Ceiling[Range[i*(n-i)]/(n-i)]],{i,n-1}],{n,6}] (* Gus Wiseman, Jan 03 2019 *)

a(n) = if(n == 0, return(1)); my(f = matrix(binomial(n+1,2), 2)); f[, 1] = primes(#f~ )~; f[, 2] = Vecrev(concat(vector(n, i, vector(n+1-i, j, i))))~; factorback(f) \\ David A. Corneth, Jan 03 2019

Partial products of A113511.
log a(n) ~ (1/3) n^3 log n. [Charles R Greathouse IV, Jan 13 2012]
A001222(a(n)) = A000292(n). - Gus Wiseman, Jan 03 2019
a(0) = 1; a(n + 1) = A002110(binomial(n + 2, 2)) * a(n). - David A. Corneth, Jan 03 2019

A319157 Smallest Heinz number of a superperiodic integer partition requiring n steps in the reduction to a multiset of size 1 obtained by repeatedly taking the multiset of multiplicities.

Original entry on oeis.org

2, 3, 9, 441, 11865091329, 284788749974468882877009302517495014698593896453070311184452244729

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

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

An integer partition is superperiodic if either it consists of a single part equal to 1 or its parts have a common divisor > 1 and its multiset of multiplicities is itself superperiodic. For example, (8,8,6,6,4,4,4,4,2,2,2,2) has multiplicities (4,4,2,2) with multiplicities (2,2) with multiplicities (2) with multiplicities (1). The first four of these partitions are periodic and the last is (1), so (8,8,6,6,4,4,4,4,2,2,2,2) is superperiodic.

Cf. A001462, A001597, A056239, A072774, A181819, A182850, A182857, A304455, A304464, A317246, A317257, A319149, A319151.

Function[m,Times@@Prime/@m]/@NestList[Join@@Table[Table[2i,{Reverse[#][[i]]}],{i,Length[#]}]&,{1},4]

A354906 Position of first appearance of n in A354579 = Number of distinct run-lengths of standard compositions.

Original entry on oeis.org

0, 1, 11, 119, 5615, 251871

Offset: 0

Gus Wiseman, Jun 23 2022

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their corresponding compositions begin:
       0: ()
       1: (1)
      11: (2,1,1)
     119: (1,1,2,1,1,1)
    5615: (2,2,1,1,1,2,1,1,1,1)
  251871: (1,1,1,2,2,1,1,1,1,2,1,1,1,1,1)

The standard compositions used here are A066099, run-sums A353847/A353932.
The version for partitions is A006939, for run-sums A002110.
For run-sums instead of run-lengths we have A246534 (firsts in A353849).
For runs instead of run-lengths we have A351015 (firsts in A351014).
These are the positions of first appearances in A354579.
A005811 counts runs in binary expansion.
A333627 ranks the run-lengths of standard compositions.
A351596 ranks compositions with distinct run-lengths, counted by A329739.
A353744 ranks compositions with equal run-lengths, counted by A329738.
A353852 ranks compositions with distinct run-sums, counted by A353850.
A353853-A353859 are sequences pertaining to composition run-sum trajectory.
A353860 counts collapsible compositions.
Cf. A003242, A029837, A071625, A124767, A182857, A238279/A333755, A325278, A333381, A333489, A333629.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pd=Table[Length[Union[Length/@Split[stc[n]]]],{n,0,10000}];
Table[Position[pd,n][[1,1]]-1,{n,0,Max@@pd}]

A325256 Number of normal multisets of size n whose adjusted frequency depth is the maximum for multisets of that size.

Original entry on oeis.org

1, 1, 1, 2, 3, 10, 12, 12, 44, 128, 228, 422, 968, 1750, 420, 2100

Offset: 0

Gus Wiseman, Apr 18 2019

A multiset is normal if its union is an initial interval of positive integers.

The adjusted frequency depth of a multiset is 0 if the multiset is empty, and otherwise it is one plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the multiset {1,1,2,2,3} has adjusted frequency depth 5 because we have {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}. The enumeration of integer partitions by adjusted frequency depth is given by A325280. The adjusted frequency depth of the integer partition with Heinz number n is A323014(n).

			The a(1) = 1 through a(7) = 12 multisets:
  {1}  {12}  {112}  {1123}  {11123}  {111123}  {1112234}
             {122}  {1223}  {11223}  {111234}  {1112334}
                    {1233}  {11233}  {112345}  {1112344}
                            {11234}  {122223}  {1122234}
                            {12223}  {122234}  {1123334}
                            {12233}  {122345}  {1123444}
                            {12234}  {123333}  {1222334}
                            {12333}  {123334}  {1222344}
                            {12334}  {123345}  {1223334}
                            {12344}  {123444}  {1223444}
                                     {123445}  {1233344}
                                     {123455}  {1233444}

Cf. A011784, A181819, A182857, A225486, A323014, A323023, A325238, A325254, A325258, A325277, A325278, A325280, A325282, A325283.

nn=10;
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
mfdm=Table[Max@@fdadj/@allnorm[n],{n,0,nn}];
Table[Length[Select[allnorm[n],fdadj[#]==mfdm[[n+1]]&]],{n,0,nn}]

A334969 Heinz numbers of alternately strong integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83

Offset: 1

Gus Wiseman, Jun 09 2020

nonn

First differs from A304678 in lacking 450.
First differs from A316529 (the totally strong version) in having 150.
A sequence is alternately strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and, when reversed, are themselves an alternately strong sequence.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence does not contain 450, the Heinz number of (3,3,2,2,1), because, while the multiplicities are weakly decreasing, their reverse (1,2,2) does not have weakly decreasing multiplicities.

The co-strong version is A317257.
The case of reversed partitions is (also) A317257.
The total version is A316529.
These partitions are counted by A332339.
Totally co-strong partitions are counted by A332275.
Alternately co-strong compositions are counted by A332338.
Cf. A000041, A100883, A181819, A182850, A182857, A304660, A305563, A316496, A317256, A332292, A332340.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altstrQ[q_]:=Or[q=={},q=={1},And[GreaterEqual@@Length/@Split[q],altstrQ[Reverse[Length/@Split[q]]]]];
Select[Range[100],altstrQ[Reverse[primeMS[#]]]&]

cp's OEIS Frontend

A182856 a(0) = 1; for n > 0, a(n) = smallest positive integer whose prime signature contains, for k = 1 to n, exactly one positive number appearing exactly k times.

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A319157 Smallest Heinz number of a superperiodic integer partition requiring n steps in the reduction to a multiset of size 1 obtained by repeatedly taking the multiset of multiplicities.

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A354906 Position of first appearance of n in A354579 = Number of distinct run-lengths of standard compositions.

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A325256 Number of normal multisets of size n whose adjusted frequency depth is the maximum for multisets of that size.

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A334969 Heinz numbers of alternately strong integer partitions.

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