A318871 -id:A318871 - cp's OEIS frontend

A318953 Maximum Heinz number of a strict factorization of n into factors > 1.

1, 3, 5, 7, 11, 15, 17, 21, 23, 33, 31, 39, 41, 51, 55, 57, 59, 69, 67, 87, 85, 93, 83, 111, 97, 123, 115, 129, 109, 165, 127, 159, 155, 177, 187, 195, 157, 201, 205, 231, 179, 255, 191, 237, 253, 249, 211, 285, 227, 319, 295, 303, 241, 345, 341, 357, 335, 327

Offset: 1

Gus Wiseman, Sep 05 2018

nonn

The Heinz number of a factorization (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The strict factorizations of 80 are (2*4*10), (2*5*8), (2*40), (4*20), (5*16), (8*10), (80), with Heinz numbers 609, 627, 519, 497, 583, 551, 409 respectively, so a(80) = 627.

Cf. A001055, A007716, A045778, A056239, A080688, A162247, A215366, A246868, A318871, A318954.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Max[Times@@Prime/@#&/@Select[facs[n],UnsameQ@@#&]],{n,100}]

A064554 a(n) = Min {k | A064553(k) = n}.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 13, 8, 9, 14, 29, 12, 37, 26, 21, 16, 53, 18, 61, 28, 39, 58, 79, 24, 49, 74, 27, 52, 107, 42, 113, 32, 87, 106, 91, 36, 151, 122, 111, 56, 173, 78, 181, 116, 63, 158, 199, 48, 169, 98, 159, 148, 239, 54, 203, 104, 183, 214, 271, 84, 281, 226, 117, 64

Offset: 1

Reinhard Zumkeller, Sep 21 2001

nonn

A064553(a(n)) = n and A064553(a(k)) <> k for k < a(n). For prime p, a(p)=prime(p-1), which is sequence A055003. - T. D. Noe, Dec 12 2004
a(n) is not multiplicative because a(7*13) = a(91) = 463, but a(7)*a(13) = 13*37 = 481 and 91 is the smallest possible such n. - Christian G. Bower, May 19 2005
a(n) = A080688(n,1). - Reinhard Zumkeller, Oct 01 2012
Minimal shifted Heinz number of a factorization of n, where the shifted Heinz number of a factorization (y_1, ..., y_k) is prime(y_1 - 1) * ... * prime(y_k - 1). - Gus Wiseman, Sep 05 2018

T. D. Noe, Table of n, a(n) for n = 1..8000
T. D. Noe, Plot of A064554

Cf. A064555, A001055, A064553.
Cf. A055003 (prime(prime(n)-1)).
Cf. A007716, A056239, A162247, A215366, A246868, A318871.

a064554 = head . a080688_row  -- Reinhard Zumkeller, Oct 01 2012

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Min[Times@@Prime/@(#-1)&/@facs[n]],{n,100}] (* Gus Wiseman, Sep 05 2018 *)

A320664 Number of non-isomorphic multiset partitions of weight n with all parts of odd size.

Original entry on oeis.org

1, 1, 2, 6, 12, 30, 82, 198, 533, 1459, 4039, 11634, 34095, 102520, 316456, 995709, 3215552, 10591412, 35633438, 122499429, 428988392, 1532929060, 5579867442, 20677066725, 78027003260, 299413756170, 1168536196157, 4635420192861, 18678567555721, 76451691937279, 317625507668759

Offset: 0

Gus Wiseman, Oct 18 2018

nonn

Also the number of non-isomorphic multiset partitions of weight n in which each vertex appears an odd number of times.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 multiset partitions with all parts of odd size:
  {{1}}  {{1},{1}}  {{1,1,1}}      {{1},{1,1,1}}
         {{1},{2}}  {{1,2,2}}      {{1},{1,2,2}}
                    {{1,2,3}}      {{1},{2,2,2}}
                    {{1},{1},{1}}  {{1},{2,3,3}}
                    {{1},{2},{2}}  {{1},{2,3,4}}
                    {{1},{2},{3}}  {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A001055, A001222, A007716, A298118, A300300, A300301, A318871, A320663, A320665.

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A), sExp((A-subst(A,x,-x))/2)))} \\ Andrew Howroyd, Jan 17 2023

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
J(q, t, k, y)={1/prod(j=1, #q, my(s=q[j], g=gcd(s,t)); (1 + O(x*x^k) - y^(s/g)*x^(s*t/g))^g)}
K(q, t, k) = Vec(J(q,t,k,1)-J(q,t,k,-1), -k)/2
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 17 2023

Terms a(11) and beyond from Andrew Howroyd, Jan 16 2023

A378175 Triangle T(n,k) read by rows in which n-th row lists in increasing order all multiplicative partitions mu of n (with factors > 1) encoded as Product_{j in mu} prime(j); n>=1, 1<=k<=A001055(n).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 27, 23, 25, 29, 33, 31, 35, 37, 39, 45, 41, 43, 51, 47, 55, 49, 53, 57, 63, 81, 59, 61, 65, 69, 75, 67, 71, 77, 87, 99, 73, 85, 79, 93, 83, 89, 91, 95, 105, 111, 117, 135, 97, 121, 101, 123, 103, 115, 125, 107, 119, 129, 153

Offset: 1

Alois P. Heinz, Nov 18 2024

			The multiplicative partitions of n=8 are {[8], [4,2], [2,2,2]}, encodings give {prime(8), prime(4)*prime(2), prime(2)^3} = {19, 7*3, 3^3} => row 8 = [19, 21, 27].
For n=1 the empty partition [] gives the empty product 1.
Triangle T(n,k) begins:
   1 ;
   3 ;
   5 ;
   7,  9 ;
  11 ;
  13, 15 ;
  17 ;
  19, 21, 27 ;
  23, 25 ;
  29, 33 ;
  31 ;
  35, 37, 39, 45 ;
  41 ;
  43, 51 ;
  47, 55 ;
  49, 53, 57, 63, 81 ;
  59 ;
  ...

Alois P. Heinz, Rows n = 1..2047, flattened

Row sums give A378176.
Row lengths give A001055.
Column k=1 gives A318871.
Rightmost elements of rows give A064988.
Sorted terms give A005408.
Cf. A000040, A006450, A215366, A377852.

b:= proc(n) option remember; `if`(n=1, {1}, {seq(map(x-> x*
      ithprime(d), b(n/d))[], d=numtheory[divisors](n) minus {1})})
    end:
T:= n-> sort([b(n)[]])[]:
seq(T(n), n=1..28);

T(prime(n),1) = T(A000040(n),1) = A006450(n).

A318660 Remainder when A064988(n) is divided by n.

Original entry on oeis.org

0, 1, 2, 1, 1, 3, 3, 3, 7, 3, 9, 9, 2, 9, 10, 1, 8, 3, 10, 19, 1, 5, 14, 15, 21, 19, 17, 13, 22, 15, 3, 19, 23, 7, 12, 9, 9, 11, 10, 17, 15, 3, 19, 15, 5, 19, 23, 21, 44, 13, 40, 5, 29, 51, 11, 11, 50, 37, 41, 15, 39, 9, 47, 25, 61, 3, 63, 55, 1, 1, 69, 27, 2, 27, 5, 71, 65, 69, 6, 11, 58, 45, 16, 9, 54, 57, 23, 45, 16, 15, 60, 11, 77, 69

Offset: 1

Altug Alkan and Antti Karttunen, Sep 08 2018

nonn

Inspired by A064988 and a 'minimum' version of it (A318871).

a(n) = 0 only for n = 1. Numbers n such that a(n) = 1 are 2, 4, 5, 16, 21, 69, 70, 181, 265, 370, 1043, 3760, 4531, ...

			a(6) = prime(2)*prime(3) mod 6 = 15 mod 6 = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A064988, A076240, A318668, A318871.

Table[Mod[If[n == 1, 1, Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 1 :> Prime[p]^e]], n], {n, 94}] (* Michael De Vlieger, Sep 10 2018 *)

A318660(n) = { my(f = factor(n)); for (k=1, #f~, f[k, 1] = prime(f[k, 1]); ); (factorback(f)%n); }; \\ After code in A064988.

a(n) = A064988(n) mod n.

a(A000040(n)) = A076240(n).

A318954 Minimum shifted Heinz number of a strict factorization of n into factors > 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 6, 13, 10, 19, 14, 29, 15, 37, 26, 21, 34, 53, 33, 61, 35, 39, 58, 79, 30, 89, 74, 57, 65, 107, 42, 113, 85, 87, 106, 91, 66, 151, 122, 111, 70, 173, 78, 181, 145, 129, 158, 199, 102, 223, 161, 159, 185, 239, 114, 203, 130, 183, 214, 271, 105

Offset: 1

Gus Wiseman, Sep 05 2018

nonn

The shifted Heinz number of a factorization (y_1, ..., y_k) is prime(y_1 - 1) * ... * prime(y_k - 1).

			The strict factorizations of 60 are (2*3*10), (2*5*6), (2*30), (3*4*5), (3*20), (4*15), (5*12), (6*10), (60), with shifted Heinz numbers 138, 154, 218, 105, 201, 215, 217, 253, 277 respectively, so a(60) = 105.

Cf. A001055, A007716, A045778, A056239, A080688, A162247, A215366, A246868, A318871, A318953.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Min[Times@@Prime/@(#-1)&/@Select[facs[n],UnsameQ@@#&]],{n,100}]

A322075 Number of factorizations of n into nonprime squarefree numbers > 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Nov 25 2018

nonn

First term greater than 1 is a(210) = 4.

			The a(420) = 3 factorizations: (6*70), (10*42), (14*30).

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000469, A001055, A001221, A001222, A005117, A007716, A318871, A318953, A320655, A320656, A320658, A320659.

sqnopfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqnopfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],!PrimeQ[#]&&SquareFreeQ[#]&]}]]
Table[Length[sqnopfacs[n]],{n,100}]

A330225 Position of first appearance of n in A290103 = LCM of prime indices.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Gus Wiseman, Mar 26 2020

nonn

Appears to be the prime numbers (A000040) with 2 replaced by 1 and 37 replaced by 35.

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 version for product instead of lcm is A318871
The version for standard compositions is A333225.
The version for binary indices is A333492.
Let q(k) be the prime indices of k:
- The product of q(k) is A003963(k).
- The sum of q(k) is A056239(k).
- The terms of q(k) are row k of A112798.
- The GCD of q(k) is A289508(k).
- The LCM of q(k) is A290103(k).
- The LCM of q(k) + 1 is A328219(k).
Cf. A000837, A074761, A074971, A076078, A285572, A289509, A290104, A319333, A324837, A328451, A331579, A333226.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[If[n==1,1,LCM@@primeMS[n]],{n,100}];
Table[Position[q,i][[1,1]],{i,First[Split[Union[q],#1+1==#2&]]}]

A353699 Heinz numbers of integer partitions whose product equals their length.

Original entry on oeis.org

2, 6, 20, 36, 56, 176, 240, 416, 864, 1088, 1344, 2432, 3200, 5888, 8448, 14848, 23040, 31744, 35840, 39936, 75776, 167936, 208896, 331776, 352256, 450560, 516096, 770048, 802816, 933888, 1736704, 2457600, 3866624, 4259840, 4521984, 7995392, 12976128, 17563648

Offset: 1

Gus Wiseman, May 19 2022

nonn

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.

			The terms together with their prime indices begin:
      2: {1}
      6: {1,2}
     20: {1,1,3}
     36: {1,1,2,2}
     56: {1,1,1,4}
    176: {1,1,1,1,5}
    240: {1,1,1,1,2,3}
    416: {1,1,1,1,1,6}
    864: {1,1,1,1,1,2,2,2}
   1088: {1,1,1,1,1,1,7}
   1344: {1,1,1,1,1,1,2,4}
   2432: {1,1,1,1,1,1,1,8}
   3200: {1,1,1,1,1,1,1,3,3}
   5888: {1,1,1,1,1,1,1,1,9}
   8448: {1,1,1,1,1,1,1,1,2,5}
  14848: {1,1,1,1,1,1,1,1,1,10}
  23040: {1,1,1,1,1,1,1,1,1,2,2,3}
  31744: {1,1,1,1,1,1,1,1,1,1,11}
  35840: {1,1,1,1,1,1,1,1,1,1,3,4}
  39936: {1,1,1,1,1,1,1,1,1,1,2,6}
  75776: {1,1,1,1,1,1,1,1,1,1,1,12}

Length is A001222, counted by A008284, distinct A001221.
Product is A003963, counted by A339095, firsts A318871.
A similar sequence is A353503, counted by A353506.
These partitions are counted by A353698.
A005361 gives product of signature, firsts A353500 (sorted A085629).
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A353394 gives product of shadows of prime indices, firsts A353397.
Cf. A000720, A003586, A114640, A182850, A320325, A325131, A325755, A353399, A353507.

Select[Range[1000],Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]==PrimeOmega[#]&]

cp's OEIS Frontend

A318953 Maximum Heinz number of a strict factorization of n into factors > 1.

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Mathematica

A064554 a(n) = Min {k | A064553(k) = n}.

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Haskell

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A320664 Number of non-isomorphic multiset partitions of weight n with all parts of odd size.

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PARI

PARI

Extensions

A378175 Triangle T(n,k) read by rows in which n-th row lists in increasing order all multiplicative partitions mu of n (with factors > 1) encoded as Product_{j in mu} prime(j); n>=1, 1<=k<=A001055(n).

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Maple

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A318660 Remainder when A064988(n) is divided by n.

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PARI

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A318954 Minimum shifted Heinz number of a strict factorization of n into factors > 1.

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Mathematica

A322075 Number of factorizations of n into nonprime squarefree numbers > 1.

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Mathematica

A330225 Position of first appearance of n in A290103 = LCM of prime indices.

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