A380218 -id:A380218 - cp's OEIS frontend

A001055 The multiplicative partition function: number of ways of factoring n with all factors greater than 1 (a(1) = 1 by convention).

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 9, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 5, 2, 1, 11, 2, 2, 2, 7, 1, 11, 2, 4, 2, 2, 2, 19, 1, 4, 4, 9, 1, 5, 1

Offset: 1

N. J. A. Sloane

From David W. Wilson, Feb 28 2009: (Start)
By a factorization of n we mean a multiset of integers > 1 whose product is n.
For example, 6 is the product of 2 such multisets, {2, 3} and {6}, so a(6) = 2.
Similarly 8 is the product of 3 such multisets, {2, 2, 2}, {2, 4} and {8}, so a(8) = 3.
1 is the product of 1 such multiset, namely the empty multiset {}, whose product is by definition the multiplicative identity 1. Hence a(1) = 1. (End)
a(n) = # { k | A064553(k) = n }. - Reinhard Zumkeller, Sep 21 2001; Benoit Cloitre and N. J. A. Sloane, May 15 2002
Number of members of A025487 with n divisors. - Matthew Vandermast, Jul 12 2004
See sequence A162247 for a list of the factorizations of n and a program for generating the factorizations for any n. - T. D. Noe, Jun 28 2009
So a(n) gives the number of different prime signatures that can be found among the integers that have n divisors. - Michel Marcus, Nov 11 2015
For n > 0, also the number of integer partitions of n with product n, ranked by A301987. For example, the a(12) = 4 partitions are: (12), (6,2,1,1,1,1), (4,3,1,1,1,1,1), (3,2,2,1,1,1,1,1). See also A380218. In general, A379666(m,n) = a(n) for any m >= n. - Gus Wiseman, Feb 07 2025

			1: 1, a(1) = 1
2: 2, a(2) = 1
3: 3, a(3) = 1
4: 4 = 2*2, a(4) = 2
6: 6 = 2*3, a(6) = 2
8: 8 = 2*4 = 2*2*2, a(8) = 3
etc.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 292-295.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 198, exercise 9 (in the third edition 2015, p. 296, exercise 211).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
D. Beckwith, Problem 10669, Amer. Math. Monthly 105 (1998), p. 559.
R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.
R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28. [A second link to the same paper.]
Marc Chamberland, Colin Johnson, Alice Nadeau, and Bingxi Wu, Multiplicative Partitions, Electronic Journal of Combinatorics, 20(2) (2013), #P57.
S. R. Finch, Kalmar's composition constant, Jun 05 2003. [Cached copy, with permission of the author]
Shamik Ghosh, Counting number of factorizations of a natural number, arXiv:0811.3479 [cs.DM], 2008.
R. K. Guy and R. J. Nowakowski, Monthly unsolved problems, 1969-1995, Amer. Math. Monthly, 102 (1995), 921-926.
John F. Hughes and J. O. Shallit, On the Number of Multiplicative Partitions, American Mathematical Monthly 90(7) (1983), 468-471.
Cao Hui-Zhong and Ku Tung-Hsin, On the Enumeration Function of Multiplicative Partitions, Math. Balkanica, Vol. 4 (1990), Fasc. 3-4.
Florian Luca, Anirban Mukhopadhyay and Kotyada Srinivas, On the Oppenheim's "factorisatio numerorum" function, arXiv:0807.0986 [math.NT], 2008.
Pankaj Jyoti Mahanta, On the number of partitions of n whose product of the summands is at most n, arXiv:2010.07353 [math.CO], 2020.
Amarnath Murthy, Generalization of Partition Function (Introducing the Smarandache Factor Partition) [Broken link]
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4.
Paul Pollack, On the parity of the number of multiplicative partitions and related problems, Proc. Amer. Math. Soc. 140 (2012), 3793-3803.
Marko Riedel, Calculating the multiplicative partition function in Maple with the Polya Enumeration Theorem
Eric Weisstein's World of Mathematics, Unordered Factorization
Wikipedia, Multiplicative Partition Function
Index entries for "core" sequences

A045782 gives the range of a(n).
For records see A033833, A033834.
Cf. A002033, A045778 (strict version), A050322, A050336, A064553, A064554, A064555, A077565, A051731, A005171, A097296, A190938, A216599, A216600, A216601, A216602.
Row sums of A316439 (for n>1).
Cf. A096276 (partial sums).
The additive version is A000041 (integer partitions), strict A000009.
Row sums of A318950.
A002865 counts partitions into parts > 1.
A069016 counts distinct sums of factorizations.
A319000 counts partitions by product and sum, row sums A319916.
A379666 (transpose A380959) counts partitions by sum and product, without 1's A379668, strict A379671.
Cf. A003963, A057567, A057568 (strict A379733), A114324, A301987, A319005, A379734.

a001055 = (map last a066032_tabl !!) . (subtract 1)
-- Reinhard Zumkeller, Oct 01 2012

public class MultiPart {
    public static void main(String[] argV) {
        for (int i=1;i<=100;++i) System.out.println(1+getDivisors(2,i));
    }
    public static int getDivisors(int min,int n) {
        int total = 0;
        for (int i=min;i=i) { ++total; if (n/i>i) total+=getDivisors(i,n/i); }
        return total;
    }
} \\ Scott R. Shannon, Aug 21 2019

with(numtheory):
T := proc(n::integer, m::integer)
        local A, summe, d:
        if isprime(n) then
                if n <= m then
                        return 1;
                end if:
                return 0 ;
        end if:
        A := divisors(n) minus {n, 1}:
        for d in A do
                if d > m then
                        A := A minus {d}:
                end if:
        end do:
        summe := add(T(n/d,d),d=A) ;
        if n <=m then
                summe := summe + 1:
        end if:
        summe ;
end proc:
A001055 := n -> T(n, n):
[seq(A001055(n), n=1..100)]; # Reinhard Zumkeller and Ulrich Schimke (ulrschimke(AT)aol.com)

c[1, r_] := c[1, r]=1; c[n_, r_] := c[n, r] = Module[{ds, i}, ds = Select[Divisors[n], 1 < # <= r &]; Sum[c[n/ds[[i]], ds[[i]]], {i, 1, Length[ds]}]]; a[n_] := c[n, n]; a/@Range[100] (* c[n, r] is the number of factorizations of n with factors <= r. - Dean Hickerson, Oct 28 2002 *)
T[, 1] = T[1, ] = 1;
T[n_, m_] := T[n, m] = DivisorSum[n, Boole[1 < # <= m] * T[n/#, #]&];
a[n_] := T[n, n];
a /@ Range[100] (* Jean-François Alcover, Jan 03 2020 *)

/* factorizations of n with factors <= m (n,m positive integers) */
fcnt(n,m) = {local(s);s=0;if(n == 1,s=1,fordiv(n,d,if(d > 1 & d <= m,s=s+fcnt(n/d,d))));s}
A001055(n) = fcnt(n,n) \\ Michael B. Porter, Oct 29 2009

\\ code using Dirichlet g.f., based on Somos's code for A007896
{a(n) = my(A, v, w, m);
if(
n<1, 0,
\\ define unit vector v = [1, 0, 0, ...] of length n
v = vector(n, k, k==1);
for(k=2, n,
m = #digits(n, k) - 1;
\\ expand 1/(1-x)^k out far enough
A = (1 - x)^ -1 + x * O(x^m);
\\ w = zero vector of length n
w = vector(n);
\\ convert A to a vector
for(i=0, m, w[k^i] = polcoeff(A, i));
\\ build the answer
v = dirmul(v, w)
);
v[n]
)
};
\\ produce the sequence
vector(100,n,a(n)) \\ N. J. A. Sloane, May 26 2014

v=vector(100, k, k==1); for(n=2, #v, v+=dirmul(v, vector(#v, k, (k>1) && n^valuation(k,n)==k)) ); v \\ Max Alekseyev, Jul 16 2014

from sympy import divisors, isprime
def T(n, m):
    if isprime(n): return 1 if n<=m else 0
    A=filter(lambda d: d<=m, divisors(n)[1:-1])
    s=sum(T(n//d, d) for d in A)
    return s + 1 if n<=m else s
def a(n): return T(n, n)
print([a(n) for n in range(1, 106)]) # Indranil Ghosh, Aug 19 2017

The asymptotic behavior of this sequence was studied by Canfield, Erdős & Pomerance and Luca, Mukhopadhyay, & Srinivas. - Jonathan Vos Post, Jul 07 2008
Dirichlet g.f.: Product_{k>=2} 1/(1 - 1/k^s).
If n = p^k for a prime p, a(n) = partitions(k) = A000041(k).
Since the sequence a(n) is the right diagonal of A066032, the given recursive formula for A066032 applies (see Maple program). - Reinhard Zumkeller and Ulrich Schimke (ulrschimke(AT)aol.com)
a(A002110(n)) = A000110(n).
a(p^k*q^k) = A002774(k) if p and q are distinct primes. - R. J. Mathar, Jun 06 2024
a(n) = A028422(n) + 1. - Gus Wiseman, Feb 07 2025

Incorrect assertion about asymptotic behavior deleted by N. J. A. Sloane, Jun 08 2009

A379320 Number of integer partitions of n whose product is a multiple of n + 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 2, 3, 0, 14, 0, 7, 15, 53, 0, 81, 0, 110, 61, 32, 0, 562, 170, 62, 621, 560, 0, 1400, 0, 3387, 569, 199, 1515, 7734, 0, 339, 1486, 13374, 0, 11926, 0, 8033, 27164, 913, 0, 85326, 15947, 47588, 8294, 25430, 0, 174779, 39748, 169009

Offset: 0

Gus Wiseman, Jan 18 2025

nonn

Also the number of integer partitions of n containing 1 whose product is a multiple of n. Without requiring a 1 we get A057568.

			The a(5) = 1 through a(11) = 14 partitions:
  (3,2)  .  (4,2,1)    (3,3,2)    (5,4)      .  (8,3)
            (2,2,2,1)  (3,3,1,1)  (5,2,2)       (4,4,3)
                                  (5,2,1,1)     (6,3,2)
                                                (6,4,1)
                                                (4,3,2,2)
                                                (4,3,3,1)
                                                (6,2,2,1)
                                                (3,2,2,2,2)
                                                (3,3,2,2,1)
                                                (4,3,2,1,1)
                                                (6,2,1,1,1)
                                                (3,2,2,2,1,1)
                                                (4,3,1,1,1,1)
                                                (3,2,2,1,1,1,1)

For n instead of n+1 we have A057568 (strict A379733), ranks A326149.
These partitions are ranked by A380217 = A379319/2 = (even case of A326149)/2.
The case of equality is A380218, see also A028422 = A001055 - 1 (ranks A325041).
A000041 counts integer partitions, strict A000009.
A379666 counts partitions by sum and product.
A380219 counts partitions of n whose product is a proper multiple of n, ranks A380216.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568, ranks A326149
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A003963, A036844, A069016, A318950, A319000, A319916, A324851, A325042, A326150, A326152, A326156, A379671, A379678, A379734.

Table[Length[Select[IntegerPartitions[n],Divisible[Times@@#,n+1]&]],{n,0,30}]

a(n) = my(nb=0); forpart(p=n, if (!(vecprod(Vec(p)) % (n+1)), nb++)); nb; \\ Michel Marcus, Jan 21 2025

A380217 Numbers whose product of prime indices is a multiple of their sum of prime indices plus one.

Original entry on oeis.org

1, 15, 42, 54, 75, 77, 95, 99, 100, 132, 182, 195, 221, 234, 245, 253, 290, 312, 315, 329, 350, 357, 405, 420, 423, 437, 450, 459, 476, 494, 510, 540, 555, 559, 560, 612, 627, 665, 715, 720, 740, 798, 816, 833, 854, 855, 858, 893, 897, 899, 979, 1026, 1064

Offset: 1

Gus Wiseman, Jan 18 2025

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 sum and product of prime indices are A056239 and A003963 respectively.

			The prime indices of 75 are {2,3,3}, with product 18 and sum 8, and since 18 is a multiple of 8+1, 75 is in the sequence.
The terms together with their prime indices begin:
     1: {}
    15: {2,3}
    42: {1,2,4}
    54: {1,2,2,2}
    75: {2,3,3}
    77: {4,5}
    95: {3,8}
    99: {2,2,5}
   100: {1,1,3,3}
   132: {1,1,2,5}
   182: {1,4,6}
   195: {2,3,6}
   221: {6,7}
   234: {1,2,2,6}
   245: {3,4,4}

The case of equality is A325041, counted by A380218 = A028422 except n=3.
Without "plus one" we get A326149, counted by A057568, see A379733, A379734, A379735.
Double all terms to get A379319.
Partitions of this type are counted by A379320.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568, ranks A326149
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A069016, A301988, A318950, A319000, A324851, A379318, A379671, A379844, A379845.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Divisible[Times@@prix[#],1+Total[prix[#]]]&]

vpind(n)=my(v=List(), f=factor(n)); for(i=1, #f~, for(j=1, f[i, 2], listput(v, primepi(f[i, 1])))); Vec(v); \\ A112798
isok(k) = my(vind = vpind(k)); (vecprod(vind) % (vecsum(vind)+1)) == 0; \\ Michel Marcus, Jan 21 2025

a(n) = A379319(n)/2.

A380219 Number of integer partitions of n whose product is a proper multiple of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 3, 3, 3, 0, 18, 0, 9, 21, 75, 0, 109, 0, 146, 83, 43, 0, 730, 224, 82, 806, 722, 0, 1782, 0, 4254, 733, 258, 1923, 9558, 0, 435, 1875, 16395, 0, 14625, 0, 9857, 33053, 1150, 0, 102070, 19391, 57326, 10157, 30702, 0, 207699, 47925, 200645

Offset: 1

Gus Wiseman, Jan 21 2025

nonn

			The partition y = (4,3,3,2) has product 72, which is a multiple of 12, so y is counted under a(12).
The a(8) = 3 through a(14) = 9 partitions:
  (44)    (63)    (532)   .  (66)       .  (743)
  (422)   (333)   (541)      (543)         (752)
  (2222)  (3321)  (5221)     (642)         (761)
                             (831)         (7322)
                             (4332)        (7421)
                             (4431)        (72221)
                             (5322)        (73211)
                             (6222)        (74111)
                             (6321)        (722111)
                             (6411)
                             (33222)
                             (43221)
                             (43311)
                             (62211)
                             (322221)
                             (332211)
                             (432111)
                             (3222111)

The non-proper version is A057568, case of equality A001055.
The case of strict partitions is A379733 - 1.
The case of partitions without 1's is A379734 - 1.
These partitions are ranked by A380216.
A000041 counts integer partitions, strict A000009.
A379666 counts partitions by sum and product.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568 (strict A379733), ranks A326149, see A379319, A380217.
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A003963, A028422, A318950, A319000, A319916, A326152, A326156, A379320, A379671, A379735, A380218, A380221.

Table[Length[Select[IntegerPartitions[n],Divisible[Times@@#,n]&&UnsameQ[Times@@#,n]&]],{n,30}]

a(n) = my(nb=0); forpart(p=n, my(vp=vecprod(Vec(p))); if (!(vp%n) && (vp>n), nb++)); nb; \\ Michel Marcus, Jan 22 2025

a(n) = A057568(n) - A001055(n).

A380221 Number of strict integer partitions of n containing 1 whose product of parts is a multiple of n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 3, 3, 0, 4, 0, 9, 6, 4, 0, 22, 5, 6, 15, 28, 0, 54, 0, 49, 30, 14, 57, 134, 0, 22, 58, 219, 0, 242, 0, 180, 349, 44, 0, 722, 113, 369, 196, 404, 0, 994, 556, 1363, 338, 111, 0, 3016, 0, 150, 2569, 3150, 1485, 2815, 0

Offset: 1

Gus Wiseman, Jan 22 2025

nonn

Also the number of strict integer partitions of n - 1 not containing 1 whose product of parts is a multiple of n. These are strict integer factorizations of multiples of n summing to n - 1.

			The a(6) = 1 through a(16) = 3 partitions:
  (3,2,1) . . . (5,4,1) . (8,3,1)   . (7,6,1)   (9,5,1)     (8,4,3,1)
                          (6,3,2,1)   (7,4,2,1) (6,5,3,1)   (8,5,2,1)
                                                (5,4,3,2,1) (6,4,3,2,1)

Positions of 0 after 9 appear to be the prime numbers A000040.
The non-strict version is A379320 shifted right, ranks A380217 = A379319/2.
Not requiring 1 gives A379733.
For n instead of n+1 we have A379735 shifted left, non-strict A379734.
Partitions of this type are ranked by A379845.
The case of equality for non-strict partitions is A380218 shifted left.
A000041 counts integer partitions, strict A000009.
A379666 counts partitions by sum and product.
A380219 counts partitions of n whose product is a proper multiple of n, ranks A380216.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568, ranks A326149
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A003963, A069016, A319000, A319916, A325042, A326150, A326152, A326156, A379671, A379678.

Table[Length[Select[IntegerPartitions[n],MemberQ[#,1]&&UnsameQ@@#&&Divisible[Times@@#,n]&]],{n,30}]

A380411 Number of integer partitions of n such that the product of parts is greater than the sum of primes indexed by the parts.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 4, 8, 14, 23, 39, 58, 85, 121, 168, 228, 308, 404, 533, 691, 892, 1136, 1449, 1820, 2291, 2857, 3553, 4387, 5418, 6646, 8144, 9931, 12086, 14649, 17733, 21379, 25747, 30905, 37049, 44282, 52863, 62936, 74841, 88792, 105202, 124387

Offset: 0

Gus Wiseman, Jan 26 2025

nonn

			The partition y = (4,3,2) has product of parts 4*3*2 = 24 and sum of corresponding primes 7+5+3 = 15, so y is counted under a(9).
The a(0) = 1 through a(10) = 14 partitions:
  ()  .  .  .  .  .  .  (322)  (44)    (54)     (55)
                               (332)   (333)    (64)
                               (422)   (432)    (433)
                               (2222)  (522)    (442)
                                       (3222)   (532)
                                       (3321)   (622)
                                       (4221)   (3322)
                                       (22221)  (3331)
                                                (4222)
                                                (4321)
                                                (5221)
                                                (22222)
                                                (32221)
                                                (33211)

For parts instead of primes on the RHS we have A114324.
The version for divisibility instead of inequality is A330954.
The version for equality is A331383, ranks A331384.
These partitions are ranked by A380410.
A000040 lists the primes, differences A001223.
A000041 counts integer partitions, strict A000009.
A001414 gives sum of prime factors.
A003963 gives product of prime indices
A379666 counts partitions by sum and product.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568 (strict A379733), ranks A326149, see A379319, A380217.
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A175508, A318950, A319000, A319916, A326152, A379320, A379734, A380218, A380219, A380344.

Table[Length[Select[IntegerPartitions[n],Times@@#>Plus@@Prime/@#&]],{n,0,30}]

cp's OEIS Frontend

A001055 The multiplicative partition function: number of ways of factoring n with all factors greater than 1 (a(1) = 1 by convention).

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A379320 Number of integer partitions of n whose product is a multiple of n + 1.

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A380217 Numbers whose product of prime indices is a multiple of their sum of prime indices plus one.

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A380219 Number of integer partitions of n whose product is a proper multiple of n.

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A380221 Number of strict integer partitions of n containing 1 whose product of parts is a multiple of n.

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A380411 Number of integer partitions of n such that the product of parts is greater than the sum of primes indexed by the parts.

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A380343 Number of strict integer partitions of n whose product of parts is a multiple of n + 1.

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