A096276 Number of partitions of n with a product <=n.

0, 1, 2, 3, 5, 6, 8, 9, 12, 14, 16, 17, 21, 22, 24, 26, 31, 32, 36, 37, 41, 43, 45, 46, 53, 55, 57, 60, 64, 65, 70, 71, 78, 80, 82, 84, 93, 94, 96, 98, 105, 106, 111, 112, 116, 120, 122, 123, 135, 137, 141, 143, 147, 148, 155, 157, 164, 166, 168, 169, 180, 181, 183, 187

Offset

0,3

Comments

The Heinz numbers of these partitions are given by A325044. - Gus Wiseman, Mar 27 2019

References

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).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)

R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

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.

A. Oppenheim, On an Arithmetic Function, Journal of the London Mathematical Society, 1926, 10, Vol. s1-1, Iss. 4, 205-211.

A. Oppenheim, On an Arithmetic Function (II), Journal of the London Mathematical Society, 1927, 04, Vol. s1-2, Iss. 2, 123-130.

Csaba Sándor and Maciej Zakarczemny, Equal Sum and Product Problem III, arXiv:2405.11600 [math.NT], 2024.

Formula

For n>1, a(n) = a(n-1)+1 iff n is prime.

Partial sums of A001055. - Vladeta Jovovic, Jun 24 2004

a(n) ~ n * exp(2*sqrt(log(n))) / (2*sqrt(Pi) * (log(n))^(3/4)) [Oppenheim, 1927]. - Vaclav Kotesovec, May 23 2020

Example

a(6)=8 as we can have 6, 51, 411, 321, 3111, 2211, 21111, 111111, rejecting 42, 33 and 222.
From Gus Wiseman, Mar 27 2019: (Start)
The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (41)     (51)      (61)       (71)
             (111)  (31)    (221)    (321)     (511)      (611)
                    (211)   (311)    (411)     (3211)     (4211)
                    (1111)  (2111)   (2211)    (4111)     (5111)
                            (11111)  (3111)    (22111)    (22211)
                                     (21111)   (31111)    (32111)
                                     (111111)  (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

Maple

g:= proc(n, k) option remember; `if`(n>k, 0, 1)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d)),
          d=numtheory[divisors](n) minus {1, n}))
    end:
a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+g(n$2)) end:
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 26 2023

Mathematica

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]; Join[{0}, Accumulate[Array[a, 100]]] (* using program from A001055, T. D. Noe, Apr 11 2011 *)
Table[Length[Select[IntegerPartitions[n], Times@@#<=n&]], {n, 0, 20}] (* Gus Wiseman, Mar 27 2019 *)

Prog

(PARI) { bla(n, m, v, z)=v=concat(v, m); if(!n, x=prod(k=1, length(v), v[k]); if (x<=z, c++), for(i=1, min(m, n), bla(n-i, i, v, z))); }
q(n)=c=0; for(i=1, n, bla(n-i, i, [], n)); print1(c, ", ");
for(i=0, 40, q(i))

Crossrefs

Cf. A001055, A028422, A114324, A301987, A319000, A319005, A319916, A325044.

Sequence in context: A348195 A280744 A331072 * A371970 A239091 A272341

Adjacent sequences: A096273 A096274 A096275 * A096277 A096278 A096279

Keyword

nonn

Author

Jon Perry, Jun 23 2004

Extensions

More terms from Vladeta Jovovic, Jun 24 2004

Status

approved