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A096276 Number of partitions of n with a product <=n. 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

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

T. D. Noe, Table of n, a(n) for n = 0..1000

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

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.

FORMULA

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)

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: A280771 A280744 A331072 * A239091 A272341 A075725

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

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Last modified July 6 16:17 EDT 2020. Contains 335478 sequences. (Running on oeis4.)