|
|
A096276
|
|
Number of partitions of n with a product <=n.
|
|
16
|
|
|
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
|
|
|
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
|
|
|
FORMULA
|
For n>1, a(n) = a(n-1)+1 iff n is prime.
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.
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:
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|