

A083710


Number of partitions of n each of which has a summand which divides every summand in the partition.


9



1, 1, 2, 3, 5, 6, 11, 12, 20, 25, 37, 43, 70, 78, 114, 143, 196, 232, 330, 386, 530, 641, 836, 1003, 1340, 1581, 2037, 2461, 3127, 3719, 4746, 5605, 7038, 8394, 10376, 12327, 15272, 17978, 22024, 26095, 31730, 37339, 45333, 53175, 64100, 75340, 90138
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OFFSET

0,3


COMMENTS

Since the summand (part) which divides all the other summands is necessarily the smallest, an equivalent definition is: "Number of partitions of n such that smallest part divides every part."  Joerg Arndt, Jun 08 2009
The first few partitions that fail the criterion are 5=3+2, 7=5+2=4+3=3+2+2. So a(5) = A000041(5)  1 = 6, a(7) = A000041(7)  3 = 12.  Vladeta Jovovic, Jun 17 2003
Starting with offset 1 = inverse Mobius transform (A051731) of the partition numbers, A000041.  Gary W. Adamson, Jun 08 2009


REFERENCES

L. M. Chawla, M. O. Levan and J. E. Maxfield, On a restricted partition function and its tables, J. Natur. Sci. and Math., 12 (1972), 95101.


LINKS

Table of n, a(n) for n=0..46.


FORMULA

Equals left border of triangle A137587 starting (1, 2, 3, 5, 6, 11,...).  Gary W. Adamson, Jan 27 2008
Comment from Joerg Arndt, Jun 08 2009: Sequence has g.f. 1 + Sum_{n>=1} x^n/eta(x^n). The g.f. for partitions into parts that are a multiple of n is x^n/eta(x^n), now sum over n.
Gary W. Adamson's comment is equivalent to the formula a(n) = Sum_{dn} p(d1) where p(i) = number of partitions of i (A000041(i)). Hence A083710 has g.f. Sum_{d>=1} p(d1)*x^d/(1x^d),  N. J. A. Sloane, Jun 08 2009


MAPLE

with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l)0 do c := c+numbpart(l[i]1) od: RETURN(c): end: for j from 0 to 60 do printf(`%d, `, a(j)) od: # Zerinvary Lajos, Apr 14 2007


CROSSREFS

Cf. A083711, A018783, A137587.
Cf. A000041, A051731.  Gary W. Adamson, Jun 08 2009
Sequence in context: A033159 A199366 A318689 * A127524 A117086 A081026
Adjacent sequences: A083707 A083708 A083709 * A083711 A083712 A083713


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, Jun 16 2003


EXTENSIONS

More terms from Vladeta Jovovic, Jun 17 2003


STATUS

approved



