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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 (list; graph; refs; listen; history; text; internal format)
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), 95-101.

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_{d|n} p(d-1) where p(i) = number of partitions of i (A000041(i)). Hence A083710 has g.f. Sum_{d>=1} p(d-1)*x^d/(1-x^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: A240949 A033159 A199366 * 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

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Last modified May 28 16:24 EDT 2017. Contains 287241 sequences.