Search: id:A083710
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%I A083710
%S A083710 1,1,2,3,5,6,11,12,20,25,37,43,70,78,114,143,196,232,330,386,530,641,
%T A083710 836,1003,1340,1581,2037,2461,3127,3719,4746,5605,7038,8394,10376,12327,
%U A083710 15272,17978,22024,26095,31730,37339,45333,53175,64100,75340,90138
%N A083710 Number of partitions of n each of which has a summand which divides every 
               summand in the partition.
%C A083710 Comment from Joerg Arndt, Jun 08 2009: 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."
%C A083710 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 (vladeta(AT)eunet.rs), Jun 17 2003
%C A083710 Starting with offset 1 = inverse Mobius transform (A051731) of the partition 
               numbers, A000041. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 
               08 2009]
%D A083710 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.
%F A083710 Equals left border of triangle A137587 starting (1, 2, 3, 5, 6, 11,...). 
               - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 27 2008
%F A083710 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.
%F A083710 Gary 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
%p A083710 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 (zerinvarylajos(AT)yahoo.com), Apr 14 2007
%Y A083710 Cf. A083711, A018783, A137587.
%Y A083710 a000041, A051731 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 08 
               2009]
%Y A083710 Sequence in context: A161715 A164523 A033159 this_sequence A127524 A117086 
               A081026
%Y A083710 Adjacent sequences: A083707 A083708 A083709 this_sequence A083711 A083712 
               A083713
%K A083710 nonn,easy
%O A083710 0,3
%A A083710 N. J. A. Sloane (njas(AT)research.att.com), Jun 16 2003
%E A083710 More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 17 2003

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