This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A083710 #23 Feb 06 2022 06:46:26
%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,
%U A083710 12327,15272,17978,22024,26095,31730,37339,45333,53175,64100,75340,90138
%N A083710 Number of integer partitions of n with a part dividing all the other parts.
%C A083710 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
%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_, Jun 17 2003
%C A083710 Starting with offset 1 = inverse Mobius transform (A051731) of the partition numbers, A000041. - _Gary W. Adamson_, 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_, 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 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
%e A083710 From _Gus Wiseman_, Apr 18 2021: (Start)
%e A083710 The a(1) = 1 through a(7) = 12 partitions:
%e A083710 (1) (2) (3) (4) (5) (6) (7)
%e A083710 (11) (21) (22) (41) (33) (61)
%e A083710 (111) (31) (221) (42) (331)
%e A083710 (211) (311) (51) (421)
%e A083710 (1111) (2111) (222) (511)
%e A083710 (11111) (321) (2221)
%e A083710 (411) (3211)
%e A083710 (2211) (4111)
%e A083710 (3111) (22111)
%e A083710 (21111) (31111)
%e A083710 (111111) (211111)
%e A083710 (1111111)
%e A083710 (End)
%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_, Apr 14 2007
%t A083710 Table[Length[Select[IntegerPartitions[n],And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}] (* _Gus Wiseman_, Apr 18 2021 *)
%Y A083710 Cf. A018783, A137587.
%Y A083710 Cf. A000041, A051731. - _Gary W. Adamson_, Jun 08 2009
%Y A083710 The case with no 1's is A083711.
%Y A083710 The strict case is A097986.
%Y A083710 The version for "divisible by" instead of "dividing" is A130689.
%Y A083710 The case where there is also a part divisible by all the others is A130714.
%Y A083710 The complement of these partitions is counted by A338470.
%Y A083710 The Heinz numbers of these partitions are dense, complement of A342193.
%Y A083710 The case where there is also no part divisible by all the others is A343345.
%Y A083710 A000005 counts divisors.
%Y A083710 A000070 counts partitions with a selected part.
%Y A083710 A006128 counts partitions with a selected position.
%Y A083710 A015723 counts strict partitions with a selected part.
%Y A083710 A018818 counts partitions into divisors (strict: A033630).
%Y A083710 A167865 counts strict chains of divisors > 1 summing to n.
%Y A083710 Cf. A001792, A098965, A264401, A339563, A343340, A343341, A343378.
%K A083710 nonn,easy
%O A083710 0,3
%A A083710 _N. J. A. Sloane_, Jun 16 2003
%E A083710 More terms from _Vladeta Jovovic_, Jun 17 2003
%E A083710 Name shortened by _Gus Wiseman_, Apr 18 2021