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 A215513 spt(n) - p(n): total number of smallest parts in all partitions of n minus the number of partitions of n. 3
 0, 1, 2, 5, 7, 15, 20, 35, 50, 77, 105, 161, 214, 305, 413, 570, 751, 1022, 1330, 1772, 2295, 2996, 3837, 4970, 6305, 8050, 10155, 12844, 16065, 20169, 25055, 31197, 38549, 47650, 58540, 71960, 87916, 107424, 130655, 158830, 192260, 232642, 280406 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Also total number of smallest parts that are not on the right border in all partitions of n. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 FORMULA a(n) = A092269(n) - A000041(n). a(n) = A000070(n-2) + A220479(n), n >= 2. a(n) ~ exp(Pi*sqrt(2*n/3)) / (2*Pi*sqrt(2*n)) * (1 - 25*Pi/(24*sqrt(6*n)) + (25/48 + 49*Pi^2/6912)/n). - Vaclav Kotesovec, Jul 31 2017 EXAMPLE For n = 6 the partitions of 6 with the smallest parts that are not in the right border in brackets are ----------------------------------------- . Partitions of 6 Value ----------------------------------------- . 6 0 . [3]+ 3 1 . 4 + 2 0 . [2]+[2]+ 2 2 . 5 + 1 0 . 3 + 2 + 1 0 . 4 +[1]+ 1 1 . 2 + 2 +[1]+ 1 1 . 3 +[1]+[1]+ 1 2 . 2 +[1]+[1]+[1]+ 1 3 . [1]+[1]+[1]+[1]+[1]+ 1 5 -------------------------------------- . Total: 15 On the other hand the total number of smallest parts in all partitions of 6 is 26 and the number of partitions of 6 is 11, so a(6) = 26 - 11 = 15. MATHEMATICA b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r == 0, q, 0] + Sum[b[n - i*j, i - 1], {j, 0, n/i}]]; a[n_] := b[n, n] - PartitionsP[n]; Array[a, 50] (* Jean-François Alcover, Jun 05 2021, using Alois P. Heinz's code for A092269 *) CROSSREFS Cf. A000041, A000070, A002865, A092269, A120452, A220479. Sequence in context: A257025 A076720 A245921 * A358878 A306956 A337655 Adjacent sequences: A215510 A215511 A215512 * A215514 A215515 A215516 KEYWORD nonn AUTHOR Omar E. Pol, Jan 13 2013 STATUS approved

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Last modified February 21 02:19 EST 2024. Contains 370219 sequences. (Running on oeis4.)