A138135 Number of parts > 1 in the last section of the set of partitions of n.

0, 1, 1, 3, 3, 8, 8, 17, 20, 34, 41, 68, 80, 123, 153, 219, 271, 382, 469, 642, 795, 1055, 1305, 1713, 2102, 2713, 3336, 4241, 5190, 6545, 7968, 9950, 12090, 14953, 18104, 22255, 26821, 32752, 39371, 47774, 57220, 69104

Offset: 1

Author

Omar E. Pol, Mar 30 2008

Keywords

• nonn

Comments

Also first differences of A096541. For more information see A135010.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Zero together with the column k=2 of A207031.

Cf. A000041, A006128, A096541, A135010, A138121, A138137.

Programs

Maple

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+`if`(i>1, g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]-b(n-1, n-1)[2]:
seq (a(n), n=1..60); # Alois P. Heinz, Apr 04 2012

Mathematica

a[n_] := DivisorSigma[0, n] - 1 + Sum[(DivisorSigma[0, k] - 1)*(PartitionsP[n - k] - PartitionsP[n - k - 1]), {k, 1, n - 1}]; Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Jan 14 2013, from 1st formula *)
Table[Length@Flatten@Select[IntegerPartitions[n], FreeQ[#, 1] &], {n, 1, 42}]  (* Robert Price, May 01 2020 *)

PARI

a(n)=numdiv(n)-1+sum(k=1,n-1,(numdiv(k)-1)*(numbpart(n-k) - numbpart(n-k-1))) \\ Charles R Greathouse IV, Jan 14 2013

Formula

a(n) = A096541(n)-A096541(n-1) = A138137(n)-A000041(n-1) = A006128(n)-A006128(n-1)-A000041(n-1).

a(n) ~ exp(Pi*sqrt(2*n/3))*(2*gamma - 2 + log(6*n/Pi^2))/(8*sqrt(3)*n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 24 2016

G.f.: Sum_{k>=1} x^(2*k)/(1 - x^k) / Product_{j>=2} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021