A183003 a(n) = A183002(n)/2.

0, 0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 7, 7, 8, 9, 11, 11, 13, 13, 15, 16, 17, 17, 20, 21, 22, 23, 25, 25, 28, 28, 30, 31, 32, 33, 37, 37, 38, 39, 42, 42, 45, 45, 47, 49, 50, 50, 54, 55, 57, 58, 60, 60, 63, 64, 67, 68, 69, 69, 74, 74, 75, 77, 80, 81, 84, 84, 86, 87, 90, 90, 95, 95, 96, 98, 100, 101, 104, 104, 108, 110, 111, 111, 116, 117, 118, 119, 122, 122, 127, 128, 130, 131, 132, 133, 138, 138, 140, 142, 146

Offset: 1

Author

Omar E. Pol, Jan 27 2011

Keywords

• nonn
• easy

Comments

For n >= 2, a(n) is the number of partitions of n-1 into 3 parts such that the largest part is greater than or equal to the product of the other two. For example, a(9) = 4 since the partitions for 8 would be 1+1+6 = 1+2+5 = 1+3+4 = 2+2+4, but not 2+3+3 since 2*3 > 3. - Wesley Ivan Hurt, Jan 03 2022

Conjecture: partial sums of A072670. - Sean A. Irvine, Jul 14 2022

Crossrefs

Cf. A000005, A001620, A072670, A183002.

Programs

Mathematica

Accumulate[Table[d = DivisorSigma[0, n]; If[OddQ[d], d - 1, d - 2], {n, 100}]]/2

PARI

a(n) = sum(k=1, n, numdiv(k) - 2 + numdiv(k)%2)/2; \\ Michel Marcus, Jan 04 2022

Formula

a(n) = Sum_{k=1..floor((n-1)/3)} Sum_{i=k..floor((n-k-1)/2)} sign(floor((n-i-k-1)/(i*k))). - Wesley Ivan Hurt, Jan 03 2022

a(n) = (1/2) * Sum_{k=1..n} (tau(k)-2 + (tau(k) mod 2)), tau = A000005. - Alois P. Heinz, Jan 04 2022

a(n) ~ n * (log(n) + 2*gamma - 3) / 2, where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 19 2024