A009641 a(n) = Product_{i=0..6} floor((n+i)/7).

0, 0, 0, 0, 0, 0, 0, 1, 2, 4, 8, 16, 32, 64, 128, 192, 288, 432, 648, 972, 1458, 2187, 2916, 3888, 5184, 6912, 9216, 12288, 16384, 20480, 25600, 32000, 40000, 50000, 62500, 78125, 93750, 112500, 135000, 162000, 194400, 233280, 279936, 326592, 381024

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

For n >= 7, a(n) is the maximal product of seven positive integers with sum n. - Wesley Ivan Hurt, Jun 29 2022

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• M. El-Mikkawy and T. Sogabe, A new family of k-Fibonacci numbers, Appl. Math. Comput. 215 (2010) 4456-4461, Table 1 k=7.
• Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 6, -12, 6, 0, 0, 0, 0, -15, 30, -15, 0, 0, 0, 0, 20, -40, 20, 0, 0, 0, 0, -15, 30, -15, 0, 0, 0, 0, 6, -12, 6, 0, 0, 0, 0, -1, 2, -1).

Crossrefs

Maximal product of k positive integers with sum n, for k = 2..10: A002620 (k=2), A006501 (k=3), A008233 (k=4), A008382 (k=5), A008881 (k=6), this sequence (k=7), A009694 (k=8), A009714 (k=9), A354600 (k=10).

Cf. A001015, A013665.

Programs

Mathematica

A009641[n_] := Product[Floor[(n + i)/7], {i, 0, 6}]; Array[A009641, 50, 0] (* Paolo Xausa, Jan 17 2025 *)

PARI

a(n) = prod(k=0, 6, (n+k)\7); \\ Georg Fischer, Nov 07 2019

Formula

a(n) = 2*a(n-1) - a(n-2) + 6*a(n-7) - 12*a(n-8) + 6*a(n-9) - 15*a(n-14) + 30*a(n-15) - 15*a(n-16) + 20*a(n-21) - 40*a(n-22) + 20*a(n-23) - 15*a(n-28) + 30*a(n-29) - 15*a(n-30) + 6*a(n-35) - 12*a(n-36) + 6*a(n-37) - a(n-42) + 2*a(n-43) - a(n-44). - Wesley Ivan Hurt, Jun 29 2022

a(7*n) = n^7 (A001015). - Bernard Schott, Nov 04 2022

Sum_{n>=7} 1/a(n) = 1 + zeta(7). - Amiram Eldar, Jan 10 2023

Extensions

a(40)-a(44) from Georg Fischer, Nov 07 2019