A335761 Nonnegative numbers that are the difference between a positive tetrahedral number and a positive cubic number.

0, 2, 3, 4, 8, 9, 12, 19, 20, 21, 27, 29, 34, 40, 43, 47, 48, 53, 55, 56, 57, 70, 76, 83, 87, 93, 95, 101, 103, 112, 119, 136, 138, 140, 144, 148, 152, 156, 157, 161, 164, 168, 174, 181, 193, 209, 212, 217, 219, 222, 239, 240, 253, 259, 275, 278, 279, 281, 285

Offset

1,2

Comments

The sequence is the difference between the tetrahedral number (A000292) and the cubic number (A000578) such that terms are of the form A000292(i)-A000578(j), where A000292(i) >= A000578(j) >= 0.

It appears that sequence terms are more scarce than prime numbers. The number of terms in this sequence (n) and the number of prime numbers up to a(n) are shown in the figure attached in the LINKS section. It can be seen that, for a(n) > 304, n is less than pi(a(n)), where pi is the prime counting function.

Links

Table of n, a(n) for n=1..59.

Ya-Ping Lu, The number of terms in a(n) and the number of prime numbers up to a(n)

Formula

The difference between the i-th tetrahedral number, t(i), and j-th cubic number, c(j) is d = i*(i+1)*(i+2)/6 - j^3, where i, j >=1 and t(i) >= c(j).

Example

a(1)=0 because t(1)-c(1)=1-1=0;
a(2)=2 because t(3)-c(2)=10-8=2;
a(7)=12 because t(4)-c(2)=20-8=12, and t(39)-c(22)=10660-10648=12;
a(19)=55 because t(6)-c(1)=56-1=55, and t(4669)-c(2570)=16974593055-16974593000=55.

Crossrefs

Cf. A000292, A000578, A230044, A328792.

Sequence in context: A332622 A085256 A364810 * A127483 A130804 A022999

Adjacent sequences: A335758 A335759 A335760 * A335762 A335763 A335764

Keyword

nonn

Author

Ya-Ping Lu, Jun 21 2020

Status

approved