A328792 -id:A328792 - cp's OEIS frontend

A328791 Triangular numbers of the form k^2 + 3.

3, 28, 903, 30628, 1040403, 35343028, 1200622503, 40785822028, 1385517326403, 47066803275628, 1598885794044903, 54315050194251028, 1845112820810490003, 62679520857362409028, 2129258596329511416903, 72332112754346025765628, 2457162575051435364614403

Offset: 1

Jon E. Schoenfield, Oct 27 2019

nonn

There exist triangular numbers of the form k^2 + j for j=0 (A001110), j=1 (A164055), j=2 (A214838), and j=3 (this sequence), but not for j=4,7,8,13,16,18,... (A328792).

Cf. A001110, A164055, A214838, A328792.
Intersection of A000217 and A117950.
Cf. A276598 (the k's).

limit = 10**7 # rough limit for k
A000217 = set(k*(k+1)//2 for k in range(14*limit//10))
A117950 = set(k**2 + 3 for k in range(limit))
print(sorted(A000217 & A117950)) # Michael S. Branicky, Mar 28 2021

a(1) = 3, a(2) = 28; for n > 2, a(n) = 34*a(n-1) - a(n-2) - 46.

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

Original entry on oeis.org

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

Ya-Ping Lu, Jun 21 2020

nonn

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.

			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.

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

Cf. A000292, A000578, A230044, A328792.

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).

cp's OEIS Frontend

A328791 Triangular numbers of the form k^2 + 3.

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