Centered tetrahedral number

Centered tetrahedral number
Total no. of terms	Infinity
Subsequence of	Polyhedral numbers
Formula
First terms	1, 5, 15, 35, 69, 121, 195
OEIS index	A005894; Centered tetrahedral;

A centered tetrahedral number is a centered figurate number that represents a tetrahedron. The centered tetrahedral number for a specific n is given by

$(2n+1)\times {(n^{2}+n+3) \over 3}$

The first such numbers are 1, 5, 15, 35, 69, 121, 195, 295, 425, 589, 791, ... (sequence A005894 in the OEIS).

Parity and divisibility

Every centered tetrahedral number is odd.
Every centered tetrahedral number with an index of 2, 3 or 4 modulo 5 is divisible by 5.
The only centered tetrahedral number which is also prime is 5. Proof: If (2*n+1)*(n^2+n+3)/3 is prime, then at least one of the factors are divisors of 3, so either n=0 or n=1, corresponding to the centered tetrahedral numbers 1 and 5, respectively.

References

Deza, E.; Deza, M. (2012). Figurate Numbers. Singapore: World Scientific Publishing. pp. 126–128. ISBN 978-981-4355-48-3.

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