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Total no. of terms Infinity Polyhedral numbers ${\displaystyle {\frac {(2n+1)\,(n^{2}+n+3)}{3))}$ 1, 5, 15, 35, 69, 121, 195 A005894Centered 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

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

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.