A309487 -id:A309487 - cp's OEIS frontend

A324795 a(n) = 2p(n)p(n+2) - p(n+1)^2 where p(k) = k-th prime.

11, 17, 61, 61, 205, 205, 421, 573, 585, 1185, 1173, 1501, 2005, 2349, 2737, 2985, 4185, 4173, 4741, 5889, 5877, 7173, 8181, 8569, 9781, 11005, 11005, 12301, 14917, 13477, 17637, 17649, 21505, 19777, 23985, 24577, 25869, 28509, 29857, 30585, 35617

Offset: 1

N. J. A. Sloane, Sep 10 2019

nonn

Theorem: a(n) > 0. Proof: Use p(n+1) <= 2 p(n)^2 for n > 4. (See Sándor et al.) QED

József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter VII, p. 247, section VII.18.b.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A056221 (if leading coefficient 2 is changed to 1), A327447 or A309487 (if 2 is changed to 4).

With[{p = Prime[Range[50]]}, 2 * p[[1;;-3]] * p[[3;;-1]] - p[[2;;-2]]^2] (* Amiram Eldar, Apr 25 2024 *)

A327447 a(n) = 4p(n-1)p(n+1) - p(n)^2, where p(k) = k-th prime.

Original entry on oeis.org

31, 59, 171, 243, 579, 699, 1203, 1675, 2011, 3331, 3715, 4683, 5859, 6907, 8283, 9451, 12091, 12835, 14523, 17107, 17995, 21235, 24283, 26547, 29763, 32619, 33459, 36483, 42603, 43083, 52435, 54067, 62331, 61755, 70771, 73803, 78307, 84907, 89643, 93211, 103995, 103251, 113259, 114819, 126667, 132987, 141859

Offset: 2

N. J. A. Sloane, Sep 14 2019

nonn

It follows from Sándor et al., Sect. VII.18(b) that a(n) > 0 for all n.

József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter VII, p. 247, section VII.18(b).

Cf. A309487, A324795.

With[{p = Prime[Range[50]]}, 4 * p[[1;;-3]] * p[[3;;-1]] - p[[2;;-2]]^2] (* Amiram Eldar, Apr 25 2024 *)

cp's OEIS Frontend

A324795 a(n) = 2p(n)p(n+2) - p(n+1)^2 where p(k) = k-th prime.

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A327447 a(n) = 4p(n-1)p(n+1) - p(n)^2, where p(k) = k-th prime.

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cp's OEIS Frontend

A324795 a(n) = 2*p(n)*p(n+2) - p(n+1)^2 where p(k) = k-th prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

A327447 a(n) = 4*p(n-1)*p(n+1) - p(n)^2, where p(k) = k-th prime.

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

A324795 a(n) = 2p(n)p(n+2) - p(n+1)^2 where p(k) = k-th prime.

A327447 a(n) = 4p(n-1)p(n+1) - p(n)^2, where p(k) = k-th prime.