A274306 -id:A274306 - cp's OEIS frontend

A274307 Numbers n such that n^2 is a term of A274306.

1, 325, 7010415003293431191312386899866982908203125

Offset: 1

N. J. A. Sloane, Jun 20 2016

nonn

The sequence is known to be infinite.
Once sufficiently many further terms are found, their indices in A274306 can give another sequence.
The terms' indices in A274306 are: 0, 3, 20, 119, 696, 4059, 23660, 137903, ... - Chai Wah Wu, Jun 20 2016
The terms' indices are equal to the sequence A001652 and thus satisfy a linear recurrence. Gürel showed that 2k^2+2k+1 = k^2 + (k+1)^2 is a square if and only if A274306(k) is a square, i.e, (k, k+1) are the 2 smaller terms of a Pythagorean triple and thus k is the index of a term if and only if it is in A001652. - Chai Wah Wu, Jun 21 2016

Erhan Gürel, On the Occurrence of Perfect Squares Among Values of Certain Polynomial Products, The American Mathematical Monthly 123.6 (2016): 597-599.

Cf. A274306.

A277354 a(n) = Product_{k=1..n} (4*k^2+1).

Original entry on oeis.org

1, 5, 85, 3145, 204425, 20646925, 2993804125, 589779412625, 151573309044625, 49261325439503125, 19753791501240753125, 9580588878101765265625, 5527999782664718558265625, 3742455852864014463945828125, 2937827844498251354197475078125

Offset: 0

Vaclav Kotesovec, Oct 10 2016

nonn

In general, for m>0, Product_{k=1..n} (m*k^2+1) is asymptotic to 2*m^(n+1/2) * n^(2*n+1) * sinh(Pi/sqrt(m)) / exp(2*n).

Cf. A079484, A101686, A101928, A274306, A277352, A277353.

Table[Product[4*k^2+1, {k, 1, n}], {n, 0, 15}]
Round@Table[2^(2 n + 1) Abs[Gamma[1 + I/2 + n]]^2 Sinh[Pi/2]/Pi, {n, 0, 15}] (* Vladimir Reshetnikov, Oct 10 2016 *)

a(n) = prod(k=1, n, (4*k^2+1)); \\ Michel Marcus, Oct 11 2016

a(n) = (-1)^(n+1) * A101928(n+2).
a(n) ~ 2^(2*n+2) * n^(2*n+1) * sinh(Pi/2) / exp(2*n).
a(n) = 2^(2*n+1) * |Gamma(1 + i/2 + n)|^2 * sinh(Pi/2)/Pi. - Vladimir Reshetnikov, Oct 10 2016

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A274307 Numbers n such that n^2 is a term of A274306.

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A277354 a(n) = Product_{k=1..n} (4*k^2+1).

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