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A236767 Numbers whose square is a fourth power plus a prime. 1
2, 10, 37, 82, 442, 577, 730, 901, 1090, 1297, 1765, 2026, 4357, 5185, 5626, 7570, 8650, 9217, 9802, 10405, 11026, 15130, 17425, 18226, 23410, 24337, 26245, 31330, 34597, 35722, 40402, 41617, 47962 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Based on a 1999 observation of Alessandro Zaccagnini (via John Robertson) intended to dissuade expectation of a finite fourth-power analogy to A020495, A045911.

It can be shown that A089001^2 + 1 are members of this sequence. David Applegate shows that they are the only members: If x^2 = y^4 + p, let a = x - y^2. Then y^4 + p = x^2 = (y^2 + a)^2 = y^4 + 2a*y^2 + a^2, so p = 2a*y^2 + a^2, and so a divides p. Since p is a prime, a must be a unit (that is, +1 or -1). But since p >= 2, a must be +1.

LINKS

Hans Havermann, Table of n, a(n) for n = 1..1000

John Robertson, Integers of the form x^2+kp (see last paragraph)

FORMULA

A089001^2 + 1

EXAMPLE

2 because 2^2 = 1^4 + 3

10 because 10^2 = 3^4 + 19

37 because 37^2 = 6^4 + 73

MATHEMATICA

r=Range[10000]^4; j=1; Do[c=i^2; k=c^2-Take[r, i]; Do[c++; j=j+2; k=k+j; If[MemberQ[PrimeQ[k], True], Print[c]], {2*i+1}], {i, 10000}] (*brute force*)

s=A089001; s^2+1 (*based on formula*)

CROSSREFS

Cf. A020495, A045911, A089001.

Sequence in context: A151021 A151022 A144895 * A154323 A191349 A073110

Adjacent sequences:  A236764 A236765 A236766 * A236768 A236769 A236770

KEYWORD

nonn

AUTHOR

Hans Havermann, Jan 30 2014

STATUS

approved

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Last modified December 12 09:36 EST 2019. Contains 329953 sequences. (Running on oeis4.)