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A198443 Records of minima of positive distance d between a square of integer y and a fifth power of positive integer x such that d = y^2 - x^5 (x <> k^2 and y <> k^5). 3
3, 4, 11, 26, 37, 368, 1828, 2180, 7825, 8177, 8217, 71393, 72481, 75154, 118409, 175485, 203697, 206370, 1049148, 1058224, 1843945, 1846618, 8186369, 8197633, 9600802, 96020524, 169503449, 294638801, 305158594, 305192969, 657099024 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Distance d is equal 0 when x = k^2 and y = k^5.

This sequence is proved for x<10^8.

For x values see A198444.

For y values see A198445.

Conjecture (*Artur Jasinski*): For any positive number x >= A198444(n) distance d between a square of integer y and a fifth power of positive integer x (such that x<>k^2 and y<>k^5) can't be less than A198443(n).

LINKS

Table of n, a(n) for n=1..31.

FORMULA

a(n) = (A198445(n))^2 - (A198444(n))^5.

MATHEMATICA

max = 1000; vecd = Table[10^100, {n, 1, max}]; vecx = Table[10^100, {n, 1, max}]; vecy = Table[10^100, {n, 1, max}]; len = 1; Do[m = Floor[(n^5)^(1/2)] + 1; k = m^2 - n^5; If[k != 0, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m], {n, 1, 100000000}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]];  AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; dd

CROSSREFS

Cf. A179406, A179407, A179408.

Sequence in context: A292623 A077900 A208176 * A041231 A042129 A141723

Adjacent sequences:  A198440 A198441 A198442 * A198444 A198445 A198446

KEYWORD

nonn,hard

AUTHOR

Artur Jasinski, Oct 25 2011

STATUS

approved

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Last modified June 25 04:15 EDT 2021. Contains 345452 sequences. (Running on oeis4.)