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A210446
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Largest integer which is both the product of two integers summing to n+1 and the product of two integers summing to n-1.
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0
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0, 0, 0, 0, 0, 6, 0, 0, 16, 18, 0, 30, 0, 36, 48, 0, 0, 70, 0, 90, 96, 90, 0, 126, 144, 126, 160, 180, 0, 210, 0, 0, 240, 216, 288, 300, 0, 270, 336, 378, 0, 420, 0, 450, 480, 396, 0, 510, 576, 594, 576, 630, 0, 700, 720, 756, 720, 630, 0, 858, 0, 720, 960, 0
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OFFSET
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1,6
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COMMENTS
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a(n) is also the difference between ((n+1)/2)^2 and Q, where Q is the smallest square which exceeds n by a square q (or by 0 if n itself is a square): ((n+1) / 2)^2 - a(n) = Q; Q - n = q; (Q, q squares of an integer if n is odd).
If n is an odd nonprime > 1, a(n)/16 is the product of two triangular numbers (see A085780).
If n is 1, a prime or a power of 2, a(n) = 0.
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LINKS
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Table of n, a(n) for n=1..64.
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FORMULA
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a(n) = (f1^2 - 1)*(f2^2 - 1)/4 (with f1 and f2 the nearest integers such that f1*f2 = n).
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EXAMPLE
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a(15) = 48 because 6*8 = 12*4 = 48 and 6 + 8 = 15 - 1; 12 + 4 = 15 + 1.
a(45) = 480 because 20*24 = 16*30 = 480 and 20 + 24 = 45 - 1; 16 + 30 = 45 + 1.
(Also 448 = 28*16 = 14*32, but 480 is larger.)
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MATHEMATICA
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a[n_] := Module[{x, y, p}, Max[p /. List@ToRules@Reduce[p == x*(n-1-x) == y*(n+1-y), {x, y, p}, Integers]]]; Table[a[n], {n, 100}] (* Giovanni Resta, Jan 22 2013 *)
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PROG
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(PARI) a(n) = {my(x=vector(n\2), y=vector(n\2)); for(k=1, n\2, x[k]=k*(n-1-k); y[k]=k*(n+1-k)); v=setintersect(x, y); if(#v>0, v[#v], 0); } \\ Jinyuan Wang, Oct 13 2019
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CROSSREFS
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Cf. A085780.
Sequence in context: A263850 A341797 A340905 * A002044 A230968 A028700
Adjacent sequences: A210443 A210444 A210445 * A210447 A210448 A210449
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KEYWORD
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nonn
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AUTHOR
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Enric Reverter i Bigas, Jan 20 2013
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STATUS
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approved
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