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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Enric Reverter i Bigas

Enric Reverter i Bigas's wiki page.

Enric Reverter i Bigas has authored 2 sequences.

A338026 a(1) = 8; for n > 1, a(n) is the largest integer m such that m = ((2*x*a(n-1)) / (x+1)) - x, with x a positive nontrivial divisor of m.

Original entry on oeis.org

8, 9, 10, 12, 15, 20, 28, 42, 66, 110, 190, 342, 630, 1190, 2278, 4422, 8646, 17030, 33670, 66822, 132900, 264758, 528034, 1053990, 2105077, 4205820, 8405840, 16803405, 33595212, 67173930, 134324628, 268616475, 537185908, 1074305622, 2148516546
Offset: 1

Views

Author

Enric Reverter i Bigas, Oct 07 2020

Keywords

Comments

There are no primes in the sequence and, excepting for a(1), no powers of 2.
For each n > 1 there are two integers f, g such that f*g = a(n) and f + g = 2*a(n-1) - a(n) - 1. (Empirical observation)
Excluding the condition that each term should be the largest one, the terms which satisfies the remaining conditions performs an irregular infinite net.
If the condition "be the largest term" is replaced for "be the smallest one" with the rest of conditions remaining, A209724 sequence is obtained. (This is true, at least, from a'(1) to a'(100))
Similar sequences are obtained with a'(1) a nonprime integer larger than 6, either a power of two or not. However, if a'(1) is not a power of two, there exist at least, one integer: a'(0) = ((a'(1)+x)*(x+1)) / (2*x) with x a positive nontrivial divisor of a'(1). Example: a'(1) = 26, a'(0) = 21, a'(-1) = 16. (As a'(-1) is a power of 2 there is not an a'(-2) term). If a'(1) = 6, a'(0) = a'(1) = a'(2) = ... = a'(k) = 6.

Examples

			a(5) = 15 = ((2*3*12) / 4) - 3 or ((2*5*12) / 6) - 5 = 15; Also 14 = ((2*2*12) / 3) - 2, but 15 is larger.

Links

Crossrefs

Cf. A209724.

Programs

  • Mathematica
    w[n_] := Module[{x, p}, Max[p /. List@ToRules@Reduce[p == (2 n*x)/(x + 1) - x == x*y && x > 1 && y > 1, p, Integers]]]; n := 8; k := {n}; m = 1; While[m < 35, {AppendTo[k, w[n]], n = w[n]}; m++]; k

A210446 Largest integer which is both the product of two integers summing to n+1 and the product of two integers summing to n-1.

Original entry on oeis.org

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
Offset: 1

Views

Author

Enric Reverter i Bigas, Jan 20 2013

Keywords

Comments

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.

Examples

			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.)

Crossrefs

Cf. A085780.

Programs

  • Mathematica
    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 *)
  • 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

Formula

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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