A255349 Numbers k such that k*(k+1) is divisible by some m*(m+1) with none of {k, k+1} divisible by any of {m, m+1}.

20, 35, 77, 84, 98, 99, 104, 119, 132, 153, 174, 175, 186, 189, 195, 216, 224, 230, 231, 245, 246, 260, 272, 279, 285, 350, 351, 363, 374, 384, 399, 425, 429, 440, 455, 459, 494, 527, 539, 551, 560, 575, 594, 608, 609, 615, 620, 644, 645, 650, 665, 696, 714, 730, 735, 759, 779, 780

Offset

1,1

Comments

A subsequence of A255346, see there for further information.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

T. Korimort, How many (x,y) satisfy x(x+1)|y(y+1),..., Number Theory group on LinkedIn.com, Feb. 2014.

Example

20 is a term since 20*21=420 is divisible by 14*15=210 and none of {20, 21} is divisible by any of {14, 15}.
35 is a term since 35*36 = 1260 is divisible by 14*15 = 210 (and also by 20*21 = 420).
69 is not a term although 69*70 = 4830 is divisible by 14*15 = 210, because 14 divides 70.

Maple

filter:= proc(n) local w, m;
  for w in select(t -> issqr(1+4*t), NumberTheory:-Divisors(n*(n+1)) minus {1, n*(n+1)}) do
    m:= (sqrt(1+4*w)-1)/2;
    if n mod m > 0 and (n+1) mod m > 0 and n mod (m+1) > 0 and (n+1) mod (m+1) > 0 then return true fi
  od;
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Jul 03 2026

Prog

(PARI) is(n)={omega(n)>=2&&omega(n+1)>=2&&fordiv(n*(n+1), x, x>=n&&return; n*(n+1)%(x*(x+1))&&next; n%x||next; (n+1)%x||next; n%(x+1)||next; (n+1)%(x+1)&&return(1))}

Crossrefs

Cf. A074851, A255346.

Sequence in context: A081962 A024755 A048022 * A254363 A229356 A384493

Adjacent sequences: A255346 A255347 A255348 * A255350 A255351 A255352

Keyword

nonn

Author

M. F. Hasler, Feb 21 2015

Status

approved