

A211510


Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2 = x*y  2n.


2



0, 0, 0, 0, 3, 0, 1, 6, 5, 4, 13, 0, 16, 12, 7, 8, 22, 10, 27, 20, 20, 8, 41, 14, 27, 32, 21, 36, 66, 0, 28, 38, 40, 36, 71, 12, 53, 60, 57, 16, 83, 14, 80, 60, 32, 64, 75, 50, 98, 62, 47, 16, 144, 36, 100, 88, 53, 52, 153, 36, 94, 76, 91, 98, 129, 20, 92, 124, 102
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OFFSET

0,5


COMMENTS

For a guide to related sequences, see A211422.
The original name was "... and w^2 = x*y + 2n", but this would yield 2 instead of 0 for a(3), as observed by Pontus von Brömssen. The corresponding sequence seems not to be in the OEIS yet.  M. F. Hasler, Jan 26 2020


LINKS

Pontus von Brömssen, Table of n, a(n) for n = 0..1024


EXAMPLE

From Bernard Schott, Jan 26 2020: (Start)
For n = 4, there are 3 ordered solutions with (1,3,3), (2,3,4) and (2,4,3) so a(4) = 3.
For n = 5, there is no solution, hence a(5) = 0.
The only solution for n = 6 is (2,4,4) with 2^2 = 4*4  2*6, hence a(6) = 1. (End)


MATHEMATICA

t[n_] := t[n] = Flatten[Table[w^2  x*y + 2 n, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}] (* A211510 *)


PROG

(Python)
import sympy
def A211510(n): return sum(x<=n and x*n>=w**2+2*n for w in range(1, n+1) for x in sympy.divisors(w**2+2*n)) # Pontus von Brömssen, Jan 26 2020
(PARI) apply( {A211510(n)=sum(w=1, n2, my(w2n=(w^21)\n+2, s); fordiv(w^2+2*n, x, x>w2nnext; x>n&&break; s++); s)}, [1..100]) \\ M. F. Hasler, Jan 26 2020


CROSSREFS

Cf. A211422.
Sequence in context: A335262 A111924 A212880 * A243984 A100485 A143397
Adjacent sequences: A211507 A211508 A211509 * A211511 A211512 A211513


KEYWORD

nonn


AUTHOR

Clark Kimberling, Apr 14 2012


EXTENSIONS

Name corrected by Pontus von Brömssen, Jan 26 2020


STATUS

approved



