

A332542


a(n) = smallest k such that n+(n+1)+(n+2)+...+(n+k) is divisible by n+k+1.


6



2, 7, 14, 3, 6, 47, 14, 4, 10, 20, 25, 11, 5, 31, 254, 15, 18, 55, 6, 10, 22, 44, 14, 23, 11, 7, 86, 27, 30, 959, 62, 16, 34, 8, 73, 35, 17, 24, 163, 39, 42, 127, 9, 22, 46, 92, 62, 19, 23, 15, 158, 51, 10, 20, 75, 28, 58, 116, 121, 59, 29, 127, 254, 11
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OFFSET

3,1


COMMENTS

Note that (n+(n+1)+(n+2)+...+(n+k))/(n+k+1) = A332544(n)/(n+k+1) = A082183(n1). See the Myers et al. link for proof.  N. J. A. Sloane, Apr 30 2020
We can always take k = n^22*n1, for then the sum in the definition becomes (n+1)*n*(n1)*(n2)/2, which is an integral multiple of n+k+1 = n*(n1). So a(n) always exists.  N. J. A. Sloane, Feb 20 2020


LINKS

Seiichi Manyama, Table of n, a(n) for n = 3..10000
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000, April 2020


EXAMPLE

n=4: we get 4 > 4+5=9 > 9+6=15 > 15+7=22 > 22+8=30 > 30+9=39 > 39+10=49 > 49+11=60, which is divisible by 12, and took k=7 steps, so a(4) = 7. Also A332543(4) = 12, A332544(4) = 60, and A082183(3) = 60/12 = 5.


MAPLE

grow2 := proc(n, M) local p, q, k; # searches out to a limit of M
# returns n, k (A332542(n)), n+k+1 (A332543(n)), p (A332544(n)), and q (which appears to match A082183(n1))
for k from 1 to M do
if ((k+1)*n + k*(k+1)/2) mod (n+k+1) = 0 then
p := (k+1)*n+k*(k+1)/2;
q := p/(n+k+1); return([n, k, n+k+1, p, q]);
fi;
od:
# if no success, return 1's
[n, 1, 1, 1, 1]; end; # N. J. A. Sloane, Feb 18 2020


MATHEMATICA

a[n_] := NestWhile[#1+1&, 0, !IntegerQ[Divide[(#+1)*n+#*(#+1)/2, n+#+1]]&]
a/@Range[3, 100] (* Bradley Klee, Apr 30 2020 *)


PROG

(Ruby)
def A(n)
s = n
t = n + 1
while s % t > 0
s += t
t += 1
end
t  n  1
end
def A332542(n)
(3..n).map{i A(i)}
end
p A332542(100) # Seiichi Manyama, Feb 19 2020


CROSSREFS

Cf. A332543, A332544, A082183.
See A332558A332561 for a multiplicative analog.
Sequence in context: A102155 A102154 A102153 * A331703 A032537 A072120
Adjacent sequences: A332539 A332540 A332541 * A332543 A332544 A332545


KEYWORD

nonn


AUTHOR

Scott R. Shannon, Feb 18 2020


STATUS

approved



