A227390 Minimum value of k such that sum_{i=k..k+n-1} i = sum_{i=k..k+n-1} i’, where i’ is the arithmetic derivative of i.

0, 16, 390, 30, 156, 93052, 1884, 2910

Offset

1,2

Comments

a(9) > 10^10. - Giovanni Resta, Jul 10 2013

Links

Table of n, a(n) for n=1..8.

Example

n = 1, k = k’.  The minimum solution is k=0 because k’=0. Other non-minimal solutions are listed in A051674.
n = 2, k + (k+1) = k’ + (k+1)’. The minimum solution is 16 because 16 + 17 = 33 and 16’ + 17’ = 32 + 1 = 33.
Other non-minimal solutions are 108, 3294172, 7975979, etc.
n = 3, k + (k+1) + (k+2) = k’ + (k+1)’ + (k+2)’. The minimum solution is 390 because 390 + 391 + 392 = 1173 and 390’ + 391’ + 392’ = 433 + 40 + 700 = 1173.

Maple

with(numtheory); ListA227390:= proc(q) local a, b, k, i, n, p;
for n from 1 to q do for k from 0 to q do a:=n*(k+(n-1)/2); b:=0;
b:=add((k+i)*add(op(2, p)/op(1, p), p=ifactors(k+i)[2]), i=0..n-1);
if a=b then print(k); break; fi; od; od; end: ListA227390(10^12);

Crossrefs

Cf. A003415, A051674.

Sequence in context: A114426 A189849 A051360 * A284854 A269412 A269643

Adjacent sequences: A227387 A227388 A227389 * A227391 A227392 A227393

Keyword

nonn,more

Author

Paolo P. Lava, Jul 10 2013

Status

approved