A308408 a(n) is the smallest k that is equal to the sum of the digits of k*(k+1)*...*(k+n-1) in base 10^n, or -1 if such a number does not exist.

1, 33, -1, 10692, 74016, 1153845, 19999998, 373722624, 3025660311, 39999999996, -1

Offset

1,2

Comments

Partition the product of n consecutive integers, from k to k+n-1, into blocks of n digits starting from right. Sequence lists, for each n, the first number k of the least product whose sum of blocks is equal to k, or -1 if such a number does not exist.

a(12) <= 6*10^12 - 6, a(13) <= 4*10^13 - 4, a(14) <= 6*10^14 - 6, a(15) <= 8*10^15 - 8, a(16) <= 6*10^16 - 6, a(17) <= 8*10^17 - 8, a(18) <= 8*10^18 - 8. - Giovanni Resta, May 25 2019

Links

Table of n, a(n) for n=1..11.

Example

a(2) = 33 because 33*34 = 1122 and 11 + 22 = 33.
a(4) = 10692 because 10692*10693*10694*10695 = 13076137707585480 and 5480 + 758 + 1377 + 3076 + 1 = 10692.

Maple

P:=proc(q) local a, b, c, j, k, n, x; c:=1; for n from 1 to q do x:=0:
for k from c to q do a:=mul(j, j=k..k+n-1); b:=0; while a>0 do
b:=b+(a mod 10^n); a:=trunc(a/10^n); od; if k>b then x:=x+1;
else if k<b then x:=0: fi; fi; if k=b then lprint(n, k); c:=n: break;
fi; if x=100 then lprint(n, -1); break; fi; od; od; end: P(10^15);

Crossrefs

Cf. A037042, A037043, A037044, A037045, A274833, A274834.

Sequence in context: A225347 A218116 A279364 * A263169 A034060 A032445

Adjacent sequences: A308405 A308406 A308407 * A308409 A308410 A308411

Keyword

sign,base,more,hard

Author

Paolo P. Lava, May 25 2019

Extensions

a(8)-a(11) from Giovanni Resta, May 25 2019

Status

approved