A190479 Let s the sum of the decimal digit of a number k. Number of integer solutions of the equation s*(s+n)- k = 0.

4, 1, 3, 6, 3, 3, 4, 4, 4, 4, 3, 2, 3, 3, 3, 5, 2, 3, 4, 4, 2, 2, 3, 3, 4, 3, 3, 5, 3, 3, 6, 1, 1, 5, 2, 3, 4, 2, 3, 3, 3, 1, 3, 1, 1, 2, 5, 2, 4, 2, 3, 0, 3, 3, 3, 1, 1, 6, 4, 1, 2, 1, 3, 6, 2, 1, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 2, 1, 2, 2, 1, 2, 4, 1, 1, 4, 3, 2, 3, 2, 3, 4, 2, 2, 3, 0, 0, 4, 3, 3, 4, 1, 3, 4, 3, 1, 2, 1, 4

Offset

1,1

Comments

The equation s*(s+n)-k = 0 has no integer solution for n = {52, 101, 102, 152, 206, 393, 408, 464, 473, 482,...} (See the proof with the sequence A190216).

Links

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

Example

a(1) = 4 because k = {12, 42, 90, 156} and :
for k = 12, s = 3 and 3*(3+1) - 12 = 0 ;
for k = 42, s = 6 and 6*(6+1) - 42 = 0 ;
for k = 90, s = 9 and 9*(9+1) - 90 = 0 ;
for k = 156, s = 12 and 12*(12+1) - 156 = 0.

Maple

with(numtheory):for n from 1 to 500 do:s1:=0:for k from 1 to 30000 do l:=length(k):n0:=k:s:=0:for m from 1 to l do q:=n0:u:=irem(q, 10):v:=iquo(q, 10):n0:=v:s:=s+u:od: if s*(s+n)=k  then s1:=s1+1:else fi:od: printf(`%d, `, s1):od:
# Alternative:
N:= 500: # to get N terms
K:= ceil(fsolve(k/(9*log[10](k)) - 9*log[10](k) = N, k = 100 .. infinity)):
sd:= k -> convert(convert(k, base, 10), `+`):
A:= Vector(N):
for k from 1 to K do
  sk:= sd(k);
  n:= k/sk - sk;
  if type(n, posint) and n <= N then
    A[n]:= A[n]+1
  fi
od:
seq(A[n], n=1..N); # Robert Israel, May 30 2014

Crossrefs

Cf. A190216.

Sequence in context: A134224 A121441 A256243 * A074813 A151861 A210583

Adjacent sequences: A190476 A190477 A190478 * A190480 A190481 A190482

Keyword

nonn,base

Author

Michel Lagneau, May 11 2011

Status

approved