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A232185
Let x(0)x(1)x(2)...x(q) denote the decimal expansion of k. Sequence lists all zeroless numbers k > 10 such that x(0)/x(1)x(2)... x(q) + x(0)x(1)/x(2)x(3)...x(q) + ... + x(0)x(1)... x(q-1)/x(q) is an integer.
0
11, 21, 22, 31, 33, 41, 42, 44, 51, 55, 61, 62, 63, 66, 71, 77, 81, 82, 84, 88, 91, 93, 99, 612, 816, 945
OFFSET
1,1
COMMENTS
Subsequence of A232184. This sequence is probably finite (no further terms less than 2*10^8).
The corresponding integers are 1, 2, 1, 3, 1, 4, 2, 1, 5, 1, 6, 3, 2, 1, 7, 1, 8, 4, 2, 1, 9, 3, 1, 31, 14, 19.
EXAMPLE
945 is in the sequence because 9/45 + 94/5 = 19 is an integer.
MAPLE
with(numtheory):U:=array(1..10):V:=array(1..10):for n from 1 to 20000 do: x:=convert(n, base, 10):n1:=nops(x): s:=product('x[i]', 'i'=1..n1):if s<>0 then s1:=0:s2:=0:for i from 1 to n1 do:s1:=s1+x[i]*10^(i-1): U[i]:=s1:od: s2:=x[n1]:V[n1]:=s2:for j from n1-1 by -1 to 1 do:s2:=s2*10+x[j]:V[j]:=s2:od:s3:=0:ii:=0:for k from n1 by -1 to 2 while(ii=0) do:if U[k-1]=0 then ii:=1: else s3:=s3+V[k]/U[k-1]:fi:od:if s3=floor(s3) and ii=0 then printf(`%d, `, n):else fi:fi:od:
PROG
(PARI) isok(k) = my(d=digits(k)); if (vecmin(d), denominator(sum(j=1, #d-1, fromdigits(Vec(d, j))/fromdigits(vector(#d, k, if (k>j, d[k])))))==1); \\ Michel Marcus, Jan 11 2025
CROSSREFS
Cf. A084906,
Intersection of A052382 and A232184.
Sequence in context: A108237 A084906 A232184 * A118853 A117841 A105956
KEYWORD
nonn,base,more
AUTHOR
Michel Lagneau, Nov 20 2013
STATUS
approved