A243024 Consider a k-digit number m = d_(k)*10^(k-1) + d_(k-1)*10^(k-2) + ... + d_(2)*10 + d_(1). Sequence lists the numbers m that divide Sum_{i=1..k-1}{d_(i)^d_(i+1)}+d_(k)^d_(1).

1, 2, 3, 4, 5, 6, 7, 8, 9, 63, 448, 1899, 1942, 4155, 4355, 8503, 28375, 44569, 73687, 1953504, 1954329, 70860598, 522169982

Offset: 1

Author

Paolo P. Lava, May 29 2014

Keywords

• nonn
• base
• fini
• full

Comments

Numbers with two consecutive zeros are not considered, to avoid the case 0^0. Nevertheless, even if we consider 0^0=1 the results do not change (at least up to m=10^8).

Examples

			For 1899 we have: 9^9 + 9^8 + 8^1 + 1^9 = 430467219.
Finally 430467219/1899 = 226681.
For 1954329 we have: 9^2 + 2^3 +3^4 + 4^5 + 5^9 + 9^1 + 1^9 = 1954329.
		

Crossrefs

Cf. A243023, A243025.

Programs

Maple

with(numtheory): P:=proc(q) local a,b,k,ok,n; for n from 10 to q do a:=[]; b:=n;
while b>0 do a:=[op(a),b mod 10]; b:=trunc(b/10); od; b:=0; ok:=1; for k from 2 to nops(a)
do if a[k-1]=0 and a[k]=0 then ok:=0; break; else b:=b+a[k-1]^a[k]; fi; od;
if ok=1 then if type((b+a[nops(a)]^a[nops(1)])/n,integer) then print(n);
fi; fi; od; end: P(10^10);

PARI

isok(n) = d = digits(n); k = #d; (sum(i=1, k-1, j=k-i+1; d[j]^d[(j-1)])+ d[1]^d[k]) % n == 0; \\ Michel Marcus, Sep 29 2014

Extensions

a(1)-a(9) prepended and a(23) from Hiroaki Yamanouchi, Sep 29 2014