A243025 -id:A243025 - cp's OEIS frontend

A243023 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+1)^d(i)}+d(1)^d(k) (see example below).

1, 2, 3, 4, 5, 6, 7, 8, 9, 63, 448, 1547, 1693, 6189068, 20443796, 67526389

Offset: 1

Paolo P. Lava, May 29 2014

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 the last number I tested, that is m=10^8).

			For 63 we have 6^3 + 3^6 = 945 and 945/63 = 15.
Obviously also with 36 we have 3^6 + 6^3 = 945 but 945/36 = 105/4.
For 6189068 we have: 6^8 + 0^6 + 9^0 + 8^9 + 1^8 + 6^1 + 8^6 = 136159496.
Finally 136159496/6189068 = 22.

Cf. A243024, A243025.

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]^a[k-1]; fi; od;
if ok=1 then if type((b+a[1]^a[nops(a)])/n,integer) then print(n);
fi; fi; od; end: P(10^10);

fQ[n_] := Block[{id = IntegerDigits@ n}, IntegerQ[ Total[ (id^RotateLeft@ id)]/n]]; k = 1; lst = {}; While[k < 1000000001, If[fQ@k, AppendTo[lst, k]; Print@ k]; k++]; lst (* Robert G. Wilson v, Jun 01 2014 *)

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).

Original entry on oeis.org

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

Paolo P. Lava, May 29 2014

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).

			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.

Cf. A243023, A243025.

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);

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

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

cp's OEIS Frontend

A243023 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+1)^d(i)}+d(1)^d(k) (see example below).

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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).

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cp's OEIS Frontend

A243023 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+1)^d(i)}+d(1)^d(k) (see example below).

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Author

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Examples

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Programs

Maple

Mathematica

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).

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

PARI

Extensions

A243023 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+1)^d(i)}+d(1)^d(k) (see example below).

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).