A063427 -id:A063427 - cp's OEIS frontend

A333923 a(n) is the smallest positive integer such that n^a(n) is divisible by n+a(n).

2, 6, 4, 20, 3, 42, 8, 18, 6, 110, 4, 156, 14, 10, 16, 272, 6, 342, 5, 6, 10, 506, 3, 100, 6, 54, 4, 812, 6, 930, 32, 48, 30, 14, 12, 1332, 26, 42, 10, 1640, 6, 1806, 20, 30, 18, 2162, 6, 294, 14, 30, 12, 2756, 10, 66, 8, 24, 6, 3422, 4, 3660, 62, 18, 64, 60, 6, 4422

Offset: 2

Scott R. Shannon, Apr 10 2020

nonn

As in A063427, if n is a prime then a(n^k) = (n-1)*n^k for k>=1. This sequence also matches A063427 for numerous other nonprime terms for small values of n.

For n below 10000 the values where n = a(n), other than n being a power of 2, are n = 14, 62, 122, 254, 508, 1018, 2038, 2042, 8182, 8186.

			a(2) = 2 as 2 ^ 2 = 4 is divisible by 2 + 2 = 4.
a(3) = 6 as 3 ^ 6 = 729 is divisible by 3 + 6 = 9.
a(4) = 4 as 4 ^ 4 = 256 is divisible by 4 + 4 = 8.
a(5) = 20 as 5 ^ 20 = 95367431640625 is divisible by 5 + 20 = 25.

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A063427, A333811, A332795, A332703.

spi[n_]:=Module[{k=1},While[PowerMod[n,k,n+k]!=0,k++];k]; Array[spi,70,2] (* Harvey P. Dale, Jan 16 2022 *)

A235268 Least integer k > n such that n*k/(n+k) is an integer, or 0 if no such k exists.

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 12, 42, 24, 18, 15, 110, 24, 156, 35, 30, 48, 272, 36, 342, 30, 28, 99, 506, 40, 100, 143, 54, 70, 812, 45, 930, 96, 66, 255, 140, 45, 1332, 323, 78, 60, 1640, 56, 1806, 77, 90, 483, 2162, 80, 294, 75, 102, 117, 2756, 108, 66, 140, 114, 783

Offset: 0

Alex Ratushnyak, Jan 05 2014

nonn

			a(3) = 6 because 6 is the smallest k > 3 such that k*3/(k+3) is an integer.

Giovanni Resta, Table of n, a(n) for n = 0..1000

Cf. A063427.

a[0]=1; a[n_] := Block[{k,s,x}, s = Reduce[k*n/(k+n) == x && k>n, {k,x}, Integers]; If[s === False, 0, Min[k /. List@ ToRules@s]]]; a/@Range[0,100] (* Giovanni Resta, Jan 20 2014 *)

a(n)=my(k=n+1);while((n*k)%(n+k)!=0,k=k+1);k \\ Ralf Stephan, Jan 15 2014

For prime p, a(p) = p*(p-1) = A002378(p-1). - Ralf Stephan, Jan 15 2014

A252669 a(n) is the smallest integer k such that n*k mod (n+k) = 1, or -1 if no such k exists.

Original entry on oeis.org

1, 3, 2, 13, 8, 31, 3, 5, 32, 91, 50, 17, 4, 183, 98, 241, 12, 7, 162, 381, 5, 75, 30, 553, 288, 651, 46, 129, 392, 23, 6, 9, 76, 55, 578, 1261, 100, 47, 722, 1561, 17, 311, 7, 105, 968, 27, 18, 413, 1152, 11, 1250, 489, 228, 2863, 34, 3081, 8, 615, 1682, 217, 1800, 707

Offset: 1

Alex Ratushnyak, Jan 02 2015

nonn

Cf. A063427.

sik[n_]:=Module[{k=1},While[Mod[n*k,n+k]!=1,k++];k]; Array[sik,70] (* Harvey P. Dale, Aug 15 2015 *)

a(n) = k=1; while ((n*k) % (n+k) != 1, k++); k; \\ Michel Marcus, Jan 08 2015

import math
for n in range(1,333):
    res=-1
    for k in range(2**31-1):
        if ((n*k) % (n+k) == 1):
            res=k
            break
    print(res, end=', ')

a(a(n)) <= n.

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A333923 a(n) is the smallest positive integer such that n^a(n) is divisible by n+a(n).

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A235268 Least integer k > n such that n*k/(n+k) is an integer, or 0 if no such k exists.

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A252669 a(n) is the smallest integer k such that n*k mod (n+k) = 1, or -1 if no such k exists.

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