A357481 - OEIS

%I #23 Oct 06 2022 04:30:54

%S 2,-1,-1,-1,-1,2,-1,6,6,8,-1,10,-1,12,12,14,-1,16,-1,18,18,20,-1,22,

%T 20,24,24,26,-1,28,-1,30,30,32,30,34,-1,36,36,38,-1,40,-1,42,42,44,-1,

%U 3,42,3,48,2,-1,52,50,54,54,56,-1,58,-1,60,2,62,60,7,-1,66,66,68,-1,70,-1,72,7

%N a(n) is the least integer b such that the digit representation of n in base b is equal to the digit representation in base b of the initial terms of the sets of divisors of n in increasing order, or -1 if no such b exists.

%C It appears that a(n) != -1 iff n in A056653.

%H Michel Marcus, <a href="/A357481/b357481.txt">Table of n, a(n) for n = 1..10000</a>

%H Peter Munn, <a href="/plot2a?name1=A000027&amp;name2=A357481&amp;tform1=asinh&amp;tform2=asinh&amp;shift=0&amp;radiop1=xy&amp;drawpoints=true">Scatterplot with asinh-scale axes</a>. (Axis labeled "A000027(n)" is n.)

%e In base 10, 12 is a term of A175252, and there is no other lesser base for which 12 works, so a(12) = 10.

%o (PARI) isok(k, b) = my(s=[]); fordiv(k, d, s=concat(s, digits(d, b)); if (fromdigits(s, b)==k, return(1)); if (fromdigits(s, b)> k, return(0)));

%o a(n) = if (n==1, 2, for (b=2, n-1, if (isok(n, b), return(b))); return(-1););

%o (Python)

%o from sympy import divisors

%o from sympy.ntheory import digits

%o def ok(n, b):

%o     target, s = digits(n, b)[1:], []

%o     if target[0] != 1: return False

%o     for d in divisors(n):

%o         s += digits(d, b)[1:]

%o         if len(s) >= len(target): return s == target

%o def a(n):

%o     if n == 1: return 2

%o     for b in range(2, n):

%o         if ok(n, b): return b

%o     return -1

%o print([a(n) for n in range(1, 76)]) # _Michael S. Branicky_, Oct 05 2022

%Y Cf. A056653, A175252, A357428, A357429.

%K sign,base

%O 1,1

%A _Michel Marcus_, Sep 30 2022