cp's OEIS Frontend

A160453 Numbers k which have a prime divisor p such that 1 is the only positive integer which divides k/p^m and is congruent to 1 modulo p, where p^(m+1) does not divide k.

%I A160453 #16 Jun 10 2025 11:00:03
%S A160453 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,19,20,21,22,23,25,26,27,28,
%T A160453 29,31,32,33,34,35,37,38,39,40,41,42,43,44,45,46,47,49,50,51,52,53,54,
%U A160453 55,57,58,59,61,62,63,64,65,66,67,68,69,70,71,73,74,75,76,77,78
%N A160453 Numbers k which have a prime divisor p such that 1 is the only positive integer which divides k/p^m and is congruent to 1 modulo p, where p^(m+1) does not divide k.
%C A160453 The solvability of a group whose order is a(n) can be reduced to the solvability of smaller group using the Sylow theorems, provided the order is not a prime.
%C A160453 80 is not a member of this sequence, but is a member of A168186. - _Franklin T. Adams-Watters_, Jan 26 2010
%H A160453 Charles R Greathouse IV, <a href="/A160453/b160453.txt">Table of n, a(n) for n = 1..10000</a>
%o A160453 (PARI) is(k)=if(k<12,return(k>0));my(f=factor(k)); for(i=1,#f~, fordiv(k/f[i,1]^f[i,2],d, if(d>1&&d%f[i,1]==1,next(2))); return(1)); 0 \\ _Charles R Greathouse IV_, Oct 27 2013
%Y A160453 All three of A023805, A160453, A168186 are different.
%K A160453 nonn
%O A160453 1,2
%A A160453 _Masahiko Shin_, May 14 2009
%E A160453 Corrected by _Charles R Greathouse IV_, Oct 27 2013