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Composites k such that p-1 divides k-1 for each prime divisor of k-1.
2

%I #23 Feb 26 2024 15:43:07

%S 9,21,25,33,49,55,65,81,85,121,129,145,161,169,201,217,221,253,273,

%T 289,295,301,321,325,343,361,385,441,469,481,501,505,513,545,589,625,

%U 649,685,721,781,801,817,841,865,901,961,973,1001,1025,1027,1081,1089,1101,1135,1177,1261,1281,1333,1345,1369,1405,1441,1501

%N Composites k such that p-1 divides k-1 for each prime divisor of k-1.

%C Composites not in A272601.

%H Charles R Greathouse IV, <a href="/A272600/b272600.txt">Table of n, a(n) for n = 1..10000</a>

%t pdkQ[n_]:=Module[{p=FactorInteger[n-1][[;;,1]]-1},AllTrue[(n-1)/p,IntegerQ]]; Select[Range[ 1550],CompositeQ[#]&&pdkQ[#]&] (* _Harvey P. Dale_, Feb 26 2024 *)

%o (PARI) forcomposite(n=4, 10^4, my(q=1, f=factor(n-1)[,1]); for(j=1, #f, if((n-1)%(f[j]-1), q=0; break)); if(q, print1(n,", ") ) );

%o (PARI) is(n)=my(f=factor(n-1)[,1]); for(i=1,#f, if((n-1)%(f[i]-1), return(0))); !isprime(n) && n>1 \\ _Charles R Greathouse IV_, May 16 2016

%Y Cf. A080062 (composite n such that p-1 divides n-1 for each prime divisor of n).

%K nonn

%O 1,1

%A _Joerg Arndt_, May 16 2016