Composites Of This Sequence

Maximilian Hasler maximilian.hasler at gmail.com
Mon Aug 25 16:59:21 CEST 2008

```On Sun, Aug 24, 2008 at 18:24, Alexander Povolotsky <apovolot at gmail.com> wrote:
> Please note that the terms given in newly submitted by you A143578 are
> coinciding (coincidentally ?) with the terms in A097605.
> A097605          Numbers n such that phi(n) divides n^2 - 1, where phi(n)
> (A000010) is number of positive integers <= n and coprime to n.
>        1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47,
> 53, 59, 61, 65, 67, 71, 73, 79, 83, 89, 91, 97, 101, 103, 107, 109,
> 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191,
> 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 255, 257

I don't agree, e.g. 65 is in the above but not in A143578 :

isA143578(n)={ local( d=divisors(n), j=(1+#d)\2 , r=d[ j ]+d[ 1+#d-j ]);
for( k=1, j, ( d[k]+d[#d+1-k] ) % r & return ); 1 }

for(n=1,300,isA143578(n) && print1(n","))
1,2,3,5,7,11,13,15,17,19,23,29,31,35,37,41,43,47,53,59,61,67,71,73,79,83,89,95,97,101,
103,107,109,113,119,127,131,137,139,143,149,151,157,163,167,173,179,181,191,193,
197,199,209,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,287,293

Below the factorization of the first composites in A143578 :

for(n=1,3000, bigomega(n)>1 | next; isA143578(n) | next;
print1(factor(n)[,1]~,","))

[3, 5],[5, 7],[5, 19],[7, 17],[11, 13],[11, 19],[7, 41],[11, 29],[17,
19],[13, 29],[17, 31],[13, 43],
[19, 41],[29, 31],[13, 71],[23, 43],[19, 53],[29, 41],[11, 109],[17,
79],[19, 71],[41, 43],
[19, 101],[17, 127],[23, 109],[31, 89],[41, 71],[41, 79],[59, 61],[43,
89],[29, 139],[17, 271]

Maximilian

```

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