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Composite numbers k such that sigma(k) - 1 is prime.
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%I #44 Jan 13 2025 19:07:35

%S 6,10,14,15,20,21,24,26,30,33,34,35,38,40,44,46,51,52,55,57,58,60,63,

%T 65,74,76,78,84,85,86,88,90,92,93,96,105,111,114,117,118,120,123,124,

%U 126,130,135,136,141,143,145,147,153,155,158,161,164,166,168,172,174

%N Composite numbers k such that sigma(k) - 1 is prime.

%C Composite numbers k such that sigma(k) = sigma(p) has a solution in the primes p. - _Jaroslav Krizek_, Feb 03 2012

%C Complement of A000040 (primes) with respect to A248792 (numbers n such that sigma(n) - 1 is prime). - _Jaroslav Krizek_, Nov 13 2014

%C Numbers k such that sigma(k) - 1 is greater than k and prime. - _Giuseppe Coppoletta_, Dec 22 2014

%H Amiram Eldar, <a href="/A066073/b066073.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harry J. Smith)

%e 30, 46, 51, and 55 are in the sequence because each is a composite number n such that sigma(n)-1 = 71, which is prime; 71 itself is excluded from the sequence by definition.

%p filter:= proc(n)

%p local s;

%p s:= numtheory:-sigma(n)-1;

%p s > n and isprime(s);

%p end proc:

%p select(filter, [$2..1000]); # _Robert Israel_, Dec 22 2014

%t Do[s=-1+DivisorSigma[1, m]; If[PrimeQ[s]&&!PrimeQ[m], Print[m]], {m, 1, 256}]

%t Select[Range[200],CompositeQ[#]&&PrimeQ[DivisorSigma[1,#]-1]&] (* _Harvey P. Dale_, Jan 13 2025 *)

%o (PARI) isA066073(n)=!isprime(n)&&isprime(sigma(n)-1) \\ _Charles R Greathouse IV_, Feb 20 2012

%o (Sage) [n for n in (2..174) if (sigma(n)-1).is_prime() and sigma(n)-1>n] # _Giuseppe Coppoletta_, Dec 22 2014

%Y Cf. A000040, A000203, A058340, A066071, A066072, A066073, A066074, A066075, A066076, A066077, A066080, A248792.

%K nonn,easy

%O 1,1

%A _Labos Elemer_, Dec 03 2001