A094949 - OEIS

%I #19 Aug 28 2024 23:11:00

%S 192,1152,20160,103680,806400,3104640,12945600,26853120,108201600,

%T 136002240,362597760,506160000,1049630400,1358807040,1536796800,

%U 2702128320,3317529600,5314118400,6323748480,9475464960,14665694400

%N Phi(m)*sigma(m), where m is the product of exactly two primes that differ by 2, where phi=A000010, sigma=A000203.

%C If m=p*q for the twin prime pair (p, q), then the relation p^2 + q^2 = 2*(m+2) is evident from equations p*(p+2)=m=q*(q-2). Now phi(m)=(p-1)*(q-1)=p^2 - 1 and sigma(m)=(p+1)*(q+1)=q^2 - 1, so that phi(m)*sigma(m)=(p*q)^2 -(p^2 + q^2)+1=m^2-2*(m+2)+1=(m-3)*(m+1).

%H Harvey P. Dale, <a href="/A094949/b094949.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n)=(m-3)*(m+1), where m=A037074(n).

%F a(n)=192*A002415(k), where k=A040040(n-1).

%F a(n) = (A120875(n))^2 - 4 = 4*((A120876(n))^2 - 1). - _Lekraj Beedassy_, Jul 09 2006

%t EulerPhi[#]DivisorSigma[1,#]&/@Times@@@Select[Partition[Prime[ Range[ 200]],2,1],#[[2]]-#[[1]]==2&] (* _Harvey P. Dale_, Apr 13 2017 *)

%o (PARI) {m=400;p=1;while(p<m,p=nextprime(p);q=nextprime(p+1);if(p+2==q,r=p*q;print1(eulerphi(r)*sigma(r),","));p=q)}

%Y Cf. A001359, A006512, A037074.

%K nonn

%O 1,1

%A _Lekraj Beedassy_, Jun 19 2004

%E Corrected and extended by _Jason Earls_, _Rick L. Shepherd_, _Vladeta Jovovic_ and _Klaus Brockhaus_, Jun 20 2004