A094949 Phi(m)*sigma(m), where m is the product of exactly two primes that differ by 2, where phi=A000010, sigma=A000203.

192, 1152, 20160, 103680, 806400, 3104640, 12945600, 26853120, 108201600, 136002240, 362597760, 506160000, 1049630400, 1358807040, 1536796800, 2702128320, 3317529600, 5314118400, 6323748480, 9475464960, 14665694400

Offset

1,1

Comments

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).

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Formula

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

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

a(n) = (A120875(n))^2 - 4 = 4*((A120876(n))^2 - 1). - Lekraj Beedassy, Jul 09 2006

Mathematica

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

Prog

(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)}

Crossrefs

Cf. A001359, A006512, A037074.

Sequence in context: A051527 A305073 A101451 * A205768 A205761 A346251

Adjacent sequences: A094946 A094947 A094948 * A094950 A094951 A094952

Keyword

nonn

Author

Lekraj Beedassy, Jun 19 2004

Extensions

Corrected and extended by Jason Earls, Rick L. Shepherd, Vladeta Jovovic and Klaus Brockhaus, Jun 20 2004

Status

approved