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COMMENTS
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The PARI program below is a generalization of this type of sequence. These numbers are rare. Are they finite? Proof?
No more terms between 217 and 3090000. - R. J. Mathar, Feb 14 2008
No more terms < 50000000 (~40 minutes computation time). - Manfred Scheucher, Jul 25 2015
No more terms < 100000000. Conjecture: sequence is full. - Jon E. Schoenfield, Jul 25 2015
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PROG
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(PARI) \prime indices such that gd of prime(x)+ k and prime(x+m) + k are equal divpm1(n, m, k) = { local(x, l1, l2, v1, v2); for(x=2, n, v1 = ifactor(prime(x)+ k); v2 = ifactor(prime(x+m)+k); l1 = length(v1); l2 = length(v2); if(v1[l1] == v2[l2], print1(x", ") ) ) } ifactor(n) = \Vector of the prime factors of n { local(f, j, k, flist); flist=[]; f=Vec(factor(n)); for(j=1, length(f[1]), for(k = 1, f[2][j], flist = concat(flist, f[1][j]) ); ); return(flist) }
(PARI) gpf(n)=if(n>1, my(f=factor(n)[, 1]); f[#f], 1)
is(n, p=prime(n))=my(q=nextprime(p+1), g=gcd(p-1, q-1)); q\=g; p\=g; forprime(r=2, gpf(g), p/=r^valuation(p, r); q/=r^valuation(q, r)); p==1 && q==1
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