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A330832 Numbers of the form p*q, where p is prime and q=(p^k-1)/(p-1) is also prime for some integer k>1.

%I A330832 #13 Jan 23 2020 16:17:19
%S A330832 6,14,39,62,155,254,3279,5219,16382,19607,70643,97655,208919,262142,
%T A330832 363023,402233,712979,1040603,1048574,1508597,2265383,2391483,4685519,
%U A330832 5207819,6728903,21243689,25239899,56328959,61035155,67977559,150508643,310747739,344964203
%N A330832 Numbers of the form p*q, where p is prime and q=(p^k-1)/(p-1) is also prime for some integer k>1.
%C A330832 Also numbers with power-spectral basis {q,p^k}. The equation q=(p^k-1)/(p-1) is equivalent to the decomposition of the identity q + p^k = pq + 1 in Z/pqZ, and it is now easily verified that {q,p^k} is the spectral basis of p*q, consisting of primes and powers.
%C A330832 The numbers p^(r^e)*q, where p, q, r are primes, and q=(p^(r^e)-1)/(p^(r^(e-1))-1), e>0, have power-spectral basis {q,p^(r^e)}. However, the primes q for e>1 are usually quite large, while e=1 is accessible. For example, the table in A003424 has 4738 entries with all primes q<10^12, but only 8 have y>1.
%H A330832 Walter Kehowski, <a href="/A330832/b330832.txt">Table of n, a(n) for n = 1..4731</a>
%F A330832 a(n) = A330833(n) * A330835(n).
%e A330832 a(5) = 5*(5^3-1)/(5-1) = 5*31 = 155. The number 155 has spectral basis {31,125}.
%Y A330832 Cf. A003424, A023194, A023195, A085104, A330833, A330834, A330835.
%K A330832 nonn,easy
%O A330832 1,1
%A A330832 _Walter Kehowski_, Jan 08 2020