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.

6, 14, 39, 62, 155, 254, 3279, 5219, 16382, 19607, 70643, 97655, 208919, 262142, 363023, 402233, 712979, 1040603, 1048574, 1508597, 2265383, 2391483, 4685519, 5207819, 6728903, 21243689, 25239899, 56328959, 61035155, 67977559, 150508643, 310747739, 344964203

Offset: 1

Author

Walter Kehowski, Jan 08 2020

Keywords

• nonn
• easy

Comments

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.

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.

Examples

			a(5) = 5*(5^3-1)/(5-1) = 5*31 = 155. The number 155 has spectral basis {31,125}.
		

Links

• Walter Kehowski, Table of n, a(n) for n = 1..4731

Crossrefs

Cf. A003424, A023194, A023195, A085104, A330833, A330834, A330835.

Formula

a(n) = A330833(n) * A330835(n).