A330834 -id:A330834 - cp's OEIS frontend

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

Walter Kehowski, Jan 08 2020

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.

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

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

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

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

A330833 a(n) = first prime factor p of the term A330832(n) = p*q.

Original entry on oeis.org

2, 2, 3, 2, 5, 2, 3, 17, 2, 7, 41, 5, 59, 2, 71, 13, 89, 101, 2, 17, 131, 3, 167, 173, 23, 29, 293, 383, 5, 13, 43, 677, 701, 743, 17, 761, 773, 827, 839, 857, 911, 1091, 1097, 5, 1163, 1181, 1193, 1217, 73, 1373, 1427, 79, 1487, 1559, 1583, 83, 2, 1709, 1811, 1847, 1931, 1973, 2129, 2273, 2309, 2339, 2411, 2663, 2729, 2789, 2957

Offset: 1

Walter Kehowski, Jan 08 2020

			a(5) = 5 and, since A330834(5) = 3, then A330835(5) = (5^3-1)/(5-1) = 31 is prime.

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

Cf. A003424, A023194, A023195, A085104, A330832, A330834, A330835.

A330835 Primes q appearing in A330832: that is, if A330832(n)=p*q, where p is prime and q=(p^k-1)/(p-1) is prime, then a(n)=q.

Original entry on oeis.org

3, 7, 13, 31, 31, 127, 1093, 307, 8191, 2801, 1723, 19531, 3541, 131071, 5113, 30941, 8011, 10303, 524287, 88741, 17293, 797161, 28057, 30103, 292561, 732541, 86143, 147073, 12207031, 5229043, 3500201, 459007, 492103, 552793, 25646167, 579883, 598303, 684757

Offset: 1

Walter Kehowski, Jan 08 2020

nonn

The terms in the b-file are the same as those of A003424 with y=1, but with an ordering based on that of A330832. The ordering allows the inclusion of the only duplicate 2^5-1=31 and (5^3-1)/(5-1)=31.

			a(5)=31 since A330833(5)=5, A330834(5)=3, and (5^3-1)/(5-1) = 31 is prime.

Walter Kehowski, Table of n, a(n) for n = 1..4731
G. Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly 108(4), April 2001.

Cf. A003424, A023194, A023195, A085104, A330832, A330833, A330834.

a(n) = (A330833(n) ^ A330834(n) - 1) / (A330833(n) - 1).

cp's OEIS Frontend

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.

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A330833 a(n) = first prime factor p of the term A330832(n) = p*q.

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A330835 Primes q appearing in A330832: that is, if A330832(n)=p*q, where p is prime and q=(p^k-1)/(p-1) is prime, then a(n)=q.

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