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A339464
a(n) = (prime(n)-1) / gpf(prime(n)-1) where gpf(m) is the greatest prime factor of m, A006530.
4
1, 2, 2, 2, 4, 8, 6, 2, 4, 6, 12, 8, 6, 2, 4, 2, 12, 6, 10, 24, 6, 2, 8, 32, 20, 6, 2, 36, 16, 18, 10, 8, 6, 4, 30, 12, 54, 2, 4, 2, 36, 10, 64, 28, 18, 30, 6, 2, 12, 8, 14, 48, 50, 128, 2, 4, 54, 12, 40, 6, 4, 18, 10, 24, 4, 30, 48, 2, 12, 32, 2, 6, 12, 54, 2
OFFSET
2,2
COMMENTS
Paul Erdős asked if there are infinitely many primes p such that (p-1)/A006530(p-1) = 2^k or = 2^q*3^r (see Richard K. Guy reference).
A074781 is the sequence of primes p such that (p-1)/A006530(p-1) = 2^k.
A339465 is the sequence of primes p such that (p-1)/A006530(p-1) = 2^q*3^r with q, r >=1.
It is not known if these two sequences are infinite.
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B46, p. 154.
LINKS
MacTutor History of Mathematics, Paul Erdős.
FORMULA
a(n) = A006093(n)/A006530(A006093(n)).
a(n) = A052126(A006093(n)). - Michel Marcus, Dec 07 2020
EXAMPLE
Prime(6) = 13 and a(6) = 12/3 = 4 = 2^2.
Prime(11) = 31 and a(11) = 30/5 = 6 = 2*3.
Prime(20) = 71 and a(20) = 70/7 =10 = 2*5.
Prime(36) = 151 and a(36) = 150/5 = 30 = 2*3*5.
MATHEMATICA
f[n_] := n/FactorInteger[n][[-1, 1]]; f /@ (Select[Range[3, 400], PrimeQ] - 1) (* Amiram Eldar, Dec 07 2020 *)
PROG
(PARI) gpf(n) = vecmax(factor(n)[, 1]); \\ A006530
a(n) = my(x=prime(n)-1); x/gpf(x); \\ Michel Marcus, Dec 07 2020
CROSSREFS
Cf. A006093 (prime(n)-1), A006530, A052126, A074781 (ratio = 2^k), A339465 (ratio = 2^q*3^r), A339466 (ratio <> 2^k and <> 2^q*3^r).
Sequence in context: A244458 A286613 A229061 * A278224 A161421 A306337
KEYWORD
nonn
AUTHOR
Bernard Schott, Dec 06 2020
STATUS
approved