A329570 - OEIS

A329570

a(n) is the least prime P such that log(P)/log(p) >= valuation(n,p) for all primes p.

2, 2, 3, 5, 5, 3, 7, 11, 11, 5, 11, 5, 13, 7, 5, 17, 17, 11, 19, 5, 7, 11, 23, 11, 29, 13, 29, 7, 29, 5, 31, 37, 11, 17, 7, 11, 37, 19, 13, 11, 41, 7, 43, 11, 11, 23, 47, 17, 53, 29, 17, 13, 53, 29, 11, 11, 19, 29, 59, 5, 61, 31, 11, 67, 13, 11, 67, 17, 23, 7, 71, 11, 73, 37, 29, 19, 11

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OFFSET

1,1

COMMENTS

Related to the inequality (54) in Ramanujan's paper about highly composite numbers A002182, also used in A199337: This is the largest prime factor of the bound A329571(n)^2 above which all highly composite numbers are divisible by n.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Srinivasa Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, Ser. 2, Vol. XIV, No. 1 (1915): 347-409. A variant of better quality with an additional footnote is available at Ramanujan Papers.

FORMULA

a(n) = A007918(A034699(n)). - Amiram Eldar, Jan 17 2025

MATHEMATICA

a[n_] := NextPrime[Max[Power @@@ FactorInteger[n]] - 1]; a[1] = 2; Array[a, 100] (* Amiram Eldar, Jan 17 2025 *)

PROG

(PARI) apply( {A329570(n, f=Col(factor(max(n, 2))), P=nextprime(vecmax([log(f[1])*f[2] | f<-f])))=[while( logint(P, f[1]) < f[2], P=nextprime(P+1)) | f<-f]; P}, [1..99])

CROSSREFS

Cf. A002182, A007918, A034699, A199337, A329571.

Sequence in context: A330404 A340575 A139171 * A279724 A050174 A115126

Adjacent sequences: A329567 A329568 A329569 * A329571 A329572 A329573

KEYWORD

nonn

AUTHOR

M. F. Hasler, Jan 03 2020

STATUS

approved