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

Offset: 1

Author

M. F. Hasler, Jan 03 2020

Keywords

• nonn

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.

Crossrefs

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

Programs

Mathematica

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

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])

Formula

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