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URL: https://oeis.org/A329570

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A329570
a(n) is the least prime P such that log(P)/log(p) >= valuation(n,p) for all primes p.
4
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
1,1
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
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
KEYWORD
nonn
AUTHOR
M. F. Hasler, Jan 03 2020
STATUS
approved