A053589 -id:A053589 - cp's OEIS frontend

A376928 The largest noncomposite number k such that n is divisible by all the primes that do not exceed k.

1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 5, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 5, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 5, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1

Offset: 1

Amiram Eldar, Oct 11 2024

First differs from A062356 and A257993 at n = 30.
The least index n such that a(n) = prime(k) is A002110(k).
Let p be a prime and prev(p) = A151799(p) if p >= 3, and prev(2) = 1 (i.e., prev(p) is the largest noncomposite number that is smaller than p). Then, the asymptotic density of the occurrences of prev(p) in this sequence is 1/prev(p)# - 1/p#, where # denotes primorial (second definition, A034386). For example, the asymptotic densities of the occurrences of 1, 2, 3, 5 and 7 are 1/2, 1/3, 2/15, 1/35 and 1/231, respectively.

			a(30) = 5 since 30 is divisible by all the primes <= 5, i.e., by 2, 3 and 5.

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

Cf. A000040, A000720, A002110, A006530, A034386, A053589, A053669, A055881, A055932, A062356, A151799, A257993, A276084, A377010.

a[n_] := Module[{p = 1}, While[Divisible[n, p], p = NextPrime[p]]; If[p > 2, NextPrime[p, -1], 1]]; Array[a, 100]

a(n) = {my(p = 1); while(!(n % p), p = nextprime(p+1)); if(p > 2, precprime(p-1), 1);}

a(n) = 1 if and only if n is odd.
a(n) = gpf(n) = A006530(n) if and only if n is in A055932.
a(n) = prime(A276084(n)) = A000040(A276084(n)) if A276084(n) > 0, and 1 otherwise.
primepi(a(n)) = A000720(a(n)) = A276084(n).
A034386(a(n)) = A053589(n).
a(n) = prev(A053669(n)), where prev(p) is defined in the Comments section.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} prev(p) * (1/prev(p)# - 1/p#) = 1.744663405017... (A377010).

A343783 a(n) is the largest primorial number (A002110) which divides phi(n).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 6, 2, 6, 2, 2, 2, 6, 6, 2, 2, 2, 6, 6, 2, 6, 2, 2, 2, 2, 6, 6, 6, 2, 2, 30, 2, 2, 2, 6, 6, 6, 6, 6, 2, 2, 6, 6, 2, 6, 2, 2, 2, 6, 2, 2, 6, 2, 6, 2, 6, 6, 2, 2, 2, 30, 30, 6, 2, 6, 2, 6, 2, 2, 6, 2, 6, 6, 6, 2, 6, 30, 6, 6, 2, 6, 2, 2, 6, 2, 6

Offset: 1

Amiram Eldar, Apr 29 2021

nonn

			a(3) = 2 since phi(3) = 2 and 2 = A002110(1).
a(5) = 2 since phi(5) = 4 and 2 = A002110(1) is the largest primorial dividing 4.
a(7) = 6 since phi(7) = 6 and 6 = A002110(2).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack and Carl Pomerance, Phi, primorials, and Poisson, Illinois J. Math., Vol. 64, No. 3 (2020), pp. 319-330; alternative link.

Cf. A000010, A002110, A034386, A053589.

prim[n_] := Times @@ Prime[Range[n]]; gp[n_] := Module[{k = 1}, While[Divisible[n, prim[k]], k++]; prim[k - 1]]; a[n_] := gp[EulerPhi[n]]; Array[a, 100]

f(n) = my(s=1); forprime(p=2, , if(n%p, return(s), s *= p)); \\ A053589
a(n) = f(eulerphi(n)); \\ Michel Marcus, May 01 2021

a(n) = A053589(A000010(n)).
Let pr(n) be the largest prime divisor of a(n) (i.e., a(n) = pr(n)# = A034386(pr(n))). Then pr(n) ~ log(log(n))/log(log(log(n))) on a set of integers of asymptotic density 1 (Pollack and Pomerance, 2020).
From Bernard Schott, May 05 2021: (Start)
a(2n) = a(n) for n>=1.
a(n) = 1 iff n = 1 or n = 2.
a(n) = 2 iff 3 does not divide phi(n) (A088232)
a(n) >= 6 iff 3 divides phi(n) (A066498). (End)

cp's OEIS Frontend

A376928 The largest noncomposite number k such that n is divisible by all the primes that do not exceed k.

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