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

Author

Amiram Eldar, Oct 11 2024

Keywords

• nonn
• easy

Comments

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.

Examples

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

Links

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

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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).