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

1,2

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.

Links

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

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).

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A078380 A062356 A257993 * A055881 A332202 A204917

Adjacent sequences: A376925 A376926 A376927 * A376929 A376930 A376931

Keyword

nonn,easy

Author

Amiram Eldar, Oct 11 2024

Status

approved