cp's OEIS Frontend

a(n) = floor(n/m), where m is the least positive nondivisor of n, as in A007978.

If n is odd, a(n) = (n+1)/2. - Robert Israel, Oct 09 2016

Beginning at n=3, a(n) represents the maximum length of consecutive numbers that are divisible by the product of their nonzero digits in base n. In particular, if n=10, the sequence of numbers that are divisible by the product of their nonzero digits is given by A055471.

Also, the largest number of distinct integers such that all their pairwise differences are coprime to n. - Max Alekseyev, Mar 17 2006

The unit 1 is not a prime number (although it has been considered so in the past). 1 is the empty product of prime numbers, thus 1 has no least prime factor. - Daniel Forgues, Jul 05 2011

a(n) = least m > 0 for which n! + m and n - m are not relatively prime. - Clark Kimberling, Jul 21 2012

For n > 1, a(n) = the smallest k > 1 that divides n. - Antti Karttunen, Feb 01 2014

For n > 1, records are at prime indices. - Zak Seidov, Apr 29 2015

The initials "lpf" might be mistaken for "largest prime factor" (A009190), using "spf" for "smallest prime factor" would avoid this. - M. F. Hasler, Jul 29 2015

n = 89 is the first index > 1 for which a(n) differs from the smallest k > 1 such that (2^k + n - 2)/k is an integer. - M. F. Hasler, Aug 11 2015

From Stanislav Sykora, Jul 29 2017: (Start)

For n > 1, a(n) is also the smallest k, 1 < k <= n, for which the binomial(n,k) is not divisible by n.

Proof: (A) When k and n are relatively prime then binomial(n,k) is divisible by n because k*binomial(n,k) = n*binomial(n-1,k-1). (B) When gcd(n,k) > 1, one of its prime factors is the smallest; let us denote it p, p <= k, and consider the binomial(n,p) = (1/p!)*Product_{i=0..p-1} (n-i). Since p is a divisor of n, it cannot be a divisor of any of the remaining numerator factors. It follows that, denoting as e the largest e > 0 such that p^e|n, the numerator is divisible by p^e but not by p^(e+1). Hence, the binomial is divisible by p^(e-1) but not by p^e and therefore not divisible by n. Applying (A), (B) to all considered values of k completes the proof. (End)

From Bob Selcoe, Oct 11 2017, edited by M. F. Hasler, Nov 06 2017: (Start)

a(n) = prime(j) when n == J (mod A002110(j)), n, j >= 1, where J is the set of numbers <= A002110(j) with smallest prime factor = prime(j). The number of terms in J is A005867(j-1). So:

a(n) = 2 when n == 0 (mod 2);

a(n) = 3 when n == 3 (mod 6);

a(n) = 5 when n == 5 or 25 (mod 30);

a(n) = 7 when n == 7, 49, 77, 91, 119, 133, 161 or 203 (mod 210);

etc. (End)

For n > 1, a(n) is the leftmost term, other than 0 or 1, in the n-th row of A127093. - Davis Smith, Mar 05 2019

From Antti Karttunen, Nov 20 2013 & Jan 26 2014: (Start)

Differs from A232098 for the first time at n=840, where a(840)=8, while A232098(840)=7. A232099 gives all the differing positions. See also the comments at A055926 and A232099.

The positions where a(n) is an odd prime is given by A017593 up to A017593(34)=414 (so far all 3's), after which comes the first 7 at a(420). (A017593 gives the positions of 3's.)

(Continued on Jan 26 2014):

Only terms of A181062 occur as values.

A235921 gives such n where a(n^2) (= A235918(n)) differs from A071222(n-1) (= A053669(n)-1). (End)

a(n) is the largest m such that A003418(m) divides n. - David W. Wilson, Nov 20 2014

a(n) is the largest number of consecutive integers dividing n. - David W. Wilson, Nov 20 2014

A051451 gives indices where record values occur. - Gionata Neri, Oct 17 2015

Yuri Matiyasevich calls this the maximum inheritable divisor of n. - N. J. A. Sloane, Dec 14 2023

This sequence is a member of an interesting class of sequences defined by the rule, "Smallest k such that b^^n is not congruent to b^^(n-1) mod k, where b^^n denotes the power tower b^b^...^b (in which b appears n times)," for some constant b. Different choices for b, with b>=2, determine a sequence of this class. The first sequence of this class is A027763, which uses b=2. This is the sequence for b=3. Adjacent sequences follow for b=4 through b=9.

Sequences of this class can contain only terms that are either prime numbers or powers of primes (A000961).

Powers of three (A000244) are commonplace as terms in sequences of this class, occurring significantly more often than powers of other primes.

For successive values of b, the sequence of first terms of all sequences of this class is A007978.

It appears that the sequence for b=x contains the term y, if and only if the sequence for b=x+A003418(y) also contains the term y, where A003418(y) is the least common multiple of all the integers from 1 to y. Can this be proved?

Sequences of this class generally share many terms with other sequences of this class, although each appears to be unique. These individual sequences are very similar to each other, so much so that among the first 7000 of them, only 120 distinct values less than 50000 occur (compare that to the 5217 available primes or powers of primes that are less than 50000.)

Sequences of this class tend to share many terms with A056637, the least prime of class n-.

a(n) = least m>0 such that gcd(n!+1+m,n-m) = 1. [Clark Kimberling, Jul 21 2012]

From Antti Karttunen, Jan 26 2014: (Start)

a(n-1)+1 = A053669(n) = Smallest k >= 2 coprime to n = Smallest prime not dividing n.

Note that a(n) is equal to A235918(n+1) for the first 209 values of n. The first difference occurs at n=210 and A235921 lists the integers n for which a(n) differs from A235918(n+1).

(End)