A235921 -id:A235921 - cp's OEIS frontend

A053669 Smallest prime not dividing n.

2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2

Offset: 1

Henry Bottomley, Feb 15 2000

Smallest prime coprime to n.
Smallest k >= 2 coprime to n.
a(#(p-1)) = a(A034386(p-1)) = p is the first appearance of prime p in sequence.
a(A005408(n)) = 2; for n > 2: a(n) = A112484(n,1). - Reinhard Zumkeller, Sep 23 2011
Average value is 2.920050977316134... = A249270. - Charles R Greathouse IV, Nov 02 2013
Differs from A236454, "smallest number not dividing n^2", for the first time at n=210, where a(210)=11 while A236454(210)=8. A235921 lists all n for which a(n) differs from A236454. - Antti Karttunen, Jan 26 2014
For k >= 0, a(A002110(k)) is the first occurrence of p = prime(k+1). Thereafter p occurs whenever A007947(n) = A002110(k). Thus every prime appears in this sequence infinitely many times. - David James Sycamore, Dec 04 2024

			a(60) = 7, since all primes smaller than 7 divide 60 but 7 does not.
a(90) = a(120) = a(150) = a(180) = 7 because 90,120,150,180 all have same squarefree kernel = 30 = A002110(3), and 7 is the smallest prime which does not divide 30. - _David James Sycamore_, Dec 04 2024

T. D. Noe, Table of n, a(n) for n = 1..10000
Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, A Prime-Representing Constant, The American Mathematical Monthly, Vol. 126, No. 1 (2019), pp. 70-73; ResearchGate link.
James Grime and Brady Haran, 2.920050977316, Numberphile video (2020)
Igor Rivin, Geodesics with one self-intersection, and other stories, Advances in Mathematics, Vol. 231, No. 5 (2012), pp. 2391-2412; arXiv preprint, arXiv:0901.2543 [math.GT], 2009-2011.

One more than A071222(n-1).
Cf. A000040, A020639, A053670, A053671, A053672, A053673, A053674, A055874, A079578, A087560, A096014, A235921, A236454, A249270, A257993 (the indices of these primes), A276086, A324895, A353526, A353528, A353529, A358754, A358755, A370125.
Cf. A002110, A007947.

a053669 n = head $ dropWhile ((== 0) . (mod n)) a000040_list
-- Reinhard Zumkeller, Nov 11 2012

f:= proc(n) local p;
p:= 2;
while n mod p = 0 do p:= nextprime(p) od:
p
end proc:
map(f, [$1..100]); # Robert Israel, May 18 2016

Table[k := 1; While[Not[GCD[n, Prime[k]] == 1], k++ ]; Prime[k], {n, 1, 60}] (* Stefan Steinerberger, Apr 01 2006 *)
With[{prs=Prime[Range[10]]},Flatten[Table[Select[prs,!Divisible[ n,#]&,1],{n,110}]]] (* Harvey P. Dale, May 03 2012 *)

a(n)=forprime(p=2,,if(n%p,return(p))) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import nextprime
def a(n):
    p = 2
    while True:
        if n%p: return p
        else: p=nextprime(p) # Indranil Ghosh, May 12 2017

# using standard library functions only
import math
def a(n):
    k = 2
    while math.gcd(n,k) > 1: k += 1
    return k # Ely Golden, Nov 26 2020

(define (A053669 n) (let loop ((i 1)) (cond ((zero? (modulo n (A000040 i))) (loop (+ i 1))) (else (A000040 i))))) ;; Antti Karttunen, Jan 26 2014

a(n) = A071222(n-1)+1. [Because the right hand side computes the smallest k >= 2 such that gcd(n,k) = gcd(n-1,k-1) which is equal to the smallest k >= 2 coprime to n] - Antti Karttunen, Jan 26 2014
a(n) = 1 + Sum_{k=1..n}(floor((n^k)/k!)-floor(((n^k)-1)/k!)) = 2 + Sum_{k=1..n} A001223(k)*( floor(n/A002110(k))-floor((n-1)/A002110(k)) ). - Anthony Browne, May 11 2016
a(n!) = A151800(n). - Anthony Browne, May 11 2016
a(2k+1) = 2. - Bernard Schott, Jun 03 2019
Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = A249270. - Amiram Eldar, Oct 29 2020
a(n) = A000040(A257993(n)) = A020639(A276086(n)) = A276086(n) / A324895(n). - Antti Karttunen, Apr 24 2022
a(n) << log n. For every e > 0, there is some N such that for all n > N, a(n) < (1 + e)*log n. - Charles R Greathouse IV, Dec 03 2022
A007947(n) = A002110(k) ==> a(n) = prime(k+1). - David James Sycamore, Dec 04 2024

More terms from Andrew Gacek (andrew(AT)dgi.net), Feb 21 2000 and James Sellers, Feb 22 2000

Entry revised by David W. Wilson, Nov 25 2006

A055874 a(n) = largest m such that 1, 2, ..., m divide n.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2

Offset: 1

Leroy Quet, Jul 16 2000

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

			a(12) = 4 because 1, 2, 3, 4 divide 12, but 5 does not.

Antti Karttunen, Table of n, a(n) for n = 1..10080
Bakir Farhi, On the average asymptotic behavior of a certain type of sequences of integers, Integers, Vol. 9 (2009), pp. 555-567.

One less than A007978.

Cf. also A053669, A055881, A055926, A017593, A064859, A181062, A126800, A232098, A232099, A233284, A235918, A235921.

a055874 n = length $ takeWhile ((== 0) . (mod n)) [1..]
-- Reinhard Zumkeller, Feb 21 2012, Dec 09 2010

N:= 1000: # to get a(1) to a(N)
A:= Vector(N,1);
for m from 2 do
  Lm:= ilcm($1..m);
  if Lm > N then break fi;
  if Lm mod (m+1) = 0 then next fi;
  for k from 1 to floor(N/Lm) do
    A[k*Lm]:=m
  od
od:
convert(A,list); # Robert Israel, Nov 28 2014

a[n_] := Module[{m = 1}, While[Divisible[n, m++]]; m - 2]; Array[a, 100] (* Jean-François Alcover, Mar 07 2016 *)

a(n) = my(m = 1); while ((n % m) == 0, m++); m - 1; \\ Michel Marcus, Jan 17 2014

from itertools import count
def A055874(n):
    for m in count(1):
        if n % m:
            return m-1 # Chai Wah Wu, Jan 02 2022

(define (A055874 n) (let loop ((m 1)) (if (not (zero? (modulo n m))) (- m 1) (loop (+ 1 m))))) ;; Antti Karttunen, Nov 18 2013

a(n) = A007978(n) - 1. - Antti Karttunen, Jan 26 2014

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A064859 (Farhi, 2009). - Amiram Eldar, Jul 25 2022

A071222 Smallest k such that gcd(n,k) = gcd(n+1,k+1).

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2

Offset: 0

Benoit Cloitre, Jun 10 2002

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)

Clark Kimberling & Antti Karttunen, Table of n, a(n) for n = 0..10001 (Terms up to n=1000 from Kimberling)

One less than A053669(n+1).

Cf. also A007978, A055874, A235918, A235921, A249270.

a071222 n = head [k | k <- [1..], gcd (n + 1) (k + 1) == gcd n k]
-- Reinhard Zumkeller, Oct 01 2014

sgcd[n_]:=Module[{k=1},While[GCD[n,k]!=GCD[n+1,k+1],k++];k]; Array[sgcd,110] (* Harvey P. Dale, Jul 13 2012 *)

PARI
```
for(n=1,140,s=1; while(gcd(s,n)
				
```

(define (A071222 n) (let loop ((k 1)) (cond ((= (gcd n k) (gcd (+ n 1) (+ k 1))) k) (else (loop (+ 1 k)))))) ;; Antti Karttunen, Jan 26 2014

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A249270 - 1. - Amiram Eldar, Jul 26 2022

Added a(0)=1. - N. J. A. Sloane, Jan 19 2014

A235918 Largest m such that 1, 2, ..., m divide n^2.

Original entry on oeis.org

Offset: 1

Michel Marcus, Jan 17 2014

nonn

Note that a(n) is equal to A071222(n-1) = A053669(n)-1 for the first 209 values of n. The first difference occurs at n=210, where a(210)=7, while A071222(209)=10. A235921 lists all n where a(n) differs from A071222(n-1). (Note also that a(n) is equal to A071222(n+29) for n=1..179.) - [Comment revised by Antti Karttunen, Jan 26 2014 because of the changed definition of A235921 and newly inserted a(0)=1 term of A071222.]
See A055874 for a similar comment concerning the difference between A055874 and A232098.
Average value is 1.9124064... = sum_{n>=1} 1/A019554(A003418(n)). - Charles R Greathouse IV, Jan 24 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

One less than A236454.

Cf. also A055874, A053669, A055926, A071222, A232098, A235921.

a[n_] := Module[{m = 1}, While[Divisible[n^2, m++]]; m - 2]; Array[a, 100] (* Jean-François Alcover, Mar 07 2016 *)

a(n) = my(m = 1); while ((n^2 % m) == 0, m++); m - 1; \\ Michel Marcus, Jan 17 2014

a(n) = A055874(n^2).

a(n) = A236454(n)-1.

A236454 Smallest number not dividing n^2.

Original entry on oeis.org

Offset: 1

Antti Karttunen, Jan 26 2014

nonn

Differs from A053669, "smallest prime not dividing n", for the first time at n=210, where a(210)=8, while A053669(210)=11. A235921 lists all n for which a(n) differs from A053669(n).

Differs from A214720 at n=2, 210, 630, 1050, 1470, 1890, 2310,.... - R. J. Mathar, Mar 30 2014

Antti Karttunen, Table of n, a(n) for n = 1..10000

One more than A235918.
Cf. A003418, A019554.
Cf. also A000290, A007978, A053669, A235921.

A236454 := proc(n)
    for m from 2 do
        if modp(n^2,m) <> 0 then
            return m;
        end if;
    end do:
end proc:# R. J. Mathar, Mar 30 2014

Join[{2,3},Table[Complement[Range[n],Divisors[n^2]][[1]],{n,3,90}]] (* Harvey P. Dale, Mar 18 2018 *)

(define (A236454 n) (A007978 (A000290 n)))

a(n) = A007978(A000290(n)) = A007978(n^2).
a(n) = A235918(n)+1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} 1/A019554(A003418(k)) = 2.91240643793540602415... . - Amiram Eldar, Jan 14 2024

A236432 a(n) = (2n-1)*210; numbers which are 210 times an odd number.

Original entry on oeis.org

210, 630, 1050, 1470, 1890, 2310, 2730, 3150, 3570, 3990, 4410, 4830, 5250, 5670, 6090, 6510, 6930, 7350, 7770, 8190, 8610, 9030, 9450, 9870, 10290, 10710, 11130, 11550, 11970, 12390, 12810, 13230, 13650, 14070, 14490, 14910, 15330, 15750, 16170, 16590, 17010

Offset: 1

Antti Karttunen, Jan 25 2014

This is a subsequence of A235921, from which it differs for the first time at n = 1062348, where a(n) = ((2*1062348)-1)*210 = 446185950, while A235921(n) = 446185740.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for linear recurrences with constant coefficients, signature (2,-1).
Index entries for sequences which agree for a long time but are different

Cf. A235921.

A236432:=n->420*n - 210; seq(A236432(n), n=1..50); # Wesley Ivan Hurt, Mar 13 2014

Table[420 n - 210, {n, 50}] (* Wesley Ivan Hurt, Mar 13 2014 *)
LinearRecurrence[{2,-1},{210,630},41] (* Ray Chandler, Jul 14 2015 *)
210*Range[1,81,2] (* Harvey P. Dale, Apr 13 2020 *)

a(n) = (2n-1) * 210 = 420*n - 210.

For all n, A236454(a(n)) = 8, while A053669(a(n)) >= 11. [Cf. comments at A235921]

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A053669 Smallest prime not dividing n.

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A055874 a(n) = largest m such that 1, 2, ..., m divide n.

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A071222 Smallest k such that gcd(n,k) = gcd(n+1,k+1).

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A235918 Largest m such that 1, 2, ..., m divide n^2.

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A236454 Smallest number not dividing n^2.

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A236432 a(n) = (2n-1)*210; numbers which are 210 times an odd number.

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