A071222 -id:A071222 - 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

A235921 Numbers n such that smallest number not dividing n^2 (A236454) is different from smallest prime not dividing n (A053669).

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 and Michel Marcus, Jan 17 2014

nonn

Equivalent definition is: numbers n such that {the largest m such that 1, 2, ..., m divide n^2 = A055874(n^2) = A235918(n)} is different from {the smallest k such that gcd(n-1,k) = gcd(n,k+1) = A071222(n-1)}.
All terms are multiples of 210 = 2*3*5*7, the fourth primorial, A002110(4).
The first term which is an even multiple of 210 (i.e., 210 times an even number), is 446185740 = 2124694 * 210 = 2*223092870 = 2*A002110(9) = 2*A034386(23). Note that 23 is the 9th prime, and 223092870 is its primorial. Thus this sequence differs from its subsequence, A236432, "the odd multiples of 210" = (2n-1)*210, for the first time at n = 1062348, where a(n) = 446185740, while A236432(n) = 446185950.
Note that a more comprehensive description for which terms are included is still lacking. Compare for example to the third definition of A055926.
At least we know the following:
If a number is not divisible by 210, then it cannot be a member, as then it is "missing" (i.e., not divisible by) one of those primes, 2, 3, 5 or 7, and thus its square is also "missing" the same prime. In more detail, this follows because:
If the least nondividing prime is 2, then A053669(n) = A236454(n) = 2. If the least nondividing prime is 3, then A053669(n) = A236454(n) = 3.
If the least nondividing prime is 5 (so 2 and 3 are present), then as 2|n and 4|(n^2), we have A053669(n) = A236454(n) = 5.
If the least nondividing prime is 7, but 2, 3 and 5 are present, then we have A053669(n) = A236454(n) = 7.
On the other hand, when n is an odd multiple of 210 (= 2*3*5*7), i.e., (2k+1)*210, so that its prime factorization is of the form 2*3*5*7*{zero or more additional odd prime factors}, then A053669(n) must be at least 11, the next prime after 7, which is certainly different from A236454(n) = A007978(n^2) which must be 8, as then 4 is the highest power of 2 dividing n^2.
In contrast to that, when n is an even multiple of 210, so that its prime factorization is of the form 2*2*3*5*7*{zero or more additional prime factors}, then also all the composites 8, 9, 10, 12, 14, 15, 16, 18 and 20 divide n^2, thus if A053669(n) is any prime from 11 to 19, A236454(n) will return the same result.
However, if n is of the form k*446185740, where k is not a multiple of 3, so that the prime factorization of n is 2*2*3*5*7*11*13*17*19*23*{zero or more additional prime factors, all different from 3}, then A053669(n) must be at least 29 (next prime after 23), but A236454(n) = 27, because then 9 is the highest power of 3 dividing n^2.
The pattern continues indefinitely: If n is of the form (2k+1)*2*3*200560490130, where 200560490130 = A002110(11), so that n has a prime factorization of the form 2*2*3*3*5*7*11*13*17*19*23*29*31*{zero or more additional odd prime factors}, then A053669(n) must be at least 37, while A236454(n) = 32 = 2^5, because then 16 is the highest power of 2 dividing n^2.

			210 (= 2*3*5*7) is a member, because A053669(210)=11, while A236454(210) = A007978(210*210) = A007978(44100) = 8.
446185740 (= 2*2*3*5*7*11*13*17*19*23) is a member, because A053669(446185740) = 29, while A236454(446185740) = 27, as there is only one 3 present in 446185740, which means that its square is only divisible by 9, but not by 27 = 3^3.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, C-program for computing the first few terms empirically
Index entries for sequences which agree for a long time but are different

Cf. A053669, A071222, A236454, A055874, A235918, A236432, A002110, A034386, A232099, A055926.

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

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

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.

A327571 Triangle T(n,k) read by rows giving the number of zeroless polydivisible numbers in base n that contains only "k" in the digits with 1 <= k <= n-1.

Original entry on oeis.org

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

Offset: 2

Seiichi Manyama, Sep 17 2019

			n | zeroless polydivisible numbers with all digits the same in base n
--+------------------------------------------------------------------
2 | [1]
3 | [1, 11], [2, 22]
4 | [1], [2, 22, 222], [3]
So T(2,1) = 1, T(3,1) = 2, T(3,2) = 2, T(4,1) = 1, T(4,2) = 3, T(4,3) = 1.
Triangle begins:
n\k  | 1  2  3  4  5  6  7  8  9 10 11 12
-----+------------------------------------
   2 | 1;
   3 | 2, 2;
   4 | 1, 3, 1;
   5 | 2, 2, 4, 2;
   6 | 1, 2, 1, 2, 1;
   7 | 4, 4, 4, 4, 6, 4;
   8 | 1, 2, 1, 2, 1, 3, 1;
   9 | 2, 2, 4, 2, 2, 4, 2, 2;
  10 | 1, 3, 1, 4, 1, 3, 1, 4, 1;
  11 | 2, 2, 6, 2, 2, 6, 2, 2, 6, 2;
  12 | 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1;
  13 | 4, 4, 4, 4, 6, 4, 4, 4, 4, 6, 4, 4;

Seiichi Manyama, Rows n = 2..141, flattened
Wikipedia, Polydivisible number.

Row sums give A327577.

Cf. A071222, A327545.

def T(k, n)
  s = 0
  (0..n - 2).each{|i|
    s += k * n ** i
    return i if s % (i + 1) > 0
  }
  n - 1
end
def A327571(n)
  (2..n).map{|i| (1..i - 1).map{|j| T(j, i)}}.flatten
end
p A327571(10)

T(n,1) = T(n,n-1) = A071222(n-2).
T(n,1) <= T(n,k).
T(n,2*m) >= 2 for m >= 1.

A213852 Least m>0 such that n+1+m and n-m are relatively prime.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 26 2012

a(n) > 1 for n == 1 mod 3, a(n) > 2 for n == 7 mod 15, a(n) > 3 for n == 52 mod 105, a(n) > 5 for n == 577 mod 1155, and so on, see A070826. - Ralf Stephan, Mar 16 2014

It appears that we get this sequence if we bisect A071222 and then divide by 2. - N. J. A. Sloane, May 17 2019

			gcd(9,6) = 3, gcd(10,5) = 5, gcd(11,4) = 1, so that a(7) = 3.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A071222, A195508.

Table[m = 1; While[GCD[n+1+m,n-m] != 1, m++]; m, {n, 1, 140}]

cp's OEIS Frontend

A053669 Smallest prime not dividing n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Python

Scheme

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Scheme

Formula

A235921 Numbers n such that smallest number not dividing n^2 (A236454) is different from smallest prime not dividing n (A053669).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A327571 Triangle T(n,k) read by rows giving the number of zeroless polydivisible numbers in base n that contains only "k" in the digits with 1 <= k <= n-1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Ruby

Formula

A213852 Least m>0 such that n+1+m and n-m are relatively prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A214073 Least m>0 such that 2^n-m and n^2-m are relatively prime.

Views

Author

Keywords

Examples

Links