A007978 -id:A007978 - cp's OEIS frontend

A331886 T(n, k) is the least positive m such that floor(n/m) = floor(k/m). Square array T(n, k) read by antidiagonals, n >= 0 and k >= 0.

1, 2, 2, 3, 1, 3, 4, 3, 3, 4, 5, 4, 1, 4, 5, 6, 5, 2, 2, 5, 6, 7, 6, 5, 1, 5, 6, 7, 8, 7, 6, 3, 3, 6, 7, 8, 9, 8, 7, 3, 1, 3, 7, 8, 9, 10, 9, 8, 7, 2, 2, 7, 8, 9, 10, 11, 10, 9, 8, 4, 1, 4, 8, 9, 10, 11, 12, 11, 10, 9, 4, 4, 4, 4, 9, 10, 11, 12, 13, 12, 11, 10

Offset: 0

Rémy Sigrist, Jan 30 2020

			Array T(n, k) begins:
   n\k|   0   1   2   3   4   5   6   7   8   9  10  11  12
   ---+----------------------------------------------------
     0|   1   2   3   4   5   6   7   8   9  10  11  12  13
     1|   2   1   3   4   5   6   7   8   9  10  11  12  13
     2|   3   3   1   2   5   6   7   8   9  10  11  12  13
     3|   4   4   2   1   3   3   7   8   9  10  11  12  13
     4|   5   5   5   3   1   2   4   4   9  10  11  12  13
     5|   6   6   6   3   2   1   4   4   5   5  11  12  13
     6|   7   7   7   7   4   4   1   2   3   5   6   6  13
     7|   8   8   8   8   4   4   2   1   3   5   6   6   7
     8|   9   9   9   9   9   5   3   3   1   2   4   4   7
     9|  10  10  10  10  10   5   5   5   2   1   3   3   7
    10|  11  11  11  11  11  11   6   6   4   3   1   2   5
    11|  12  12  12  12  12  12   6   6   4   3   2   1   5
    12|  13  13  13  13  13  13  13   7   7   7   5   5   1

Rémy Sigrist, Table of n, a(n) for n = 0..10010 (antidiagonals 0..140)
Rémy Sigrist, Colored representation of T(n, k) for n, k = 0..1000 (where the hue is function of T(n, k))
Rémy Sigrist, Colored representation of floor(n/T(n, k)) for n, k = 0..1000 (where the hue is function of floor(n/T(n, k)), red pixels correspond to 0's)

Cf. A007978, A331902.

T(n,k) = for (m=1, oo, if (n\m==k\m, return (m)))

T(n, k) = T(k, n).
T(n, k) = 1 iff n = k.
T(n, k) <= 1 + max(n, k) with equality iff max(n, k) >= 2*min(n, k).
T(n, n+1) = A007978(n+1).

A360805 Nonnegative integers k such that k! mod nextprime(k) is larger than k.

Original entry on oeis.org

0, 31, 120, 283, 293, 712, 2872, 3287, 5028, 5129, 7088, 9553, 13229, 14232, 14799, 15113, 20153, 20830, 23239, 30233, 31430, 31667, 34443, 40654, 44298, 50184, 78877, 105834, 115281, 125120, 164253, 192103, 201590, 227747, 239910, 241910, 282230, 322550, 374370

Offset: 1

Alois P. Heinz, Feb 22 2023

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..213

Cf. A000142, A007978, A151800, A360825.

q:= n-> is(n! mod nextprime(n)>n):
select(q, [$0..20000])[];

from itertools import count, islice
from functools import reduce
from sympy import nextprime
def A360805_gen(startvalue=0): # generator of terms >= startvalue
    n = max(startvalue,0)
    m = nextprime(n)
    while True:
        a = m-1
        klist = []
        for i in range(m-1,n,-1):
            a = a*pow(i,-1,m)%m
            if a>i-1:
                klist.append(i-1)
        yield from sorted(klist)
        n, m = m, nextprime(m)
A360805_list = list(islice(A360805_gen(),30)) # Chai Wah Wu, Feb 24 2023

{ k >= 0 : k! mod nextprime(k) > k }.

A360825(a(n)) > a(n).

A061853 Difference between smallest prime not dividing n and smallest nondivisor of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Henry Bottomley, May 10 2001

nonn

a(12m+6) is always positive since it involves subtracting 4 from a larger number; the first case where a term not of this form is positive is a(420).
Primorials from A002110(2)=6 onward seem to give the positions of records. - Antti Karttunen, Jul 28 2017
Difference between the smallest prime coprime to n and the smallest non-divisor of n. - Michael De Vlieger, Jul 28 2017

			a(29)=2-2=0; a(30)=7-4=3; a(420)=11-8=3.

Antti Karttunen, Table of n, a(n) for n = 1..30030

Cf. A002110, A007978, A053669.

a(n) = {my(f = factor(n), d = divisors(f), res, p = 2, i = 1, j); while(i<=#f~ && f[i, 1]==p, i++; p = nextprime(p+1)); res = p; for(j=2, #d, if(d[j]!=j, return(res - d[j-1] - 1)))} \\ David A. Corneth, Jul 29 2017

from sympy import nextprime
def a053669(n):
    p=2
    while n%p==0: p=nextprime(p)
    return p
def a007978(n):
    p=2
    while n%p==0: p+=1
    return p
def a(n): return a053669(n) - a007978(n)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 29 2017

a(n) = A053669(n) - A007978(n).

Description corrected by Michael De Vlieger, Jul 28 2017

A066482 The smallest anti-divisor of n.

Original entry on oeis.org

2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 8, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 8, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 64, 2, 3, 2, 3

Offset: 3

Robert G. Wilson v, Jan 02 2002

nonn

Almost identical to A007978, least non-divisor of n, but there are some subtle differences.

See A066272 for definition of anti-divisor.

Seiichi Manyama, Table of n, a(n) for n = 3..10000

Cf. A066481.

antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 &], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 &], 2n/Select[ Divisors[2*n], OddQ[ # ] && # != 1 &]]], # < n & ]; Table[ First[ antid[n]], {n, 3, 100} ]

A208570 LCM of n and smallest nondivisor of n.

Original entry on oeis.org

2, 6, 6, 12, 10, 12, 14, 24, 18, 30, 22, 60, 26, 42, 30, 48, 34, 36, 38, 60, 42, 66, 46, 120, 50, 78, 54, 84, 58, 60, 62, 96, 66, 102, 70, 180, 74, 114, 78, 120, 82, 84, 86, 132, 90, 138, 94, 240, 98, 150, 102, 156, 106, 108, 110, 168, 114, 174, 118, 420, 122

Offset: 1

J. Lowell, Feb 28 2012

nonn

a(n) = 2*n for all odd numbers.

			a(6) = 12 because the divisors of 6 are 1,2,3,6; 4 is the smallest number not a divisor of 6; the LCM of 6 and 4 is 12.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A007978.

Cf. A258115.

a208570 n = lcm n $ a007978 n  -- Reinhard Zumkeller, May 22 2015

a:= proc(n) local t;
      for t from 2 do
        if irem (n, t)<>0 then return ilcm(t, n) fi
      od
    end:
seq(a(n), n=1..100); # Alois P. Heinz, Mar 13 2012

Table[LCM[n, Min[Complement[Range[n + 1], Divisors[n]]]], {n, 61}] (* Ivan Neretin, May 20 2015 *)

a(n) = {my(k=2); while(!(n % k), k++); lcm(n, k); } \\ Michel Marcus, Mar 13 2018

From Robert Israel, May 20 2015: (Start)
a(n) = lcm(n, A007978(n)).
For primes p let nu_p(n) be the p-adic order of n.
a(n) = p * n where p is the prime that minimizes p^(1+nu_p(n)). (End)

More terms from Alois P. Heinz, Mar 13 2012

A242342 a(n) = binomial(n, smallest non-divisor of n).

Original entry on oeis.org

0, 0, 3, 4, 10, 15, 21, 56, 36, 120, 55, 792, 78, 364, 105, 560, 136, 3060, 171, 1140, 210, 1540, 253, 42504, 300, 2600, 351, 3276, 406, 27405, 465, 4960, 528, 5984, 595, 376992, 666, 8436, 741, 9880, 820, 111930, 903, 13244, 990, 15180, 1081, 1712304, 1176

Offset: 1

Reinhard Zumkeller, May 11 2014

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A014105.

a242342 n = if n <= 2 then 0 else a007318' n (a007978 n)

Join[{0,0},Table[Binomial[n,Complement[Range[DivisorSigma[0,n]], Divisors[ n]][[1]]],{n,3,50}]] (* Harvey P. Dale, May 23 2019 *)

a(n) = A007318(n, A007978(n)) for n > 2.

a(2*n-1) = A014105(n).

A258115 a(n) = A208570(n)/n.

Original entry on oeis.org

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

Offset: 1

Robert Israel, May 20 2015

nonn

The prime p that minimizes p^(1+nu_p(n)), where nu_p(n) is the p-adic order of n.

It satisfies p^(1+nu_p(n)) = A007978(n).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007978, A208570.

a258115 n = a208570 n `div` n  -- Reinhard Zumkeller, May 22 2015

Primes:= select(isprime,[2,seq(2*i+1,i=1..100)]):
seq(Primes[min[index](map(p -> p^(1+padic[ordp](n,p)), Primes))],n=1..200);

Table[(LCM[n, Min[Complement[Range[n+1], Divisors[n]]]]/n), {n, 100}] (* Vincenzo Librandi, May 21 2015 *)

A266620 a(n) = least non-divisor of n!.

Original entry on oeis.org

2, 3, 4, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 17, 17, 17, 19, 19, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 31, 31, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41, 43, 43, 47, 47, 47, 47, 53, 53, 53, 53, 53, 53, 59, 59, 59, 59, 59, 59, 61, 61, 67, 67, 67, 67, 67, 67, 71

Offset: 1

Jeffrey Shallit, Jan 01 2016

It appears that a(n) = A151800(n) with the exception of n = 3. - Robert Israel, Jan 13 2016

			For n = 4 the least non-divisor of 4! = 24 = 2^3 * 3 is 5.
For n = 5 the least non-divisor of 5! = 120 = 2^3 * 3 * 5 is 7.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007918, A007978, A066169, A115627, A151800.

N:= 100: # to get a(1)..a(N)
m:= 1 + numtheory:-pi(N):
Primes:= [seq(ithprime(i),i=1..m)]:
for i from 1 to m do pindex[Primes[i]]:= i od:
V:= Vector(m):
k:= 0:
for n from 1 to N do
  for f in ifactors(n)[2] do
    q:= pindex[f[1]];
    V[q]:= V[q] + f[2];
    k:= max(k, q);
  od:
  a[n]:= min(seq(Primes[i]^(1+V[i]),i=1..k),Primes[k+1]);
od:
seq(a[n],n=1..N); # Robert Israel, Jan 13 2016

Table[Complement[Range[2n], Divisors[n!]][[1]], {n, 30}] (* Alonso del Arte, Sep 23 2017 *)
Table[Block[{m = n!, k = n + 1}, While[Divisible[m, k], k++]; k], {n, 67}] (* Michael De Vlieger, Sep 23 2017 *)

from sympy import nextprime
def A266620(n): return 4 if n == 3 else nextprime(n) # Chai Wah Wu, Feb 22 2023

a(n) = min_{k >= 1} prime(k)^(1 + v(n!, prime(k))) where v(m, p) is the p-adic order of m. - Robert Israel, Jan 13 2016

a(n) = prime(pi(n) + 1) except for n = 3, in which case the least non-divisor of 3! is 4, not 5. - Alonso del Arte, Sep 23 2017

A374369 Triangle T(n, k), n > 0, k = 0..n-1, read by rows; T(n, k) is the least m such that n and k differ modulo m.

Original entry on oeis.org

2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 2, 4, 2, 3, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 2, 3, 2, 4, 2, 3, 2, 3, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2

Offset: 1

Rémy Sigrist, Jul 06 2024

			Triangle T(n, k) begins:
  n   n-th row
  --  ----------------------------------
   1  2
   2  3, 2
   3  2, 3, 2
   4  3, 2, 3, 2
   5  2, 3, 2, 3, 2
   6  4, 2, 3, 2, 3, 2
   7  2, 4, 2, 3, 2, 3, 2
   8  3, 2, 4, 2, 3, 2, 3, 2
   9  2, 3, 2, 4, 2, 3, 2, 3, 2
  10  3, 2, 3, 2, 4, 2, 3, 2, 3, 2
  11  2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2
  12  5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2

Cf. A007978, A374381, A374383.

T[n_,k_]:=Module[{m=2},While[Mod[n,m]==Mod[k,m], m++]; m]; Table[T[n,k],{n,13},{k,0,n-1}]//Flatten (* Stefano Spezia, Jul 12 2024 *)

T(n, k) = { for (m = 2, oo, if ((n%m) != (k%m), return (m););); }

T(n, k) = A007978(n-k).

A382445 Lexicographically least increasing sequence of distinct positive integers such that for any n > 1, a(n) does not divide the concatenation of the earlier terms.

Original entry on oeis.org

1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Rémy Sigrist, Mar 25 2025

			a(1) = 1.
a(2) must not divide 1; we can take a(2) = 2.
a(3) must not divide 12; we can take a(3) = 5.

Rémy Sigrist, PARI program

Cf. A007978, A096098, A382441.

PARI
```
\\ See Links section.
```

from itertools import count, islice
def agen(): # generator of terms
    an = t = 1
    while True:
        yield an
        an = next(k for k in count(an+1) if t%k != 0)
        t = t*10**len(str(an)) + an
print(list(islice(agen(), 54))) # Michael S. Branicky, Mar 26 2025

cp's OEIS Frontend

A331886 T(n, k) is the least positive m such that floor(n/m) = floor(k/m). Square array T(n, k) read by antidiagonals, n >= 0 and k >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A360805 Nonnegative integers k such that k! mod nextprime(k) is larger than k.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Python

Formula

A061853 Difference between smallest prime not dividing n and smallest nondivisor of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Formula

Extensions

A066482 The smallest anti-divisor of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A208570 LCM of n and smallest nondivisor of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A242342 a(n) = binomial(n, smallest non-divisor of n).

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A258115 a(n) = A208570(n)/n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

A266620 a(n) = least non-divisor of n!.