A007978 -id:A007978 - cp's OEIS frontend

A199969 a(n) = the greatest non-divisor h of n (1 < h < n), or 0 if no such h exists.

0, 0, 2, 3, 4, 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, 43, 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, 71, 72, 73, 74, 75

Offset: 1

Jaroslav Krizek, Nov 26 2011

From Paul Curtz, Feb 09 2015: (Start)
The nonnegative numbers with 0 instead of 1. See A254667(n), which is linked to the Bernoulli numbers A164555(n)/A027642(n), an autosequence of the second kind.
Offset 0 could be chosen.
An autosequence of the second kind is a sequence whose main diagonal is the first upper diagonal multiplied by 2. If the first upper diagonal is
s0, s1, s2, s3, s4, s5, ...,
the sequence is
Ssk(n) = 2*s0, s0, s0 + 2*s1, s0 +3*s1, s0 + 4*s1 + 2*s2, s1 + 5*s1 + 5*s2, etc.
The corresponding coefficients are A034807(n), a companion to A011973(n).
The binomial transform of Ssk(n) is (-1)^n*Ssk(n).
Difference table of a(n):
   0,  0, 2, 3, 4, 5, 6, 7, ...
   0,  2, 1, 1, 1, 1, 1, ...
   2, -1, 0, 0, 0, 0 ...
  -3,  1, 0, 0, 0, ...
   4, -1, 0, 0, ...
  -5,  1, 0, ...
   6, -1, ...
   7, ...
  etc.
a(n) is an autosequence of the second kind. See A054977(n).
The corresponding autosequence of the first kind (a companion) is 0, 0 followed by the nonnegative numbers (A001477(n)). Not in the OEIS.
Ssk(n) = 2*Sfk(n+1) - Sfk(n) where Sfk(n) is the corresponding sequence of the first kind (see A254667(n)).
(End)
Number of binary sequences of length n-1 that contain exactly one 0 and at least one 1. - Enrique Navarrete, May 11 2021

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A199968 (the smallest non-divisor h of n (1A199970. A001477, A011973, A034807, A054977, A254667.
Cf. A007978.
Essentially the same as A000027, A028310, A087156 etc.

Join[{0,0},Table[Max[Complement[Range[n],Divisors[n]]],{n,3,70}]] (* or *) Join[{0,0},Range[2,70]] (* Harvey P. Dale, May 31 2014 *)

if(n>2,n-1,0) \\ Charles R Greathouse IV, Sep 02 2015

a(n) = n-1 for n >= 3.

E.g.f.: 1-x^2/2+(x-1)*exp(x). - Enrique Navarrete, May 11 2021

A213636 Remainder when n is divided by its least nondivisor.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jun 16 2012

nonn

Experimentation suggests that every positive integer occurs in this sequence and that
2 occurs only in even numbered positions,
3 occurs in only in positions that are multiples of 12,
4 occurs only in positions that are multiples of 12,
5 occurs only in positions that are multiples of 60,
6 occurs only in positions that are multiples of 60,
7 occurs only in positions that are multiples of 2520, etc.
See A213637 for positions of 1 and A213638 for positions of 2.
From Robert Israel, Jul 28 2017: (Start)
Given any positive number m, let q be a prime > m and r = A003418(q-1).  Then a(n) = m if n == m (mod q) and n == 0 (mod r).  By the Chinese Remainder Theorem, such n exists.
On the other hand, if a(n) = m, we must have A007978(n) > m, and then n must be divisible by A003418(q-1) where q = A007978(n) is a member of A000961 greater than m.  Moreover, if q=p^j with j>1, n is divisible by p^(j-1) so m must be divisible by p^(j-1).  Thus:
For m=2, A003418(2)=2.
For m=3, A007978(n) can't be 4 because m is odd, so A007978(n)>= 5 and n must be divisible by A003418(4)=12.
For m=4, A003418(4)=12.
For m=5 or 6, A003418(6)=60.
For m=7, A007978(n) can't be 8 because m is odd, and can't be 9 because m is not divisible by 3, so n must be divisible by A003418(10)=2520. (End)

			a(10) = 10-3*[10/3] = 1.

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

Cf. A000961, A003418, A007978, A213633, A213635, A213637, A213638.

f:= proc(n) local k;
  for k from 2 do if n mod k <> 0 then return n mod k fi od
end proc:
map(f, [$1..100]); # Robert Israel, Jul 27 2017

y=120; z=2000;
t = Table[k := 1; While[Mod[n, k] == 0, k++];
   k, {n, 1, z}]  (*A007978*)
Table[Floor[n/t[[n]]], {n, 1, y}] (*A213633*)
Table[n - Floor[n/t[[n]]], {n, 1, y}] (*A213634*)
Table[t[[n]]*Floor[n/t[[n]]], {n, 1, y}] (*A213635*)
t1 = Table[n - t[[n]]*Floor[n/t[[n]]],
   {n, 1, z}] (* A213636 *)
Flatten[Position[t1, 1]] (* A213637 *)
Flatten[Position[t1, 2]] (* A213638 *)
rem[n_]:=Module[{lnd=First[Complement[Range[n],Divisors[n]]]},Mod[n,lnd]]; Join[{1,2},Array[rem,100,3]] (* Harvey P. Dale, Mar 26 2013 *)
Table[Mod[n, SelectFirst[Range[n + 1], ! Divisible[n, #] &]], {n, 105}] (* Michael De Vlieger, Jul 29 2017 *)

def a(n):
    k=2
    while n%k==0: k+=1
    return n%k
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 28 2017

def A213636(n): return next(filter(None, (n%d for d in range(2,n)))) if n>2 else n # Chai Wah Wu, Feb 22 2023

(define (A213636 n) (modulo n (A007978 n))) ;; Antti Karttunen, Jul 27 2017

a(n) = n - A213635(n).

a(n) = n - m(n)*floor(n/m(n)), where m(n) = A007978(n).

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.

A193574 Smallest divisor of sigma(n) that does not divide n.

Original entry on oeis.org

3, 2, 7, 2, 4, 2, 3, 13, 3, 2, 7, 2, 3, 2, 31, 2, 13, 2, 3, 2, 3, 2, 5, 31, 3, 2, 8, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 31, 3, 3, 2, 7, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 127, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 5, 2, 4, 2, 3

Offset: 2

Franklin T. Adams-Watters, Aug 27 2011

nonn

a(n) = 2 iff n is an odd number that is not a perfect square.
From Hartmut F. W. Hoft, May 05 2017: (Start)
(1)  Every a(n) > n is a prime: Because of the minimality of a(n), a(n) = u*v with gcd(u,v)=1 leads to the contradiction (u*v)|n. Similarly, a(n)=p^k with p prime an k>1 leads to the contradiction (p^k-1)/(p-1) | n.
(2)  n=p^(2*k), k>=1 and 2*k+1 prime, when a(n) = sigma(n) for n>2: Because n having two distinct prime factors implies sigma(n) composite, and if n is an odd power of a prime then 2|sigma(n). Finally, if 2*k+1=u*v  with  u,v > 1 then sigma(p^(u-1)) divides sigma(p^(2*k)), but not p^(2k), for any prime p, contradicting minimality of a(n). For example, no number sigma(p^8) for any prime p is in the sequence.
(3)  The converse of (2) is false since, e.g. sigma(7^2) = 3*19 so that a(7^2) = 3, and sigma(2^10) = 23*89 so that a(2^10) = 23.
(4)  Conjecture: a(n) > n implies a(n) = sigma(n); tested through n = 20000000.
(5)  Subsequences are: A053183 (sigma(p^2) is prime for prime p), A190527 (sigma(p^4) is prime for prime p), A194257 (sigma(p^6) is prime for prime p), A286301 (sigma(p^10) is prime for prime p)
(6)  Subsequences are: A000668 (primes of form 2^p-1), A076481 (primes of form (3^p-1)/2), A086122 (primes of form (5^p-1)/4), A102170 (primes of form (7^p-1)/6), all when p is prime.
(End)
Up to n = 10^6, there are 89 distinct elements. For those n, a(n) is prime. If it's not, it's a power of 2, a power of 3 or a perfect square <= 121. - David A. Corneth, May 10 2017

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A000203, A007978, A027750, A135718.

Subsequences: A000668, A053183, A076481, A086122, A102170, A190527, A194257, A286301.

import Data.List ((\\))
a193574 n = head [d | d <- [1..sigma] \\ nDivisors, mod sigma d == 0]
   where nDivisors = a027750_row n
         sigma = sum nDivisors
-- Reinhard Zumkeller, May 20 2015, Aug 28 2011

a193574[n_] := First[Select[Divisors[DivisorSigma[1, n]], Mod[n, #]!=0&]]
Map[a193574, Range[2, 80]] (* data *) (* Hartmut F. W. Hoft, May 05 2017 *)

a(n)=local(ds);ds=divisors(sigma(n));for(k=2,#ds,if(n%ds[k],return(ds[k])))

A236454 Smallest number not dividing n^2.

Original entry on oeis.org

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

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

A199968 a(n) = the smallest non-divisor h of n (1

Original entry on oeis.org

0, 0, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 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

Offset: 1

Jaroslav Krizek, Nov 26 2011

nonn

a(n) = A173540(n,1). - Reinhard Zumkeller, Oct 02 2015

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

Cf. A199969 (the greatest non-divisor h of n (1A199970.

Cf. A173540.

a199968 = head . a173540_row  -- Reinhard Zumkeller, Oct 02 2015

a(n) = A007978(n) for n>=3.

A213637 Values of n for which A213636(n) = 1.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 10, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 31, 33, 34, 35, 36, 37, 39, 40, 41, 43, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 67, 69, 70, 71, 73, 75, 76, 77, 79, 81, 82, 83, 85, 87, 88, 89, 91, 93, 94, 95, 96, 97, 99

Offset: 1

Clark Kimberling, Jun 16 2012

a(n) is the position of the n-th 1 in the sequence of remainders when each positive integer is divided by its least nondivisor (as in A007978).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

(See A213636.)
rem[n_]:=Module[{lnd=First[Complement[Range[n],Divisors[n]]]},Mod[n,lnd]];Flatten[Position[Join[{1,2},Array[rem,100,3]],1]] (* Harvey P. Dale, Feb 07 2014 *)

A337686 a(n) is the least multiplier k such that n*k has twice as many divisors as n.

Original entry on oeis.org

2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 6, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4

Offset: 1

Michel Marcus, Sep 15 2020

nonn

The zeros in A139315 are the missing values in this sequence (see A337709).
There are no 1's in this sequence. a(n) = 2 for all odd n and a(n) >= 3 for all even n. - J. Lowell, Sep 15 2020
Empirical observation: A007978(n) - a(n) = 1 for n = 60*A206547(n), = 2 for n = 420*A007310(n), else = 0. - Hugo Pfoertner, Sep 30 2020

			a(1) = 2 because 1 has 1 divisor, 1*2 has 2 divisors, so 2 is the least multiplier to apply to 1 to get twice as many divisors.

Michel Marcus, Table of n, a(n) for n = 1..10000
Hugo Pfoertner, Illustration of ratio A007978(n) / a(n), using Plot 2.

Cf. A000005, A129902, A139315, A337709 (missing values).

nn = 105; Do[d[i] = DivisorSigma[0, i], {i, 12 nn}]; Reap[Do[m = 2; While[d[m i] != 2 d[i], m++]; Sow[m ], {i, nn}]][[-1, -1]] (* Michael De Vlieger, Jan 10 2022 *)

a(n) = {my(k=1); while (numdiv(n*k) != 2*numdiv(n), k++); k;}

a(n) = A129902(n)/n.

A067391 a(n) is the least common multiple of numbers in {1,2,3,...,n-1} which do not divide n.

Original entry on oeis.org

1, 1, 2, 3, 12, 20, 60, 210, 840, 504, 2520, 27720, 27720, 51480, 360360, 180180, 720720, 4084080, 12252240, 232792560, 232792560, 21162960, 232792560, 5354228880, 5354228880, 2059318800, 26771144400, 80313433200, 80313433200

Offset: 1

Labos Elemer, Jan 22 2002

nonn

			For n=10: non-divisors = {3,4,6,7,8,9}, lcm(3,4,6,7,8,9) = 8*9*7 = 504 = a(10).
For n=18, a(18) = lcm(4,5,7,8,10,11,12,13,14,15,16,17) = 4084080.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A049820 [count], A007978 [min], A024816 [sum], A055067 [product].
Cf. A003418, A066574, A038610.
Cf. A173540.

a067391 n | n <= 2    = 1
          | otherwise = foldl lcm 1 $ a173540_row n
-- Reinhard Zumkeller, Apr 04 2012

a[n_] := LCM@@Select[Range[1, n-1], Mod[n, # ]!=0& ]
Join[{1,1},Table[LCM@@Complement[Range[n],Divisors[n]],{n,3,30}]] (* Harvey P. Dale, Mar 27 2013 *)

Let f(n) = lcm(1, 2, ..., n-1) = A003418(n-1). If n = 2*p^k for some prime p, then a(n) = f(n)/p; otherwise a(n) = f(n).

A089090 a(n) is the smallest composite number coprime to n.

Original entry on oeis.org

4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 49, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 49, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 49, 4, 9, 4, 9, 4, 25, 4

Offset: 1

Labos Elemer, Nov 26 2003

If n is the n-th primorial, then a(n) = prime(n+1)^2.

			n=30: below 30 coprimes to 30 phi(30)=8 numbers are relevant but each 1 or primes; so a(8)>30; the first suitable number is a(30)=49.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A053669, A007978, A053670-A053674, A090091-A090093.

m=0;Table[fla=1;Do[s=GCD[n, k];If[Equal[s, 1]&&!PrimeQ[n]&&!Equal[n, 1]&& Equal[fla, 1], m=m+1;Print[n];fla=0], {n, 1, 130}], {k, 1, 256}]

A089090(n) = forprime(p=2, , if(n%p, return(p*p))); \\ Antti Karttunen, Dec 19 2018

a(n) = A053669(n)^2.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} ((p^2*(p-1)/Product_{q prime <= p} q)) = 10.3344588090... . - Amiram Eldar, Jul 25 2022

Offset corrected by Antti Karttunen, Dec 19 2018

cp's OEIS Frontend

A199969 a(n) = the greatest non-divisor h of n (1 < h < n), or 0 if no such h exists.

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A213636 Remainder when n is divided by its least nondivisor.

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

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A193574 Smallest divisor of sigma(n) that does not divide n.

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A236454 Smallest number not dividing n^2.

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Maple

Mathematica

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A199968 a(n) = the smallest non-divisor h of n (1

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Haskell

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A213637 Values of n for which A213636(n) = 1.

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Mathematica

A337686 a(n) is the least multiplier k such that n*k has twice as many divisors as n.

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