A019554 -id:A019554 - cp's OEIS frontend

A015049 Let m = A013929(n); then a(n) = smallest k such that m divides k^2.

2, 4, 3, 6, 4, 6, 10, 12, 5, 9, 14, 8, 6, 20, 22, 15, 12, 7, 10, 26, 18, 28, 30, 21, 8, 34, 12, 15, 38, 20, 9, 42, 44, 30, 46, 24, 14, 33, 10, 52, 18, 28, 58, 39, 60, 11, 62, 25, 42, 16, 66, 45, 68, 70, 12, 21, 74, 30, 76, 51, 78, 40, 18, 82, 84, 13, 57, 86

Offset: 1

R. Muller

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Henry Ibstedt, Surfing on the Ocean of Numbers, Erhus Univ. Press, Vail, 1997.

Cf. A013929, A015050, A015051, A019554.

f[p_, e_] := p^Ceiling[e/2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; s /@ Select[Range[200], !SquareFreeQ[#] &] (* Amiram Eldar, Feb 10 2021 *)

lista(kmax) = {my(f); for(k = 2, kmax, f = factor(k); if(!issquarefree(f), print1(k/core(f, 1)[2], ", ")));} \\ Amiram Eldar, Jan 06 2024

a(n) = A019554(A013929(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2)*(zeta(3)-1)/(zeta(2)-1)^2 = 0.799082... . - Amiram Eldar, Jan 06 2024

Description corrected by Diego Torres (torresvillarroel(AT)hotmail.com), Jun 23 2002

Offset corrected by Amiram Eldar, Feb 10 2021

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.

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

A254732 a(n) is the least k > n such that n divides k^2.

Original entry on oeis.org

2, 4, 6, 6, 10, 12, 14, 12, 12, 20, 22, 18, 26, 28, 30, 20, 34, 24, 38, 30, 42, 44, 46, 36, 30, 52, 36, 42, 58, 60, 62, 40, 66, 68, 70, 42, 74, 76, 78, 60, 82, 84, 86, 66, 60, 92, 94, 60, 56, 60, 102, 78, 106, 72, 110, 84, 114, 116, 118, 90, 122, 124, 84, 72

Offset: 1

Peter Kagey, Feb 06 2015

A073353(n) <= a(n) <= 2*n. Any prime that divides n must also divide a(n), and because n divides (2*n)^2.

Are all terms even? -Harvey P. Dale, Aug 07 2025

			a(12) = 18 because 12 divides 18^2, but 12 does not divide 13^2, 14^2, 15^2, 16^2, or 17^2.

Peter Kagey, Table of n, a(n) for n = 1..5000

Cf. A254733 (similar, with k^3), A254734 (similar, with k^4), A073353 (similar, with limit m->infinity of k^m).

Cf. A253905.

a254732 n = head [k | k <- [n + 1 ..], mod (k ^ 2) n == 0]
-- Reinhard Zumkeller, Feb 07 2015

lk[n_]:=Module[{k=n+1},While[!Divisible[k^2,n],k++];k]; Array[lk,70] (* Harvey P. Dale, Nov 05 2017 *)
Table[Module[{k=n+1},While[PowerMod[k,2,n]!=0,k++];k],{n,70}] (* Harvey P. Dale, Aug 07 2025 *)

a(n)=for(k=n+1,2*n,if(k^2%n==0,return(k)))
vector(100,n,a(n)) \\ Derek Orr, Feb 06 2015

a(n)=my(t=factorback(factor(n)[,1])); forstep(k=n+t,2*n,t,if(k^2%n==0, return(k))) \\ Charles R Greathouse IV, Feb 07 2015

def A254732(n):
    k = n + 1
    while pow(k,2,n):
        k += 1
    return k # Chai Wah Wu, Feb 15 2015

def a(n)
  (n+1..2*n).find { |k| k**2 % n == 0 }
end

a(n) = sqrt(n*A072905(n)).
a(n) = A019554(n)*(A000188(n)+1).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 + zeta(3)/zeta(2) = 1 + A253905 = 1.73076296940143849872... . - Amiram Eldar, Feb 17 2024

A365491 The number of divisors of the smallest number whose 4th power is divisible by n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 3, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 3, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Sep 05 2023

First differs from A365210 at n = 25 and from A034444 at n = 32.

The number of divisors of the smallest 4th divisible by n, A053167(n), is A365492(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms).

Cf. A000005, A001620, A005117, A053166, A053167, A365492, A365499.

Cf. A019554, A034444, A322483, A365210.

f[p_, e_] := Ceiling[e/4] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
With[{c=Range[100]^4},Table[DivisorSigma[0,Surd[SelectFirst[c,Mod[#,n]==0&],4]],{n,90}]] (* Harvey P. Dale, Jul 09 2024 *)

a(n) = vecprod(apply(x -> (x-1)\4 + 2, factor(n)[, 2]));

a(n) = A000005(A053166(n)).
Multiplicative with a(p^e) = ceiling(e/4) + 1.
a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 + 1/p^s - 1/p^(4*s)).
From Vaclav Kotesovec, Sep 06 2023: (Start)
Dirichlet g.f.: zeta(s)^2 * zeta(4*s) * Product_{p prime} (1 - 1/p^(2*s) - 1/p^(4*s) + 1/p^(5*s)).
Let f(s) = Product_{p prime} (1 - 1/p^(2*s) - 1/p^(4*s) + 1/p^(5*s)).
Sum_{k=1..n} a(k) ~ zeta(4) * f(1) * n * (log(n) + 2*gamma - 1 + 4*zeta'(4)/zeta(4) + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.57615273538566705952061107826411727540624711680289618854325028459572487...,
f'(1) = f(1) * Sum_{p prime} (-5 + 4*p + 2*p^3) * log(p) / (1 - p - p^3 + p^5) = f(1) * 1.3011434396559802378314782600747661399223385669839998680418996210...
and gamma is the Euler-Mascheroni constant A001620. (End)
a(n) = A322483(A019554(n)) (the number of exponentially odd divisors of the smallest number whose square is divisible by n). - Amiram Eldar, Sep 08 2023

A065887 Smallest number whose square is divisible by n!.

Original entry on oeis.org

1, 1, 2, 6, 12, 60, 60, 420, 1680, 5040, 5040, 55440, 332640, 4324320, 8648640, 43243200, 172972800, 2940537600, 8821612800, 167610643200, 335221286400, 7039647014400, 14079294028800, 323823762662400, 647647525324800, 3238237626624000, 6476475253248000

Offset: 0

Henry Bottomley, Nov 27 2001

nonn

			a(10) = 5040 since 10! = 3628800 and the smallest square divisible by this is 25401600 = 3628800*7 = 5040^2.

Alois P. Heinz, Table of n, a(n) for n = 0..735 (first 101 terms from Kevin P. Thompson)

Cf. A000142, A019554, A055772, A065886.

a:= n-> mul(i[1]^ceil(i[2]/2), i=ifactors(n!)[2]):
seq(a(n), n=0..26);  # Alois P. Heinz, Jan 24 2022

f[p_, e_] := p^Ceiling[e/2]; a[0] = a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 30, 0] (* Amiram Eldar, Feb 11 2024 *)

a(n) = A019554(A000142(n)) = sqrt(A065886(n)) = A000142(n)/A055772(n).

Missing a(0) inserted, formula corrected, and a(25)-a(26) added by Kevin P. Thompson, Jan 24 2022

A066069 a(n) is the smallest positive integer m such that n divides (n + m)^m.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 4, 3, 10, 11, 6, 13, 14, 15, 4, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 4, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 4, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Dec 02 2001

nonn

Essentially the same as A019530. - R. J. Mathar, Sep 30 2008

			a(8) = 4 because (8+1)^1 mod 8 = 1, (8+2)^2 mod 8 = 4, (8+3)^3 mod 8 = 3 and (8+4)^4 mod 8 = 0.

Harry J. Smith, Table of n, a(n) for n = 1..1000

First 23 terms are the same as A019554.

{ for (n=1, 1000, m=1; while ((n+m)^m % n, m++); write("b066069.txt", n, " ", m) ) } \\ Harry J. Smith, Nov 09 2009

A083482 Square root of smallest square of the type n(n+1)*k.

Original entry on oeis.org

2, 6, 6, 10, 30, 42, 28, 12, 30, 110, 66, 78, 182, 210, 60, 68, 102, 114, 190, 210, 462, 506, 276, 60, 130, 234, 126, 406, 870, 930, 248, 264, 1122, 1190, 210, 222, 1406, 1482, 780, 820, 1722, 1806, 946, 330, 690, 2162, 564, 84, 70, 510, 1326, 1378, 954, 990

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 03 2003

nonn

Squares pertaining to A083481.

a(n) == (p*q*r... ) where p,q,r are prime factors of n(n+1).

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A083481.

Table[Times @@ ((a = Transpose[FactorInteger[n (n + 1)]])[[1]]^Quotient[a[[2]] + 1, 2]), {n, 54}] (* Ivan Neretin, May 20 2015 *)

PARI
```
a(n)=sqrt(n*(n+1)*core(n*(n+1)))
```

from math import prod
from sympy import factorint
def A083482(n): return n*(n+1)//prod(p**(q>>1) for p, q in factorint(n*(n+1)).items()) # Chai Wah Wu, Mar 20 2023

a(n) = sqrt(A002378(n)*A083481(n)) = sqrt(A002378(n)*A007913(A002378(n))). a(n) = A019554(A002378(n)). - David Wasserman, Nov 16 2004

More terms from Benoit Cloitre, May 04 2003

More terms from David Wasserman, Nov 16 2004

A327170 Number of divisors d of n such that A327171(d) (= phi(d)*core(d)) is equal to n.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 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, 2, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Antti Karttunen, Sep 28 2019

nonn

From any solution (*) to A327171(d) = d*phi(d) = n, we obtain a solution for core(d')*phi(d') = n by forming a "pumped up" version d' of d, by replacing each exponent e_i in the prime factorization of d = p_1^e_1 * p_2^e_2 * ... * p_k^e_k, with exponent 2*e_i - 1 so that d' = p_1^(2*e_1 - 1) * p_2^(2*e_2 - 1)* ... * p_k^(2*e_k - 1) = A102631(d) = d*A003557(d), and this d' is also a divisor of n, as n = d' * A173557(d). Generally, any product m = p_1^(2*e_1 - x) * p_2^(2*e_2 - y)* ... * p_k^(2*e_k - z), where each x, y, ..., z is either 0 or 1 gives a solution for core(m)*phi(m) = n, thus every nonzero term in this sequence is a power of 2, even though not all such m's might be divisors of n.
  (* by necessity unique, see Franz Vrabec's Dec 12 2012 comment in A002618).
On the other hand, if we have any solution d for core(d)*phi(d) = n, we can find the unique such divisor e of d that e*phi(e) = n by setting e = A019554(d).
Thus, it follows that the nonzero terms in this sequence occur exactly at positions given by A082473.
Records (1, 2, 4, 8, 16, ...) occur at n = 1, 12, 504, 223200, 50097600, ...

			For n = 504 = 2^3 * 3^2 * 7, it has 24 divisors, out of which four divisors: 42 (= 2^1 * 3^1 * 7^1), 84 (= 2^2 * 3^1 * 7^1), 126 (= 2^1 * 3^2 * 7^1), 252 (= 2^2 * 3^2 * 7^1) are such that A007913(d)*A000010(d) = 504, thus a(504) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A002618, A003557, A007913, A019554, A082473, A102631, A173557, A194507, A327171.

Cf. also A327153, A327166, A327169.

With[{s = Array[EulerPhi[#] (Sqrt@ # /. (c_: 1) a_^(b_: 0) :> (c a^b)^2) &, 120]}, Table[DivisorSum[n, 1 &, s[[#]] == n &], {n, Length@ s}]] (* Michael De Vlieger, Sep 29 2019, after Bill Gosper at A007913 *)

A327170(n) = sumdiv(n,d,eulerphi(d)*core(d) == n);

a(n) = Sum_{d|n} [A000010(d)*A007913(d) == n], where [ ] is the Iverson bracket.

A335437 Numbers k with a partition into two distinct parts (s,t) such that k | s*t.

Original entry on oeis.org

9, 16, 18, 25, 27, 32, 36, 45, 48, 49, 50, 54, 63, 64, 72, 75, 80, 81, 90, 96, 98, 99, 100, 108, 112, 117, 121, 125, 126, 128, 135, 144, 147, 150, 153, 160, 162, 169, 171, 175, 176, 180, 189, 192, 196, 198, 200, 207, 208, 216, 224, 225, 234, 240, 242, 243, 245, 250, 252, 256

Offset: 1

Wesley Ivan Hurt, Jun 10 2020

nonn

All values of this sequence are nonsquarefree (A013929).
From Peter Munn, Nov 23 2020: (Start)
Numbers whose square part is greater than 4. [Proof follows from s and t having to be multiples of A019554(k), the smallest number whose square is divisible by k.]
Compare with A116451, numbers whose odd part is greater than 3. The self-inverse function A225546(.) maps the members of either one of these sets 1:1 onto the other set.
Compare with A028983, numbers whose squarefree part is greater than 2.
(End)
The asymptotic density of this sequence is 1 - 15/(2*Pi^2). - Amiram Eldar, Mar 08 2021
From Bernard Schott, Jan 09 2022: (Start)
Numbers of the form u*m^2, for u >= 1 and m >= 3 (union of first 2 comments).
A geometric application: in trapezoid ABCD, with AB // CD, the diagonals intersect at E. If the area of triangle ABE is u and the area of triangle CDE is v, with u>v, then the area of trapezoid ABCD is w = u + v + 2*sqrt(u*v); in this case, u, v, w are integer solutions iff (u,v,w) = (k*s^2,k*t^2,k*(s+t)^2), with s>t and k positives; hence, w is a term of this sequence (see IMTS link). (End)

			16 is in the sequence since it has a partition into two distinct parts (12,4), such that 16 | 12*4 = 48.

S. Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 1 of International Mathematical Talent Search, round 7, page 285.

Amiram Eldar, Table of n, a(n) for n = 1..10000.
International Mathematical Talent Search, Problem 1/7, Round 7.
Eric Weisstein's World of Mathematics, Square part.
Index to sequences related to Olympiads and other Mathematical Competitions.
Index entries for sequences related to partitions.

Complement of A133466 within A013929.
Cf. A019554, A028983, A335234, A335438.
A038838, A046101, A062312\{1}, A195085 are subsequences.
Related to A116451 via A225546.

Table[If[Sum[(1 - Ceiling[(i*(n - i))/n] + Floor[(i*(n - i))/n]), {i, Floor[(n - 1)/2]}] > 0, n, {}], {n, 300}] // Flatten
f[p_, e_] := p^(2*Floor[e/2]); sqpart[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[256], sqpart[#] > 4 &] (* Amiram Eldar, Mar 08 2021 *)

cp's OEIS Frontend

A015049 Let m = A013929(n); then a(n) = smallest k such that m divides k^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A236454 Smallest number not dividing n^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Scheme

Formula

A254732 a(n) is the least k > n such that n divides k^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Python

Ruby

Formula

A365491 The number of divisors of the smallest number whose 4th power is divisible by n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A065887 Smallest number whose square is divisible by n!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A066069 a(n) is the smallest positive integer m such that n divides (n + m)^m.

Views

Author

Keywords

Comments

Examples