A000188 -id:A000188 - cp's OEIS frontend

A056191 Characteristic cube divisor of n: cube of g = gcd(K,F), where K is the largest square root divisor of n (A000188) and F = n/(KK) = A007913(n) is its squarefree part; g^2 divides K^2 = A008833(n) = g^2L^2 and g divides F = gf.

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Labos Elemer, Aug 02 2000

This is not the largest cube which divides n. It is canonical, since the decomposition n = KKgggf is unique (factors are defined above and dependent on n).

			If n=24, largest square divisor is 4, squarefree part is 6, g=2, a(24)=8; n=81, largest square divisor is 81, both F and g is 1, a(81)=1.

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A055229, A000188, A008833, A007913, A055231, A056192.

a[n_]:=With[{sf=Times@@Power@@@({#[[1]], Mod[#[[2]], 2]}&/@FactorInteger[n])}, GCD[sf, n/sf]]; Table[a[n]^3, {n, 1, 100}] (* Vincenzo Librandi, Oct 08 2017 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1 || !(f[i,2]%2), 1,  f[i,1]^3));} \\ Amiram Eldar, Sep 05 2023

a(n) = A055229(n)^3 = g^3 = ggg; n = (KK)*(ggg)*f = K^2*g^3*f = KK*a(n)^3*f.

Multiplicative with a(p^e)=1 for even e, a(p)=1, a(p^e)=p^3 for odd e > 1. - Vladeta Jovovic, May 01 2002

A056622 a(n) = A000188(n)/A055229(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 4, 7, 5, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 8, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 4, 9, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 7, 3, 10, 1, 1, 1, 1

Offset: 1

Labos Elemer, Aug 08 2000

Previous name: "Square root of largest unitary square divisor of n." The previous name was incorrect for numbers that have an odd exponent in their prime factorization that is larger than 3. For the correct square root of largest unitary square divisor of n see A071974. - Amiram Eldar, Jul 26 2024

Multiplicative because quotient of two multiplicative sequences. - Christian G. Bower, May 16 2005

			For n = 125: A000188(125) = 5, A055229(125) = 5, so a(125) = 1.
For n = 360: A000188(360) = 6, A055229(360) = 2, so a(360) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000188, A055229, A034444, A056623, A071974.

f[p_, e_] := If[EvenQ[e], p^(e/2), If[e == 1, 1, p^((e - 3)/2)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)

A000188(n) = core(n, 1)[2]; \\ Michel Marcus, Feb 27 2013
A055229(n) = { my(c=core(n)); gcd(c, n/c); }; \\ Charles R Greathouse IV, Nov 20 2012
A056622(n) = (A000188(n)/A055229(n)); \\ Antti Karttunen, Nov 19 2017

Multiplicative with a(p^e) = p^(e/2) if e even, a(p) = 1, and a(p^e) = p^((e-3)/2) for odd e > 1. - Amiram Eldar, Sep 14 2020
Dirichlet g.f.: zeta(2*s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-1) + 1/p^(3*s)). - Amiram Eldar, Dec 18 2023
a(n) = sqrt(A056623(n)). - Amiram Eldar, Jul 26 2024
From Vaclav Kotesovec, Jan 27 2025: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s) - 1/p^(3*s-1) - 1/p^(4*s) + 1/p^(4*s-1)).
Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s) - 1/p^(3*s-1) - 1/p^(4*s) + 1/p^(4*s-1)).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
f(1) = Product_{p prime} (1 - 2/p^2 + 2/p^3 - 1/p^4) = 0.490798286634728225909154323920711804307234495196201399106047774...,
f'(1) = f(1) * Sum_{p prime} (5*p^2 - 7*p + 4) * log(p) / (p^4 - 2*p^2 + 2*p - 1) = f(1) * 1.94788222046256567576552118452630646598176999674201755783...
and gamma is the Euler-Mascheroni constant A001620. (End)

Name replaced with a formula by Amiram Eldar, Jul 26 2024

A120486 Partial sums of A000188.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 13, 14, 15, 17, 18, 19, 20, 24, 25, 28, 29, 31, 32, 33, 34, 36, 41, 42, 45, 47, 48, 49, 50, 54, 55, 56, 57, 63, 64, 65, 66, 68, 69, 70, 71, 73, 76, 77, 78, 82, 89, 94, 95, 97, 98, 101, 102, 104, 105, 106, 107, 109, 110, 111, 114, 122, 123, 124, 125, 127, 128

Offset: 1

Gerry Myerson, Nov 21 2007

nonn

This sequence can also be described as the number of 3-term nondecreasing geometric progressions with no term exceeding n.

a(n) = A132188(n) - A132345(n). - Reinhard Zumkeller, Apr 21 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio
Gerry Myerson, Trifectas in Geometric Progression, Australian Mathematical Society Gazette, 35 (3) 2008, p 189-194.

Cf. A000188, A132188, A132189, A132345, A010052.

a120486 n = a120486_list !! (n - 1)
a120486_list = scanl1 (+) a000188_list
-- Reinhard Zumkeller, Apr 22 2012

with(numtheory): seq(add(phi(k)*floor(n/k^2), k=1..floor(sqrt(n))), n=1..100); # Ridouane Oudra, Aug 18 2019

a(n) = 3n log(n) / Pi^2 + O(n). - Griffin N. Macris, Jan 28 2017
a(n) ~ 3*n*((log(n) + (3*gamma - 1))/ Pi^2 - 12*(Zeta'(2)/Pi^4)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 30 2019
a(n) = Sum_{k=1..floor(sqrt(n))} phi(k)*floor(n/k^2), where phi is the Euler totient function A000010. - Ridouane Oudra, Aug 18 2019
G.f.: (1/(1 - x)) * Sum_{k>=1} phi(k) * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Aug 26 2021
From Ridouane Oudra, Oct 05 2024: (Start)
a(n) = Sum_{i=1..n} Sum_{j=1..i} A010052(i*j).
a(n) = A132345(n) + n.
a(n) = (1/2)*A132189(n) + n.
a(n) = (1/2)*(A132188(n) + n). (End)

A283983 Square root of the largest square dividing prime factorization representation of the n-th Stern polynomial: a(n) = A000188(A260443(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 5, 3, 1, 15, 1, 1, 1, 3, 5, 15, 7, 45, 5, 15, 1, 15, 35, 15, 1, 105, 1, 1, 1, 3, 5, 105, 7, 225, 35, 525, 11, 1575, 175, 1125, 7, 1575, 35, 105, 1, 105, 35, 525, 77, 1575, 35, 525, 1, 105, 385, 105, 1, 1155, 1, 1, 1, 3, 5, 1155, 7, 1575, 385, 3675, 11, 7875, 1225, 275625, 77, 55125, 2695, 5775, 13, 17325, 13475, 275625, 539

Offset: 0

Antti Karttunen, Mar 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4096

Cf. A000188, A000196, A001222, A003961, A260443, A283989, A284264, A283484 (odd bisection).

Cf. A023758 (positions of ones).

A003961[p_?PrimeQ] := A003961[p] = Prime[ PrimePi[p] + 1]; A003961[1] = 1; A003961[n_] := A003961[n] = Times @@ ( A003961[First[#]] ^ Last[#] & ) /@ FactorInteger[n]  (* after Jean-François Alcover, Dec 01 2011 *); A260443[n_]:= If[n<2, n + 1, If[EvenQ[n], A003961[A260443[n/2]], A260443[(n - 1)/2] * A260443[(n + 1)/2]]];  A000188[n_]:= Sum[Boole[Mod[i^2, n] == 0], {i, n}]; Table[A000188[A260443[n]], {n, 0, 50}] (* Indranil Ghosh, Mar 28 2017 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2)))); \\ Cf. Charles R Greathouse IV's code for "ps" in A186891 and A277013.
A000188(n) = core(n, 1)[2]; \\ This function from Michel Marcus, Feb 27 2013
A283983(n) = A000188(A260443(n));

(define (A283983 n) (A000188 (A260443 n)))
(define (A283983 n) (A000196 (A283989 n)))

a(n) = A000188(A260443(n)).
a(n) = A000196(A283989(n)).
Other identities. For all n >= 0:
a(2n) = A003961(a(n)).
A001222(a(n)) = A284264(n).

A293219 a(n) = A000188(A292270(n)).

Original entry on oeis.org

1, 1, 2, 1, 1, 5, 6, 1, 1, 9, 2, 1, 2, 11, 14, 1, 1, 1, 18, 1, 4, 5, 1, 7, 2, 1, 26, 1, 1, 29, 30, 1, 2, 33, 1, 2, 1, 1, 3, 1, 1, 41, 2, 1, 2, 1, 1, 22, 1, 1, 50, 1, 2, 53, 2, 2, 2, 7, 1, 3, 5, 4, 2, 1, 1, 65, 1, 1, 2, 69, 1, 17, 2, 1, 74, 1, 1, 7, 2, 1, 1, 81, 1, 1, 6, 7, 86, 1, 1, 89, 90, 26, 2, 1, 1, 1, 1, 1, 98, 3, 3, 6, 2, 1, 1, 105

Offset: 0

Antti Karttunen, Oct 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000188, A292270, A292947, A293218.

Cf. A163782 (after a(1) = 1 gives the other positions where a(n) = n).

A000188(n) = core(n, 1)[2]; \\ This function from Michel Marcus, Feb 27 2013
A000265(n) = (n >> valuation(n, 2));
A006519(n) = 2^valuation(n, 2);
A292270(n) = { my(x = n+n+1, z = ((1+x)/A006519(1+x)), m = A000265(1+x)); while(m!=1, z += ((x+m)/A006519(x+m)); m = A000265(x+m)); z; };
A293219(n) = A000188(A292270(n));

(define (A293219 n) (A000188 (A292270 n)))

a(n) = A000188(A292270(n)).

A335425 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000188(i) = A000188(j) and A335424(i) = A335424(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 5, 2, 7, 4, 8, 2, 9, 2, 5, 7, 7, 2, 10, 11, 7, 9, 5, 2, 12, 2, 13, 7, 7, 4, 14, 2, 7, 7, 15, 2, 16, 2, 5, 9, 7, 2, 13, 17, 18, 7, 5, 2, 19, 7, 15, 7, 7, 2, 10, 2, 7, 9, 20, 7, 16, 2, 5, 7, 16, 2, 21, 2, 7, 18, 5, 4, 16, 2, 13, 22, 7, 2, 15, 7, 7, 7, 15, 2, 23, 7, 5, 7, 7, 7, 24, 2, 25, 9, 26, 2, 16, 2, 15, 12

Offset: 1

Antti Karttunen, Jun 13 2020

nonn

Restricted growth sequence transform of the ordered pair [A000188(n), A046523(A335423(n))].

For all i, j: A305800(i) = A305800(j) => a(i) = a(j) => A001222(i) = A001222(j).

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

Cf. A000188, A001222, A046523, A162642, A248663, A305800, A335423, A335424.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A000188(n) = core(n, 1)[2]; \\ From A000188
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A248663(n) = A048675(core(n));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A335423(n) = A005940(1+A248663(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
Aux335425(n) = [A000188(n),A046523(A335423(n))];
v335425 = rgs_transform(vector(up_to,n,Aux335425(n)));
A335425(n) = v335425[n];

A347120 Square root of the largest square dividing A005940(1+(3*A156552(n))): a(n) = A000188(A332449(n)).

Original entry on oeis.org

1, 2, 3, 1, 5, 4, 7, 1, 1, 6, 11, 1, 13, 10, 9, 3, 17, 2, 19, 5, 15, 14, 23, 1, 1, 22, 1, 7, 29, 8, 31, 3, 21, 26, 25, 1, 37, 34, 33, 5, 41, 12, 43, 11, 1, 38, 47, 3, 1, 2, 39, 13, 53, 2, 35, 7, 51, 46, 59, 1, 61, 58, 7, 9, 55, 20, 67, 17, 57, 18, 71, 1, 73, 62, 3, 19, 49, 28, 79, 15, 5, 74, 83, 11, 65, 82, 69, 11, 89

Offset: 1

Antti Karttunen, Aug 22 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12000
Index entries for sequences computed from indices in prime factorization

Cf. A000188, A000196, A005940, A007913, A156552, A332449, A347119.

A000188(n) = core(n, 1)[2]; \\ This function from A000188
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A332449(n) = A005940(1+(3*A156552(n)));
A347120(n) = A000188(A332449(n));

a(n) = A000188(A332449(n)).
a(n) = A000196(A332449(n)/A347119(n)).
a(p) = p for all primes p.

A326038 Square root of the largest square dividing the sum of divisors of n: a(n) = A000188(sigma(n)).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 2, 1, 3, 1, 2, 1, 4, 6, 2, 2, 1, 1, 2, 2, 1, 6, 4, 3, 4, 3, 4, 1, 1, 2, 2, 3, 1, 4, 2, 2, 1, 6, 4, 2, 1, 1, 6, 7, 3, 2, 6, 2, 4, 3, 2, 2, 1, 4, 2, 1, 2, 12, 2, 3, 4, 12, 6, 1, 1, 1, 2, 2, 4, 2, 4, 1, 11, 3, 2, 4, 6, 2, 2, 6, 3, 3, 4, 2, 8, 12, 2, 6, 7, 3, 2, 1, 1, 6, 2, 1, 8

Offset: 1

Antti Karttunen, Jun 05 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000188, A000196, A326038, A326039, A326040.

Table[Last[Select[Sqrt[#]&/@Divisors[DivisorSigma[1,n]],IntegerQ]],{n,120}] (* Harvey P. Dale, Jun 25 2022 *)

A000188(n) = core(n, 1)[2]; \\ From A000188
A326038(n) = A000188(sigma(n));

a(n) = A000188(A000203(n)) = A000196(A326039(n)).

A355261 a(n) = largest-nth-power(n, 2) * radical(n) = A000188(n) * A007947(n), where largest-nth-power(n, e) is the largest positive integer b such that b^e divides n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 8, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 9, 28, 29, 30, 31, 8, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 24, 49, 50, 51, 52, 53, 18, 55, 28, 57, 58, 59, 60, 61, 62, 63, 16, 65, 66, 67, 68

Offset: 1

Peter Luschny, Jul 12 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000188, A002117, A007947, A064549, A355263.

with(NumberTheory): seq(LargestNthPower(n, 2)*Radical(n), n = 1..68);

Array[Apply[Times, #[[All, 1]]]*Apply[Times, #1^Floor[#2/2] & @@ Transpose@ #] &@ FactorInteger[#] &, 68] (* Michael De Vlieger, Jul 12 2022 *)

from math import prod
from sympy import factorint
def A355261(n): return prod(p**((e>>1)+1) for p, e in factorint(n).items()) # Chai Wah Wu, Jul 13 2022

Multiplicative with a(p^e) = p^(1+floor(e/2)). - Amiram Eldar, Jul 13 2022

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)/2) * Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.447583182004... . - Amiram Eldar, Nov 13 2022

A253196 Irregular array read by rows. T(n,k) is the number of divisors d of n such that k^2 is the greatest square that divides d, n>=1, 1<=k<=A000188(n).

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 4, 2, 2, 2, 2, 0, 1, 4, 2, 4, 2, 2, 4, 4, 2, 2, 0, 1, 2, 4, 0, 2, 2, 4, 2, 4, 4, 2, 4, 4, 2, 0, 0, 0, 1, 4, 2, 0, 2, 4, 2, 2, 8, 2, 2, 2, 0, 2, 4, 4, 4, 4, 2, 2, 0, 0, 1, 2, 4, 4, 4, 4, 2, 8, 2, 4, 2, 4, 0, 2, 4, 2, 4, 4, 0, 2, 2, 0, 0, 0, 0, 0, 1, 4, 0, 0, 0, 2, 4, 4, 2, 2, 4, 0, 4, 4, 4, 4, 4, 4, 2, 8, 4

Offset: 1

Geoffrey Critzer, Mar 24 2015

Row sums are A000005.

Column 1 is A034444.

			1
2
2
2,1
2
4
2
2,2
2,0,1
4
2
4,2
2
4
4
2,2,0,1
2
4,0,2
For n=18, The divisors are: 1,2,3,6,9,18.  T(18,1)=4 because 1 is the largest square that divides 1,2,3,6.  T(18,3) = 2 because 9 is the largest square that divides 9,18.

Alois P. Heinz, Rows n = 1..6000, flattened

Cf. A000005, A000188, A034444.

with(numtheory):
T:= n-> (p-> seq(coeff(p, x, j), j=1..degree(p)))(add(
    x^mul(i[1]^iquo(i[2], 2), i=ifactors(d)[2]), d=divisors(n))):
seq(T(n), n=1..70);  # Alois P. Heinz, Mar 25 2015

nn = 60;g[list_] := list /. {j___, 0 ...} -> {j}; f[list_, i_] := list[[i]];Map[g, Transpose[Table[a = Table[If[n == k^2, 1, 0], {n, 1, nn}]; b = Table[2^PrimeNu[n], {n, 1, nn}];Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}], {k,1, nn}]]] // Grid

Dirichlet g.f. for column k: 1/k^(2*s) * zeta(s)^2/zeta(2*s).

cp's OEIS Frontend

A056191 Characteristic cube divisor of n: cube of g = gcd(K,F), where K is the largest square root divisor of n (A000188) and F = n/(K*K) = A007913(n) is its squarefree part; g^2 divides K^2 = A008833(n) = g^2*L^2 and g divides F = gf.

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A056622 a(n) = A000188(n)/A055229(n).

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A120486 Partial sums of A000188.

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A283983 Square root of the largest square dividing prime factorization representation of the n-th Stern polynomial: a(n) = A000188(A260443(n)).

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A293219 a(n) = A000188(A292270(n)).

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A335425 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000188(i) = A000188(j) and A335424(i) = A335424(j) for all i, j >= 1.

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A347120 Square root of the largest square dividing A005940(1+(3*A156552(n))): a(n) = A000188(A332449(n)).

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A326038 Square root of the largest square dividing the sum of divisors of n: a(n) = A000188(sigma(n)).

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A056191 Characteristic cube divisor of n: cube of g = gcd(K,F), where K is the largest square root divisor of n (A000188) and F = n/(KK) = A007913(n) is its squarefree part; g^2 divides K^2 = A008833(n) = g^2L^2 and g divides F = gf.