A008833 -id:A008833 - cp's OEIS frontend

A306454 a(n) = A261327(n)/A013946(n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 25, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 169, 1, 1, 1, 1, 1, 1, 25, 1, 1, 25, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 25, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 289, 1, 1, 1, 1, 1, 841, 1, 1, 1, 25, 1, 1, 25, 1, 1, 1

Offset: 1

Paul Curtz, Feb 16 2019

nonn

Are all terms odd squares?
b(n) = A013946(n)*A261327(n) = 25, 4, 169, 25, 841, 100, 2809, 289, 7225, 676, 625, ... . Are all terms squares?
a(n) = A008833(n^2+4) if n is odd and A008833((n^2+4)/4) if n is even, so a(n) is always an odd square. - Jianing Song, Feb 27 2019
Are the square roots only primes?
The sequence of period 4: repeat [25, 1, 1, 25] appears apparently every 25 terms.
From Robert Israel, Mar 20 2019: (Start)
The first term whose square root is not 1 or a prime is a(261) = 25^2.
a(11+25*k) is divisible by 25. The first term where a(11+25*k) > 25 is a(261)=a(11+25*10)=625.
The first term where a(12+25*k) > 1 is a(1212)=a(12+25*48)=169.
The first term where a(13+25*k) > 1 is a(213)=a(13+25*8)=289.
a(14+25*k) is divisible by 25. The first term where a(14+25*k) > 25 is a(364)=a(14+25*14)=625.
All prime factors of members of the sequence are in A002144. For any p in A002144, there is k with 1 <= k < p^2/2 such that p^2 | a(n) if and only if n == k or -k (mod p^2). (End)

			A261327(n) = 5, 2, 13, 5, 29, 10, 53, 17, 85, 26, 125, 37, 173, 50, ... .
A013946(n) = 5, 2, 13, 5, 29, 10, 53, 17, 85, 26,   5, 37, 173,  2, ... .

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

Cf. A013946, A261327.

Cf. A002144, A008833.

core:= proc(n) local t; mul(t[1],t=select(s -> s[2]::odd, ifactors(n)[2])) end proc:
map(n -> numer((4+n^2)/4)/core(n^2+4), [$1..100]); # Robert Israel, Mar 20 2019

core[n_] := Times @@ Select[FactorInteger[n], OddQ[#[[2]]]&][[All, 1]];
a[n_] := Numerator[(n^2+4)/4]/core[n^2+4];
Array[a, 100] (* Jean-François Alcover, Jan 04 2022 *)

A013946(n) = core(n^2+4); \\ From A013946
A261327(n) = if(n%2, n^2+4, (n/2)^2+1); \\ From A261327
A306454(n) = (A261327(n)/A013946(n)); \\ Antti Karttunen, Feb 28 2019

A306454(n) = { my(k=((n^2)+4)/if(n%2,1,4)); k/core(k); }; \\ Antti Karttunen, Feb 28 2019, after Jianing Song's formula

A357684 The squarefree part (A007913) of numbers whose squarefree part is a unitary divisor (A335275).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 1, 26, 7, 29, 30, 31, 33, 34, 35, 1, 37, 38, 39, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 55, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 73, 74, 3, 19, 77, 78, 79

Offset: 1

Amiram Eldar, Oct 09 2022

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts, International Mathematical Forum, Vol. 12, No. 7 (2017), pp. 331-338.

Cf. A000290, A005117, A007913, A008833, A065465, A069891, A335275.

s[n_] := If[AllTrue[(f = FactorInteger[n])[[;; , 2]], # == 1 || EvenQ[#] &], i = Position[f[[;; , 2]], 1] // Flatten; Times @@ f[[i, 1]], Nothing]; Array[s, 100]

s(n) = {my(f = factor(n), ans = 1); for(k = 1, #f~, if(f[k,2] > 1 && f[k,2]%2, ans = 0)); if(ans, ans = prod(k = 1, #f~, if(f[k,2] == 1, f[k,1], 1))) };
for(n = 1, 100, if(s(n) > 0, print1(s(n), ", ")))

a(n) = A007913(A335275(n)).
a(n) = 1 iff A335275(n) is a square (A000290).
a(n) = A335275(n) iff A335275(n) is squarefree (A005117).
Sum_{k, a(k) <= x} ~ c*x^2 + o(x^2), where c = (3/Pi^2) * Sum_{k>=1} f(k)/k^4 = 0.32103327852028541131..., and f(k) = Product_{p prime | k} (p/(p+1)) (Jakimczuk, 2017).
Sum_{k=1..n} a(k) ~ c'*x^2 + o(x^2), where c' = c / (A065465)^2 = 0.41313480468422995583... .

A362412 The number of prime factors of the square root of the largest square dividing n, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0

Offset: 1

Amiram Eldar, Sep 28 2023

First differs from A366073 at n = 64.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000188, A000290, A001222, A004526, A005117, A008833, A154945, A162642, A349326, A366073.

f[p_, e_] := Floor[e/2]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI
```
a(n) = vecsum(factor(n)[, 2]\2);
```

a(n) = A001222(A000188(n)).
a(n) = A001222(A008833(n))/2.
Additive with a(p^e) = floor(e/2) = A004526(e).
a(n) >= 0, with equality if and only if n is squarefree (A005117).
a(n) <= A001222(n)/2, with equality if and only if n is square (A000290).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p^2-1) = 0.551693... (A154945).
a(n) = (A001222(n) - A162642(n))/2. - Ridouane Oudra, Apr 19 2025

A365837 Largest proper square divisor of n, for n >= 2; a(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 4, 1, 1, 1, 16, 1, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 16, 1, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 16, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 25, 4, 1, 1, 1, 16, 9, 1, 1, 4, 1, 1, 1, 4, 1, 9, 1, 4, 1, 1, 1, 16, 1, 49, 9, 25

Offset: 1

Ilya Gutkovskiy, Oct 17 2023

nonn

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

Cf. A008833, A032742, A063928, A085392, A089384, A093803, A366524.

f:= proc(n) local F, t;
  if issqr(n) then
    n/min(numtheory:-factorset(n))^2
  else
    F:= ifactors(n)[2];
    mul(t[1]^(2*floor(t[2]/2)),t=F)
  fi
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Nov 20 2023

Join[{1}, Table[Last[Select[Divisors[n], # < n && IntegerQ[Sqrt[#]]  &]], {n, 2, 100}]]
f[p_, e_] := p^(2*Floor[e/2]); a[n_] := Module[{fct = FactorInteger[n]}, Times @@ f @@@ fct/If[AllTrue[fct[[;; , 2]], EvenQ], fct[[1, 1]]^2, 1]]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

a(n) = if (n==1, 1, my(d=divisors(n)); vecmax(select(issquare, Vec(d, #d-1)))); \\ Michel Marcus, Oct 17 2023

from math import prod
from sympy import factorint
def A365837(n):
    if n<=1: return 1
    f = factorint(n)
    return prod(p**(e&-2) for p, e in f.items())//(min(f)**2 if all(e&1^1 for e in f.values()) else 1) # Chai Wah Wu, Oct 20 2023

A380168 Nonsquares whose square part is greater than their squarefree part.

Original entry on oeis.org

8, 12, 18, 27, 32, 45, 48, 50, 54, 63, 72, 75, 80, 96, 98, 108, 112, 125, 128, 147, 150, 160, 162, 175, 176, 180, 192, 200, 208, 216, 224, 240, 242, 243, 245, 250, 252, 275, 288, 294, 300, 320, 325, 338, 343, 350, 360, 363, 375, 384, 392, 396, 405, 425, 432, 448

Offset: 1

Felix Huber, Jan 25 2025

nonn

Numbers A000037(k) for which A007913(A000037(k)) < A008833(A000037(k)).

			8 is in the sequence because its square part is 4 and its squarefree part is 2.
150 is in the sequence because its square part is 25 and its squarefree part is 6.

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A000037, A000290, A007913, A008833, A056623.

A380168:=proc(n)
    option remember;
    local a,r,i;
    if n=1 then
        8
    else
        for a from procname(n-1)+1 do
            r:=1;
            for i in ifactors(a)[2] do
                if is(i[2],odd) then
                    r:=r*i[1]
                fi
            od;
            if r>1 and rA380168(n),n=1..56);

select(x->(x>sqr(core(x))), select(x->(!issquare(x)), [1..500])) \\ Michel Marcus, Feb 10 2025

A056629 a(n) = A034444(A056627(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 4, 4, 4, 8, 8, 8, 8, 16, 8, 8, 8, 8, 8, 16, 8, 16, 16, 16, 16, 32, 32, 64, 64, 64, 64, 64, 32, 64, 64, 64, 64, 128, 64, 64, 64, 64, 64, 128, 128, 256, 256, 256, 256, 256, 128, 256, 256, 256, 256, 256, 128, 256, 256, 256, 256, 512, 512, 512, 512, 512

Offset: 1

Labos Elemer, Aug 08 2000

nonn

Previous name, "Number of unitary square divisors of n!." was incorrect. See A375187 for the correct sequence with this name. - Amiram Eldar, Aug 03 2024

			a(10) = A034444(A056627(10)) = A034444(720) = 8.

Cf. A008833, A034444, A055229, A056627, A375187.

A008833[n_] := First[Select[Reverse[Divisors[n]], IntegerQ[Sqrt[#]] &, 1]]; A055229[n_] := With[{sf = Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n])}, GCD[sf, n/sf]]; Table[2^(PrimeNu[Sqrt[A008833[n!]]/A055229[n!]]), {n, 1, 50}] (* G. C. Greubel, May 19 2017 *)
f[p_, 1] := 1; f[p_, e_] := If[EvenQ[e], p^(e/2), p^((e-3)/2)]; a[1] = 1; a[n_] := 2^PrimeNu[Times @@ f @@@ FactorInteger[n!]]; Array[a, 66] (* Amiram Eldar, Aug 03 2024 *)

a(n) = {my(f = factor(n!)); 2^omega(prod(i = 1, #f~, if(f[i, 2] == 1, 1, f[i, 1]^if(f[i, 2]%2, (f[i, 2]-3)/2, f[i, 2]/2))));} \\ Amiram Eldar, Aug 03 2024

Incorrect name replaced with a formula by Amiram Eldar, Aug 03 2024

A056630 a(n) = A055993(n) - A034444(A056627(n)).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 4, 8, 22, 22, 28, 28, 56, 88, 120, 120, 172, 172, 284, 292, 584, 584, 848, 1136, 2272, 2656, 4304, 4304, 5312, 5312, 6080, 6112, 12992, 16256, 19376, 19376, 38752, 43136, 47936, 47936, 63936, 63936, 100672, 132928, 278528, 278528

Offset: 1

Labos Elemer, Aug 08 2000

nonn

Previous name, "Number of non-unitary square divisors of n!." was incorrect. See A375188 for the correct sequence with this name. - Amiram Eldar, Aug 03 2024

			example: a(10) = A055993(10) - A034444(A056627(10)) = 30 - A034444(720) = 30 - 8 = 22.

Cf. A008833, A034444, A046951, A055993, A055229, A056627, A375188.

A046951[n_] := Length[Select[Divisors[n], IntegerQ[Sqrt[#]] &]]; A008833[n_] := First[Select[Reverse[Divisors[n]], IntegerQ[Sqrt[#]] &, 1]]; A055229[n_] := With[{sf = Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n])}, GCD[sf, n/sf]]; Table[A046951[n!] - 2^(PrimeNu[Sqrt[A008833[n!]]/A055229[n!]]), {n,1,50}] (* G. C. Greubel, May 20 2017 *)
f1[p_, e_] := 1 + Floor[e/2]; f2[p_, 1] := 1; f2[p_, e_] := If[EvenQ[e], p^(e/2), p^((e-3)/2)]; ; a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n!]) - 2^PrimeNu[Times @@ f2 @@@ fct]; Array[a, 60] (* Amiram Eldar, Aug 03 2024 *)

a(n) = {my(f = factor(n!)); prod(i = 1, #f~, 1 + f[i, 2]\2) - 2^omega(prod(i = 1, #f~, if(f[i, 2] == 1, 1, f[i, 1]^if(f[i, 2]%2, (f[i, 2]-3)/2, f[i, 2]/2))));} \\ Amiram Eldar, Aug 03 2024

Incorrect name replaced with a formula by Amiram Eldar, Aug 03 2024

A056648 a(n) = A034444(A056647(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 4, 2, 4, 2, 2, 4, 2, 2, 1, 2, 2, 4, 4, 2, 2, 4, 2, 1, 2, 1, 2, 4, 4, 8, 8, 4, 4, 2, 2, 4, 2, 1, 1, 2, 1, 2, 2, 2, 4, 4, 2, 4, 2, 4, 4, 2, 1, 2, 2, 1, 4, 2, 4, 8, 2, 4, 8, 4, 2, 1, 2, 4, 8, 4, 2, 4, 4, 8, 4, 4, 8, 16, 8, 4, 4, 2, 1, 2, 2, 1

Offset: 1

Labos Elemer, Aug 09 2000

nonn

Previous name, "Number of unitary square divisors of central binomial coefficient", was incorrect. See A376555 for the correct sequence with this name. - Amiram Eldar, Sep 28 2024

			a(28) = A034444(A056647(28)) = A034444(25) = 2.

Cf. A000188, A001221, A001405, A008833, A034444, A055229, A056056, A056057, A056059, A056061, A376555.

A008833[n_] := First[Select[Reverse[Divisors[n]], IntegerQ[Sqrt[#]] &, 1]]; A055229[n_] := With[{sf = Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n])}, GCD[sf, n/sf]]; Table[2^(PrimeNu[ Sqrt[A008833[Binomial[n, Floor[n/2]]]]/A055229[Binomial[n, Floor[n/2]]]]), {n, 1, 25}] (* G. C. Greubel, May 20 2017 *)

a(n) = 2^r, where r = A001221(A000188(A001405(n))/A055229(A001405(n))).

Incorrect name replaced with a formula by Amiram Eldar, Sep 28 2024

A056649 a(n) = A056061(n) - A034444(A056647(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 4, 6, 2, 2, 0, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 6, 8, 0, 0, 0, 4, 4, 6, 2, 2, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 4, 4, 0, 0, 4, 8, 2, 3, 6, 8, 4, 8, 2, 2, 4, 4, 8, 8, 0, 0, 0, 4, 2, 4, 3, 4, 2, 3, 4

Offset: 1

Labos Elemer, Aug 09 2000

nonn

Previous name, "Number of non-unitary square divisors of central binomial coefficient", was incorrect. See A376556 for the correct sequence with this name. - Amiram Eldar, Sep 28 2024

			a(28) = A056061(28) - A034444(A056647(28)) = A056061(28) - A034444(25) = 8 - 2 = 6.

Cf. A000188, A001221, A001405, A008833, A034444, A055229, A056056, A056057, A056059, A056061, A376556.

A056061[n_] := Count[Divisors@Binomial[n, Floor[n/2]], d_ /; IntegerQ@Sqrt@d]; A008833[n_] := First[Select[Reverse[Divisors[n]], IntegerQ[Sqrt[#]] &, 1]]; A055229[n_] := With[{sf = Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n])}, GCD[sf, n/sf]];
Table[A056061[n] - 2^(PrimeNu[Sqrt[A008833[Binomial[n, Floor[n/2]]]]/ A055229[Binomial[n, Floor[n/2]]]]), {n, 1, 15}] (* G. C. Greubel, May 20 2017 *)

a(n) = A056061(n) - 2^r, where r = A001221(A000188(A001405(n))/A055229(A001405(n))).

Incorrect name replaced with a formula by Amiram Eldar, Sep 28 2024

A175384 A positive integer k is included if the largest square dividing k is not equal to the largest square that, when written in binary, occurs as a substring in binary k.

Original entry on oeis.org

17, 19, 27, 33, 34, 35, 37, 38, 39, 41, 45, 51, 54, 57, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 89, 90, 97, 99, 101, 102, 103, 105, 108, 113, 114, 115, 117, 125, 126, 129, 130, 131, 132, 133, 134, 135, 137, 138, 139, 141, 142, 143, 145, 146

Offset: 1

Leroy Quet, Apr 24 2010

A008833(a(n)) does not equal A162400(a(n)).

			The largest square dividing 17 is 1. However, 17 in binary is 10001; and the largest square occurring, in its binary representation, within 10001 is 4 (100 in binary). Since 1 does not equal 4, then 17 is in this sequence.

Cf. A008833, A162400.

From R. J. Mathar, Aug 31 2010: (Start)
A008833 := proc(n) local b; b := floor(sqrt(n)) ; while b >= 1 do if n mod (b^2) = 0 then return b^2 ; end if; b := b-1 ; end do: end proc:
A162400 := proc(n) local b,nbin,a; a := 1 ; nbin := convert(n,base,2) ; for b from 1 to floor(sqrt(n)) do convert(b^2,base,2) ; if verify(%,nbin,'sublist') then a := b^2 ; end if; end do: a ; end proc:
isA162400 := proc(n) A008833(n) <> A162400(n) ; end proc:
for n from 1 to 300 do if isA162400(n) then printf("%d,",n) ; end if; end do: (End)

More terms from R. J. Mathar, Aug 31 2010

cp's OEIS Frontend

A306454 a(n) = A261327(n)/A013946(n).

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A357684 The squarefree part (A007913) of numbers whose squarefree part is a unitary divisor (A335275).

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A362412 The number of prime factors of the square root of the largest square dividing n, counted with multiplicity.

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A365837 Largest proper square divisor of n, for n >= 2; a(1) = 1.

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A380168 Nonsquares whose square part is greater than their squarefree part.

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A056629 a(n) = A034444(A056627(n)).

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A056630 a(n) = A055993(n) - A034444(A056627(n)).

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A056648 a(n) = A034444(A056647(n)).