A008833 -id:A008833 - cp's OEIS frontend

A334747 Let p be the smallest prime not dividing the squarefree part of n. Multiply n by p and divide by the product of all smaller primes.

2, 3, 6, 8, 10, 5, 14, 12, 18, 15, 22, 24, 26, 21, 30, 32, 34, 27, 38, 40, 42, 33, 46, 20, 50, 39, 54, 56, 58, 7, 62, 48, 66, 51, 70, 72, 74, 57, 78, 60, 82, 35, 86, 88, 90, 69, 94, 96, 98, 75, 102, 104, 106, 45, 110, 84, 114, 87, 118, 120, 122, 93, 126, 128, 130, 55

Offset: 1

Peter Munn, May 09 2020

A bijection from the positive integers to the nonsquares, A000037.
A003159 (which has asymptotic density 2/3) lists index n such that a(n) = 2n. The sequence maps the terms of A003159 1:1 onto A036554, defining a bijection between them.
Similarly, bijections are defined from A007417 to A325424, from A325424 to A145204\{0}, and from the first in each of the following pairs to the nonsquare integers in the second: (A145204\{0}, A036668), (A036668, A007417), (A036554, A003159), (A332820, A332821), (A332821, A332822), (A332822, A332820). Note that many of these are between sets where membership depends on whether a number's squarefree part divides by 2 and/or 3.
Starting from 1, and iterating the sequence as a(1) = 2, a(2) = 3, a(3) = 6, a(6) = 5, a(5) = 10, etc., runs through the squarefree numbers in the order they appear in A019565. - Antti Karttunen, Jun 08 2020

			168 = 42*4 has squarefree part 42 (and square part 4). The smallest prime absent from 42 = 2*3*7 is 5 and the product of all smaller primes is 2*3 = 6. So a(168) = 168*5/6 = 140.

Permutation of A000037.
Row 2, and therefore column 2, of A331590. Cf. A334748 (row 3).
A007913, A034386, A053669, A225546 are used in formulas defining the sequence.
The formula section details how the sequence maps the terms of A002110, A003961, A019565; and how f(a(n)) relates to f(n) for f = A008833, A048675, A267116; making use of A003986.
Subsequences: A016825 (odd bisection), A036554, A329575.
Bijections are defined that relate to A003159, A007417, A036668, A145204, A325424, A332820, A332821, A332822.
Cf. also binary trees A334860, A334866 and A334870 (a left inverse).

a(n) = {my(c=core(n), m=n); forprime(p=2, , if(c % p, m*=p; break, m/=p)); m;} \\ Michel Marcus, May 22 2020

a(n) = n * m / A034386(m-1), where m = A053669(A007913(n)).
a(n) = A331590(2, n) = A225546(2 * A225546(n)).
a(A019565(n)) = A019565(n+1).
a(k * m^2) = a(k) * m^2.
a(A003961(n)) = 2 * A003961(n).
a(2 * A003961(n)) = A003961(a(n)).
a(A002110(n)) = prime(n+1).
A048675(a(n)) = A048675(n) + 1.
A008833(a(n)) = A008833(n).
A267116(a(n)) = A267116(n) OR 1, where OR denotes the bitwise operation A003986.
a(A003159(n)) = A036554(n) = 2 * A003159(n).
A334870(a(n)) = n. - Antti Karttunen, Jun 08 2020

A055460 Number of primes with odd exponents in the prime power factorization of n!.

Original entry on oeis.org

0, 1, 2, 2, 3, 1, 2, 3, 3, 1, 2, 3, 4, 4, 4, 4, 5, 4, 5, 4, 6, 6, 7, 5, 5, 5, 6, 5, 6, 5, 6, 7, 9, 7, 7, 7, 8, 8, 8, 8, 9, 10, 11, 10, 9, 7, 8, 7, 7, 8, 10, 9, 10, 8, 10, 12, 14, 12, 13, 11, 12, 12, 11, 11, 13, 12, 13, 12, 12, 13, 14, 13, 14, 14, 15, 14, 14, 11, 12, 13, 13, 13, 14, 16, 16, 14

Offset: 1

Labos Elemer, Jun 26 2000

nonn

The products of the corresponding primes form A055204.
Also, the number of primes dividing the squarefree part of n! (=A055204(n)).
Also, the number of prime factors in the factorization of n! into distinct terms of A050376. See the references in A241289. - Vladimir Shevelev, Apr 16 2014

			For n = 100, the exponents of primes in the factorization of n! are {97,48,24,16,9,7,5,5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1}, and there are 17 odd values: {97,9,7,5,5,3,3,1,1,1,1,1,1,1,1,1,1}, so a(100) = 17.
The factorization of 6! into distinct terms of A050376 is 5*9*16 with only one prime, so a(6)=1. - _Vladimir Shevelev_, Apr 16 2014

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (in Russian; MR 2000f: 11097, pp. 3912-3913).

Max Alekseyev, Table of n, a(n) for n = 1..100000
S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Cf. A000142, A000720, A007913, A008833, A162642, A348841.

Cf. A249016 (indices of records), A249017 (values of records)

Table[Count[FactorInteger[n!][[All, -1]], m_ /; OddQ@ m] - Boole[n == 1], {n, 100}] (* Michael De Vlieger, Feb 05 2017 *)

PARI
```
a(n) = omega(core(n!))
```

a(n) = A001221(A055204(n)). - Max Alekseyev, Oct 19 2014
From Wolfdieter Lang, Nov 06 2021: (Start)
a(n) = A162642(A000142(n)).
a(n) = A000720(n) - A348841(n), (End)

Edited by Max Alekseyev, Oct 19 2014

A068310 n^2 - 1 divided by its largest square divisor.

Original entry on oeis.org

3, 2, 15, 6, 35, 3, 7, 5, 11, 30, 143, 42, 195, 14, 255, 2, 323, 10, 399, 110, 483, 33, 23, 39, 3, 182, 87, 210, 899, 15, 1023, 17, 1155, 34, 1295, 38, 1443, 95, 1599, 105, 1763, 462, 215, 506, 235, 138, 47, 6, 51, 26, 2703, 78, 2915, 21, 3135, 203, 3363, 870, 3599

Offset: 2

Lekraj Beedassy, Feb 25 2002

In other words, squarefree part of n^2-1.

Least m for which x^2 - m*y^2 = 1 has a solution with x = n.

			a(6) = 35, as 6^2 - 1 = 35 itself is squarefree.
7^2-1 = 48 = A005563(6), whose largest square divisor is A008833(48) = 16, so a(7) = 48/16 = 3.

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

Cf. A002350, A007913, A067872, A033314, A027746, A175607.

a068310 n = f 1 $ a027746_row (n^2 - 1) where
   f y [] = y
   f y [p] = y*p
   f y (p:ps'@(p':ps)) | p == p' = f y ps
                       | otherwise = f (y*p) ps'
-- Reinhard Zumkeller, Nov 26 2011

a[n_] := Times@@(#[[1]] ^ Mod[ #[[2]], 2]&/@FactorInteger[n^2-1])
Table[(n^2-1)/Max[Select[Divisors[n^2-1],IntegerQ[Sqrt[#]]&]],{n,2,60}] (* Harvey P. Dale, Dec 08 2019 *)

a(n) = core(n*n - 1); \\ David Wasserman, Mar 07 2005

a(n) = A007913(n^2-1).

a(n) = A005563(n-1) / A008833(n^2 - 1). - Reinhard Zumkeller, Nov 26 2011; corrected by Georg Fischer, Dec 10 2022

Edited by Dean Hickerson, Mar 19 2002

Entry revised by N. J. A. Sloane, Apr 27 2007

A071773 a(n) = gcd(rad(n), n/rad(n)), where rad(n) = A007947(n) is the squarefree kernel of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 5, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 7, 5, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 5, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 7, 3, 10, 1, 1, 1, 2

Offset: 1

Reinhard Zumkeller, Jun 24 2002

n is squarefree iff a(n)=1.
Product of primes dividing n more than once. - Charles R Greathouse IV, Aug 08 2013
Squarefree kernel of the square part of n. - Peter Munn, Jun 12 2020

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Eric Weisstein's World of Mathematics, Squarefree.

Cf. A003415, A003557, A005117, A007947, A007948, A008833, A057521, A166486 (parity of terms), A359433 (Dirichlet inverse).

Cf. A065464.

Table[With[{r = Apply[Times, FactorInteger[n][[All, 1]]]}, GCD[r, n/r]], {n, 104}] (* Michael De Vlieger, Aug 20 2017 *)

a(n)=my(f=factor(n));prod(i=1,#f~,f[i,1]^(f[i,2]>1)) \\ Charles R Greathouse IV, Aug 08 2013

;; With memoization-macro definec.
(definec (A071773 n) (if (= 1 n) n (* (if (zero? (modulo n (expt (A020639 n) 2))) (A020639 n) 1) (A071773 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

a(n) = gcd(A007947(n), A003557(n)).
Multiplicative with p^e -> p^ceiling((e-1)/e), p prime.
a(n) = rad(n/rad(n)) = A007947(A003557(n)). - Velin Yanev, Antti Karttunen, Aug 20 2017, Nov 28 2017
a(n) = A007947(A057521(n)). - Antti Karttunen, Nov 28 2017
a(n) = A007947(A008833(n)). - Peter Munn, Jun 12 2020
a(n) = gcd(A003415(n), A007947(n)). - Antti Karttunen, Jan 02 2023
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(2*s-1) - 1/p^(2*s)). - Amiram Eldar, Nov 09 2023
From Vaclav Kotesovec, May 06 2025: (Start)
Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(4*s-1) - 1/p^(4*s-2)).
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * f(s).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
f(1) = A065464 = Product_{p prime} (1 - 2/p^2 + 1/p^3) = 0.428249505677094440218765707581823546121298513355936144031901379532123...
f'(1) = f(1) * Sum_{p prime} 2*(3*p-2)*log(p) / (p^3-2*p+1) = f(1) * 2.939073481649229666406787986900328729326669597518287791424059647447664...
and gamma is the Euler-Mascheroni constant A001620. (End)

A252849 Numbers with an even number of square divisors.

Original entry on oeis.org

4, 8, 9, 12, 18, 20, 24, 25, 27, 28, 36, 40, 44, 45, 49, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 84, 88, 90, 92, 98, 99, 100, 104, 108, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 144, 147, 148

Offset: 1

Walker Dewey Anderson, Mar 22 2015

nonn

Closed lockers in the locker problem where the student numbers are the set of perfect squares.
The locker problem is a classic mathematical problem. Imagine a row containing an infinite number of lockers numbered from one to infinity. Also imagine an infinite number of students numbered from one to infinity. All of the lockers begin closed. The first student opens every locker that is a multiple of one, which is every locker. The second student closes every locker that is a multiple of two, so all of the even-numbered lockers are closed. The third student opens or closes every locker that is a multiple of three. This process continues for all of the students.
A variant on the locker problem is when not all student numbers are considered; in the case of this sequence, only the square-numbered students open and close lockers. The sequence here is a list of the closed lockers after all of the students have gone.
From Amiram Eldar, Jul 07 2020: (Start)
Numbers k such that the largest square dividing k (A008833) is not a fourth power.
The asymptotic density of this sequence is 1 - Pi^2/15 = 1 - A182448 = 0.342026... (Cesàro, 1885). (End)
Closed under application of A331590: for n, k >= 1, A331590(a(n), k) is in the sequence. - Peter Munn, Sep 18 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ernest Cesàro, Le plus grand diviseur carré, Annali di Matematica Pura ed Applicata, Vol. 13, No. 1 (1885), pp. 251-268, entire volume.
K. A. P. Dagal, Generalized Locker Problem, arXiv:1307.6455 [math.NT], 2013.
B. Torrence and S. Wagon, The Locker Problem, Crux Mathematicorum, 2007, 33(4), 232-236.

Complement of A252895.
A046951, A335324 are used in a formula defining this sequence.
Disjoint union of A336593 and A336594.
A030140, A038109, A082293, A217319 are subsequences.
Cf. A000290, A008833, A182448, A331590.
Ordered 3rd trisection of A225546.

a252849 n = a252849_list !! (n-1)
a252849_list = filter (even . a046951) [1..]
-- Reinhard Zumkeller, Apr 06 2015

Position[Length@ Select[Divisors@ #, IntegerQ@ Sqrt@ # &] & /@ Range@ 150, Integer?EvenQ] // Flatten (* _Michael De Vlieger, Mar 23 2015 *)

isok(n) = sumdiv(n, d, issquare(d)) % 2 == 0; \\ Michel Marcus, Mar 22 2015

From Peter Munn, Sep 18 2020: (Start)
Numbers k such that A046951(k) mod 2 = 0.
Numbers k such that A335324(k) > 1.
(End)

A056060 The powerfree part of the central binomial coefficients.

Original entry on oeis.org

1, 2, 3, 6, 10, 5, 35, 70, 14, 7, 462, 231, 429, 429, 715, 1430, 24310, 12155, 92378, 46189, 88179, 88179, 1352078, 676039, 52003, 52003, 7429, 7429, 1077205, 1077205, 33393355, 66786710, 43214930, 21607465, 181502706, 90751353, 176726319, 176726319, 7658140490

Offset: 1

Labos Elemer, Jul 26 2000

nonn

			n=14, binomial(14,7) = 3432 = 8*3*11*13. The largest square divisor is 4, and the squarefree part is 858. So GCD(4,858) = 2 and a(14) = 858/2 = 429.

Amiram Eldar, Table of n, a(n) for n = 1..3388

Cf. A001405, A000188, A008833, A007913, A055229, A055231, A046098, A034974, A056056, A056057, A056058, A056059, A056060, A056061.

a[n_] := Denominator[(b = Binomial[n, Floor[n/2]])/(Times @@ First /@ FactorInteger[b])^2]; Array[a, 36] (* Amiram Eldar, Sep 05 2020 *)

a(n) = A055231(A001405(n)).

New name and more terms from Amiram Eldar, Sep 05 2020

A337533 1 together with nonsquares whose square part's square root is in the sequence.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68

Offset: 1

Peter Munn, Aug 31 2020

The appearance of a number is determined by its prime signature.
Every squarefree number is present, as the square root of the square part of a squarefree number is 1. Other 4th-power-free numbers are present if and only if they are nonsquare.
If the square part of nonsquarefree k is a 4th power, k does not appear.
Every positive integer k is the product of a unique subset S_k of the terms of A050376, which are arranged in array form in A329050 (primes in column 0, squares of primes in column 1, 4th powers of primes in column 2 and so on). k > 1 is in this sequence if and only if the members of S_k occur in consecutive columns of A329050, starting with column 0.
If the qualifying condition in the previous paragraph was based on the rows instead of the columns of A329050, we would get A055932. The self-inverse function defined by A225546 transposes A329050. A225546 also has multiplicative properties such that if we consider A055932 and this sequence as sets, A225546(.) maps the members of either set 1:1 onto the other set.

			4 is square and not 1, so 4 is not in the sequence.
12 = 3 * 2^2 is nonsquare, and has square part 4, whose square root (2) is in the sequence. So 12 is in the sequence.
32 = 2 * 4^2 is nonsquare, but has square part 16, whose square root (4) is not in the sequence. So 32 is not in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square part
Index to sequences related to prime signature

Complement of A337534.
Closed under A000188(.).
A209229, A267116 are used in a formula defining this sequence.
Subsequences: A005117, A036537, A179646, A179666, A179669, A189987, A191554, A191555, A298207, A317090.
Subsequence of A164514.
A007913, A008833, A008835, A335324 give the squarefree, square and comparably related parts of a number.
Cf. A050376, A329050.
Related to A055932 via A225546.

S:= {1}:
for n from 2 to 100 do
  if not issqr(n) then
    F:= ifactors(n)[2];
    s:= mul(t[1]^floor(t[2]/2),t=F);
    if member(s,S) then S:= S union {n} fi
  fi
od:
sort(convert(S,list)); # Robert Israel, Jan 07 2025

pow2Q[n_] := n == 2^IntegerExponent[n, 2]; Select[Range[100], # == 1 || pow2Q[1 + BitOr @@ (FactorInteger[#][[;; , 2]])] &] (* Amiram Eldar, Sep 18 2020 *)

Numbers m such that A209229(A267116(m) + 1) = 1.
If A008835(a(n)) > 1 then A335324(a(n)) > 1.
If A008833(a(n)) > 1 then A007913(a(n)) > 1.

A055230 Greatest common divisor of largest square dividing n! and squarefree part of n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 6, 10, 10, 10, 5, 5, 1, 21, 42, 42, 7, 7, 14, 42, 6, 6, 5, 5, 10, 330, 165, 231, 231, 231, 462, 2002, 5005, 5005, 4290, 4290, 390, 78, 39, 39, 13, 13, 26, 1326, 102, 102, 17, 935, 13090, 746130, 373065, 373065, 24871, 24871

Offset: 1

Labos Elemer, Jun 21 2000

nonn

			a(5) = 2 because 5! = 120; largest square divisor is 4, squarefree part is 30; GCD(4, 30) = 2.
a(7) = 1 because 7! = 5040; the largest square divisor is 144 and the squarefree part is 35 and these are coprime.

Michael De Vlieger, Table of n, a(n) for n = 1..5000

Cf. A008833, A007913, A000188, A000720.

Table[GCD[Times @@ Flatten@ Map[Table[#1, 2 Floor[#2/2]] & @@ # &, #], Times @@ Flatten@ Map[Table[#1, Floor[Mod[#2, 2]]] & @@ # &, #]] &@ FactorInteger[n!], {n, 61}] (* Michael De Vlieger, Jul 26 2016 *)

a(n) = my(fn=n!, cn=core(fn)); gcd(cn, fn/cn); \\ Michel Marcus, Dec 10 2013

a(n) = GCD(A008833(n!), A007913(n!)) = GCD(A055071(n), A055204(n)).

A056623 If n=LLgggf (see A056192) and a(n) = LL, then its complementary divisor n/LL = gggf and gcd(L^2, n/LL) = 1.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 1, 25, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1, 1, 4, 1, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 4, 1, 49, 9

Offset: 1

Labos Elemer, Aug 08 2000

The part of the name "Largest unitary square divisor of n" was removed because it is correct only for numbers whose odd exponents in their prime factorization are all smaller than 5. For the correct largest unitary square divisor of n see A350388. - Amiram Eldar, Jul 26 2024

			a(200) = A008833(200)/A055229(200)^2 = 100/2^2 = 25.
a(250) = A008833(250)/A055229(250)^2 = 25/5^2 = 1.

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

Cf. A008833, A055229, A000188, A046951, A034444, A056192, A056622, A350388.

f[p_, 1] := 1; f[p_, e_] := If[EvenQ[e], p^e, p^(e-3)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)

A055229(n) = { my(c=core(n)); gcd(c, n/c); }; \\ Charles R Greathouse IV, Nov 20 2012
A008833(n) = n/core(n) \\ Michael B. Porter, Oct 17 2009
A056623(n) = (A008833(n)/(A055229(n)^2)); \\ Antti Karttunen, Nov 19 2017

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(e == 1, 1, if(e%2, p^(e-3), p^e)));} \\ Amiram Eldar, Dec 25 2023

(definec (A056623 n) (if (= 1 n) n (let ((e (A067029 n)) (rest (A056623 (A028234 n)))) (cond ((even? e) (* (A028233 n) rest)) ((= 1 e) rest) (else (* (expt (A020639 n) (- e 3)) rest)))))) ;; After Jovovic's multiplicative formula, using memoization-macro definec - Antti Karttunen, Nov 19 2017

a(n) = A008833(n)/A055229(n)^2 = K^2/g^2, which coincides with the largest square divisor iff the g-factor is 1.
Multiplicative with a(p^e)=p^e for even e, a(p)=1, a(p^e)=p^(e-3) for odd e > 1. - Vladeta Jovovic, Apr 30 2002
From Amiram Eldar, Dec 25 2023 (Start)
Dirichlet g.f.: zeta(2*s-2) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-2) + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = Product_{p prime} (1 + 1/p^(3/2) - 1/p^(5/2) + 1/p^(9/2)) = 1.81133051934001073532... . (End)
a(n) = A056622(n)^2. - Amiram Eldar, Jul 26 2024

Name edited by Amiram Eldar, Jul 26 2024

A365171 The number of divisors d of n such that gcd(d, n/d) is a square.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Aug 25 2023

The sum of these divisors is A365172(n).

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

Cf. A000005, A000290, A000583, A004524, A005117, A008833, A034444, A046101, A046951, A182448, A365172, A365173.

f[p_, e_] := Floor[(e + 3)/4] + Floor[(e + 4)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> (x+3)\4 + (x+4)\4, factor(n)[, 2]));

for(n=1, 100, print1(direuler(p=2, n, 1/((1 - X)^2 * (1 + X^2)))[n], ", ")) \\ Vaclav Kotesovec, Jan 20 2024

Multiplicative with a(p^e) = floor((e + 3)/4) + floor((e + 4)/4) = A004524(e+3).
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
a(n) >= A034444(n), with equality if and only if n is not a biquadrateful number (A046101).
a(n) == 1 (mod 2) if and only if n is a fourth power (A000583).
From Vaclav Kotesovec, Jan 20 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/(p^(2*s) + 1)).
Let f(s) = Product_{p prime} (1 - 1/(p^(2*s) + 1)).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/(p^2 + 1)) = Pi^2/15 = A182448,
f'(1) = f(1) * Sum_{p prime} 2*log(p) / (p^2 + 1) = f(1) * 0.8852429263675811068149340172820329246145172848406469350087715037483367369...
and gamma is the Euler-Mascheroni constant A001620. (End)

cp's OEIS Frontend

A334747 Let p be the smallest prime not dividing the squarefree part of n. Multiply n by p and divide by the product of all smaller primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A055460 Number of primes with odd exponents in the prime power factorization of n!.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A068310 n^2 - 1 divided by its largest square divisor.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A071773 a(n) = gcd(rad(n), n/rad(n)), where rad(n) = A007947(n) is the squarefree kernel of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

Formula

A252849 Numbers with an even number of square divisors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A056060 The powerfree part of the central binomial coefficients.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A337533 1 together with nonsquares whose square part's square root is in the sequence.

Views

Author

Keywords

Comments

Examples