A008833 -id:A008833 - cp's OEIS frontend

A372602 The maximal exponent in the prime factorization of the largest square dividing n.

0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 2, 2, 0, 2, 2, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0

Offset: 1

Amiram Eldar, May 07 2024

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

Cf. A008833, A051903, A052928.

Similar sequences: A007424, A368781, A372601, A372603, A372604.

f[n_] := 2 * Floor[n/2]; a[n_] := f[Max[FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100]

s(n) = n \ 2 * 2;
a(n) = if(n>1, s(vecmax(factor(n)[,2])), 0);

a(n) = A051903(A008833(n)).
a(n) = A052928(A051903(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 * Sum_{i>=1} (1 - (1/zeta(2*i))) = 0.98112786070359477197... .

A145737 a(n) = square part of A145609(n).

Original entry on oeis.org

1, 5, 7, 1, 11, 13, 1, 17, 19, 1, 23, 1, 1, 29, 31, 1, 1, 37, 1, 41, 43, 1, 47, 1, 1, 53, 1, 1, 59, 61, 1, 1, 67, 1, 71, 73, 1, 1, 79, 1, 83, 1, 1, 89, 1, 1, 1, 97, 1, 101, 103, 1, 107, 109, 1, 113, 1, 1, 1, 1, 1, 1, 127, 1, 131, 1, 1, 137, 139, 1, 1, 1, 1, 149, 151, 1, 1, 157, 1, 1, 163

Offset: 1

Artur Jasinski, Oct 17 2008

nonn

For squarefree parts see A145738. A128059 is a very similar sequence.

Cf. A008833, A128059, A145609, A145738.

seq(denom(n!*3*n*(n+1)/(2*(2*n+1))), n=1..81); # Gary Detlefs, Oct 18 2011

m = 1; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; b = Sqrt[Numerator[k]] /. Sqrt[] -> 1; AppendTo[aa, b], {r, 1, 137}]; aa (* _Artur Jasinski *)

from sympy import isprime
def A145737(n): return a if isprime(a:=(n<<1)+1) and n>1 else 1 # Chai Wah Wu, Feb 26 2024

a(n) = 2n+1 if 2n+1 is prime, 1 otherwise, for n > 1.
From Gary Detlefs, Oct 18 2011: (Start)
a(n) = Denominator(n!*(Sum_{k=1..n} k^3)/(Sum_{k=1..n} k^2))
     = Denominator(n!*3*n*(n+1)/(2*(2*n+1))). (End)

A337534 Nontrivial squares together with nonsquares whose square part's square root is in the sequence.

Original entry on oeis.org

4, 9, 16, 25, 32, 36, 48, 49, 64, 80, 81, 96, 100, 112, 121, 144, 160, 162, 169, 176, 196, 208, 224, 225, 240, 243, 256, 272, 289, 304, 324, 336, 352, 361, 368, 400, 405, 416, 441, 464, 480, 484, 486, 496, 512, 528, 529, 544, 560, 567, 576, 592, 608, 624, 625

Offset: 1

Peter Munn, Aug 31 2020

The appearance of a number is determined by its prime signature.
No terms are squarefree, as the square root of the square part of a squarefree number is 1.
If the square part of k is a 4th power, other than 1, k appears.
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 is in this sequence if and only if there is m >= 1 such that column m of A329050 contains a member of S_k, but column m - 1 does not.

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

Eric Weisstein's World of Mathematics, Square part
Index to sequences related to prime signature

Complement of A337533.
Subsequences: A000290\{0,1}, A082294.
Subsequence of: A013929, A162643.
A209229, A267116 are used in a formula defining this sequence.
Cf. A008833, A050376, A329050.

A337534 := proc(n)
    option remember ;
    if n =1  then
        4;
    else
        for a from procname(n-1)+1 do
            if A209229(A267116(a)+1) = 0 then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A337534(n),n=1..80) ; # R. J. Mathar, Feb 16 2021

pow2Q[n_] := n == 2^IntegerExponent[n, 2]; Select[Range[625], ! pow2Q[1 + BitOr @@ (FactorInteger[#][[;; , 2]])] &] (* Amiram Eldar, Sep 18 2020 *)

Numbers k such that A209229(A267116(k) + 1) = 0.

A008833(a(n)) > 1.

A365331 The number of divisors of the largest square dividing n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 3, 3, 1, 3, 3, 1, 1, 1, 5, 1, 1, 1, 9, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 5, 3, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 3, 3, 1, 1, 1, 5, 5, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Sep 01 2023

All the terms are odd.
The sum of these divisors is A365332(n).
The number of divisors of the square root of the largest square dividing n is A046951(n).

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

Cf. A000005, A005117, A008833, A046951, A365332.

a:= n-> mul(2*iquo(i[2], 2)+1, i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 01 2023

f[p_, e_] := e + 1 - Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

a(n) = numdiv(n/core(n)); \\ Michel Marcus, Sep 02 2023

a(n) = A000005(A008833(n)).
a(n) = 1 if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = e + 1 - (e mod 2).
Dirichlet g.f.: zeta(s)*zeta(2*s)^2/zeta(4*s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5/2.
More precise asymptotics: Sum_{k=1..n} a(k) ~ 5*n/2 + 3*zeta(1/2)*sqrt(n)/Pi^2 * (log(n) + 4*gamma - 2  - 24*zeta'(2)/Pi^2 + zeta'(1/2)/zeta(1/2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 02 2023

A382889 The largest square dividing the n-th cubefree number.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 4, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 25, 1, 4, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 49, 25, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 1, 1, 25, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 1, 49, 9, 100

Offset: 1

Amiram Eldar, Apr 07 2025

Also, the powerful part of the n-th cubefree number.

All the terms are squares of squarefree numbers (A062503).

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

Cf. A002117, A004709, A008833, A057521, A062503, A371188 (positions of 1's).

Similar sequences: A382888, A382890, A382891.

f[p_, e_] := p^If[e == 1, 0, 2]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]

list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 3, print1(prod(i = 1, #f~, f[i, 1]^if(f[i, 2] == 1, 0, 2)), ", ")));}

a(n) = A008833(A004709(n)).
a(n) = A057521(A004709(n)).
a(n) = A382890(n)^2.
a(n) = A004709(n)/A382891(n).
a(n) = (A004709(n)/A382888(n))^2.
a(A371188(n)) = 1.
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(3)^(3/2) * Product_{p prime} (1 + 1/p^(3/2) - 1/p^2 - 1/p^(5/2)) = 1.48513488319516447978... .

A386469 The largest divisor of n whose exponents in its prime factorization are squares.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 16, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 16, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 48, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 16, 65, 66, 67, 34, 69, 70

Offset: 1

Amiram Eldar, Jul 22 2025

The largest term in A197680 that divides n.

The number of these divisors is A386470(n) and their sum is A386471(n).

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

Cf. A048760, A197680, A386470, A386471.

Similar sequences: A008833, A350390, A365683.

f[p_, e_] := p^(Floor[Sqrt[e]]^2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(sqrtint(f[i, 2])^2)); }

Multiplicative with a(p^e) = p^A048760(e).
a(n) <= n, with equality if and only if n is in A197680.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} Sum_{k>=2} (1/p^(k^2-1) - 1/p^(k^2-2)) = 0.74491327356409794092... .

A055654 Difference between n and the result of "Phi-summation" over unitary divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 2, 0, 0, 3, 0, 0, 0, 7, 0, 4, 0, 5, 0, 0, 0, 9, 4, 0, 8, 7, 0, 0, 0, 15, 0, 0, 0, 15, 0, 0, 0, 15, 0, 0, 0, 11, 10, 0, 0, 21, 6, 8, 0, 13, 0, 16, 0, 21, 0, 0, 0, 15, 0, 0, 14, 31, 0, 0, 0, 17, 0, 0, 0, 37, 0, 0, 12, 19, 0, 0, 0, 35, 26, 0, 0, 21, 0, 0, 0, 33, 0, 20, 0, 23

Offset: 1

Labos Elemer, Jun 07 2000

Squarefree numbers are roots of a(n)=0 equation, while Min n for which a(n)=k is k^2. See also A000188, A008833.

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

Cf. A000188, A008833, A055653, A053570, A053571, A055631, A055632, A055653, A065465.

a055654 n = a055654_list !! (n-1)
a055654_list = zipWith (-) [1..] a055653_list
-- Reinhard Zumkeller, Mar 11 2012

Table[n - DivisorSum[n, EulerPhi[#] &, CoprimeQ[#, n/#] &], {n, 92}] (* Michael De Vlieger, Oct 26 2017 *)
f[p_, e_] := p^e - p^(e-1) + 1; a[1] = 0; a[n_] := n - Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 04 2024 *)

a(n) = n - sumdiv(n, d, if (gcd(d, n/d)==1, eulerphi(d))); \\ Michel Marcus, Oct 27 2017

a(n) = {my(f = factor(n)); n - prod(k = 1, #f~, f[k,1]^f[k,2] - f[k,1]^(f[k,2] - 1) + 1);} \\ Amiram Eldar, Oct 04 2024

a(n) = n - Sum_{u|n, gcd(u,n/u) = 1} phi(u), i.e. when u is a unitary divisor of n.
a(n) = n - A055653(n). - Sean A. Irvine, Mar 30 2022
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 - A065465 = 0.11848616... . - Amiram Eldar, Oct 04 2024

A056627 a(n) = A056622(n!).

Original entry on oeis.org

1, 1, 1, 1, 1, 12, 12, 12, 36, 720, 720, 480, 480, 1680, 3024, 12096, 12096, 145152, 145152, 7257600, 345600, 1900800, 1900800, 136857600, 684288000, 4447872000, 4447872000, 435891456000, 435891456000, 3138418483200, 3138418483200, 6276836966400, 190207180800

Offset: 1

Labos Elemer, Aug 08 2000

nonn

Previous name "Square root of largest unitary square divisor of n!" was incorrect. See A374989 for the correct sequence with this name. - Amiram Eldar, Jul 26 2024

			a(12) = A056622(12!) = A000188(12!)/A055229(12!) = 1440/3 = 480.

Cf. A055772, A055230, A000188, A055229, A056622, A056628, A374989.

f[p_, 1] := 1; f[p_, e_] := If[EvenQ[e], p^(e/2), p^((e-3)/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 33] (* Amiram Eldar, Jul 08 2024 *)

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) = sqrtint(A056623(n!)); \\ Michel Marcus, Aug 16 2020

a(n) = A055772(n)/A055230(n) = A000188(n!)/A055229(n!).
a(n) = A056622(n!). - Michel Marcus, Aug 16 2020
a(n) = sqrt(A056628(n)). - Amiram Eldar, Jul 08 2024

More terms from Michel Marcus, Aug 16 2020

Incorrect name replaced with a formula by Amiram Eldar, Jul 26 2024

A331736 The largest odd divisor of A225546(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 9, 1, 1, 3, 1, 1, 1, 5, 1, 9, 1, 3, 1, 1, 1, 3, 81, 1, 9, 3, 1, 1, 1, 5, 1, 1, 1, 27, 1, 1, 1, 3, 1, 1, 1, 3, 9, 1, 1, 5, 6561, 81, 1, 3, 1, 9, 1, 3, 1, 1, 1, 3, 1, 1, 9, 15, 1, 1, 1, 3, 1, 1, 1, 27, 1, 1, 81, 3, 1, 1, 1, 5, 25, 1, 1, 3, 1, 1, 1, 3, 1, 9, 1, 3, 1, 1, 1, 5, 1, 6561, 9, 243, 1, 1, 1, 3, 1

Offset: 1

Antti Karttunen, Feb 02 2020

Antti Karttunen, Table of n, a(n) for n = 1..1680

Cf. A000265, A005117, A008833, A052928, A225546.

Array[#/2^IntegerExponent[#, 2] &@ If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 105] (* Michael De Vlieger, Feb 12 2020 *)

A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
A331736(n) = { my(f=factor(n)); for (i=1, #f~, my(p=f[i, 1]); f[i, 1] = A019565((f[i, 2]>>1)<<1); f[i, 2] = 2^(primepi(p)-1); ); factorback(f); };

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331736(n) = if(1==n,1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=2,e); for(i=2,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); prod(i=2,u,prime(i)^A048675(prods[i])));

Multiplicative, with a(prime(i)^j) = A000265(A019565(j))^A000079(i-1).
Equally, with a(prime(i)^j) = A019565(A052928(j))^A000079(i-1).
a(n) = A000265(A225546(n)).
a(n) = A225546(A008833(n)).

A334205 Under the isomorphism defined in A329329, of polynomials in GF(2)[x,y] to positive integers, a(n) is the image of the polynomial that results when x+1 is substituted for x in the polynomial with image n.

Original entry on oeis.org

1, 2, 6, 4, 10, 3, 210, 8, 36, 5, 22, 24, 858, 105, 15, 16, 1870, 72, 9699690, 40, 35, 11, 46, 12, 100, 429, 216, 840, 4002, 30, 7130, 32, 33, 935, 21, 9, 160660290, 4849845, 143, 20, 20746, 70, 1008940218, 88, 360, 23, 2569288370, 96, 44100, 200, 2805, 3432, 32589158477190044730, 108, 55, 420, 1616615, 2001, 118, 60, 21594, 3565

Offset: 1

Antti Karttunen and Peter Munn, May 04 2020

nonn

Under the isomorphism (defined in A329329), A059897(.,.), A329329(.,.) and A003961(.) represent polynomial addition, multiplication and multiplication by x respectively; prime(i+1) represents the polynomial x^i.
The equivalent sequence with y+1 substituted for y is A268385.
Self-inverse permutation of natural numbers. Squarefree numbers are mapped to squarefree numbers, squares are mapped to squares, and in general the sequence permutes {m : A267116(m) = k} for any k.
From Peter Munn, May 31 2020: (Start)
The odd numbers represent the polynomials that have x as a factor. So the odd bisection's terms represent polynomials with (x+1) as a factor. They are a permutation of A268390.
A193231 is an equivalent sequence with respect to GF(2)[x]. See the formula showing A019565 as the related injective homomorphism, mapping the usual encoding of GF(2) polynomials in x to their equivalent A329329-defined representation.
(End)

			Calculation for n = 5. 5 = prime(3) = prime(2+1) is the image of the polynomial x^2. Substituting x+1 for x, this becomes (x+1)^2 = x^2 + (1+1)x + 1 = x^2 + 1, as 1 + 1 = 0 in GF(2). The image of x^2 + 1 is A059897(prime(3), prime(1)) = A059897(5, 2) = 10. So a(5) = 10. (Note that A059897 gives the same result as multiplication when its operands are different terms of A050376, such as prime numbers.)

Antti Karttunen, Table of n, a(n) for n = 1..3670
Wikipedia, Polynomial ring
Index entries for sequences that are permutations of the natural numbers

Cf. A123098, A267116.
Equivalent GF(2)[x] sequence is A193231 (via A019565).
Equivalent sequences for other substitutions: x -> 0: A006519, (x -> y, y -> x): A225546, y -> y+1: A268385, x -> x^2: A319525.
Cf. A268390 (ordered odd bisection).
A003961, A007913, A008833, A059897, A329329 are used to express relationship between terms of this sequence.

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A225546(n) = if(1==n,1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); prod(i=1,u,prime(i)^A048675(prods[i])));
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ From A193231
A268385(n) = if(1==n, n, my(f=factor(n)); prod(i=1,#f~,f[i,1]^A193231(f[i,2])));
A334205(n) = A225546(A268385(A225546(n)));

\\ This program is better for larger values. A048675 and A193231 as in above:
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A334205(n) = if(1==n, n, if(issquare(n), A334205(sqrtint(n))^2, A019565(A193231(A048675(core(n)))) * A334205(n/core(n)))); \\ Antti Karttunen, May 24 2020

a(prime(i)^j) = A123098(i-1)^j, a(A059897(n, k)) = A059897(a(n), a(k)).
a(n) = A225546(A268385(A225546(n))).
a(A003961(n)) = A059897(a(n), A003961(a(n))) = A329329(6, a(n)).
a(n^2) = a(n)^2.
a(n) = a(A007913(n)) * a(A008833(n)).
a(A019565(n)) = A019565(A193231(n)).
A267116(a(n)) = A267116(n).

cp's OEIS Frontend

A372602 The maximal exponent in the prime factorization of the largest square dividing n.

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A145737 a(n) = square part of A145609(n).

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A337534 Nontrivial squares together with nonsquares whose square part's square root is in the sequence.

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A365331 The number of divisors of the largest square dividing n.

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A382889 The largest square dividing the n-th cubefree number.

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A386469 The largest divisor of n whose exponents in its prime factorization are squares.

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A055654 Difference between n and the result of "Phi-summation" over unitary divisors of n.

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