A078434 -id:A078434 - cp's OEIS frontend

A335324 Square part of 4th-power-free part of n.

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

Offset: 1

Peter Munn, May 31 2020

Equivalently, biquadratefree (4th-power-free) part of square part of n.
Multiplicative. The terms are squares of squarefree numbers (A062503).
Every positive integer n is the product of a unique subset S_n of the terms of A050376 (sometimes called Fermi-Dirac primes). a(n) is the product of the members of S_n that are squares of prime numbers (A001248).

			Removing the 4th powers from 192 = 2^6 * 3^1 gives 2^(6 - 4) * 3^1 = 2^2 * 3 = 12. So the 4th-power-free part of 192 is 12. The square part of 12 (largest square dividing 12) is 4. So a(192) = 4.

Michel Marcus, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Biquadratefree.
Eric Weisstein's World of Mathematics, Square part.

A007913, A008833, A008835, A053165 are used in formulas defining the sequence.
Column 1 of A352780.
Range of values is A062503.
Positions of 1's: A252895.
Related to A038500 by A225546.
The formula section details how the sequence maps the terms of A003961, A331590.
Cf. A001248, A050376, A083730.
Cf. A002117, A013662, A078434.

f[p_, e_] := p^(2*Floor[e/2] - 4*Floor[e/4]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 01 2020 *)

A053165(n)=my(f=factor(n)); f[, 2]=f[, 2]%4; factorback(f);
a(n) = my(m=A053165(n)); m/core(m); \\ Michel Marcus, Jun 01 2020

from math import prod
from sympy import factorint
def A335324(n): return prod(p**(e&2) for p, e in factorint(n).items()) # Chai Wah Wu, Aug 07 2024

a(n) = A053165(A008833(n)) = A008833(A053165(n)).
a(n) = A053165(n) / A007913(n).
a(n) = A008833(n) / A008835(n).
n = A007913(n) * a(n) * A008835(n).
a(n) = A225546(A038500(A225546(n))).
a(n^2) = A007913(n)^2.
a(A003961(n)) = A003961(a(n)).
a(A331590(n, k)) = A331590(a(n), a(k)).
a(p^e) = p^(2*floor(e/2) - 4*floor(e/4)). - Amiram Eldar, Jun 01 2020
From Amiram Eldar, Sep 21 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * zeta(4*s)/(zeta(2*s) * zeta(4*s-4)).
Sum_{k=1..n} a(k) ~ (4*zeta(3/2)*zeta(4))/(21*zeta(3)) * n^(3/2). (End)

A370903 Partial alternating sums of the powerful part function (A057521).

Original entry on oeis.org

1, 0, 1, -3, -2, -3, -2, -10, -1, -2, -1, -5, -4, -5, -4, -20, -19, -28, -27, -31, -30, -31, -30, -38, -13, -14, 13, 9, 10, 9, 10, -22, -21, -22, -21, -57, -56, -57, -56, -64, -63, -64, -63, -67, -58, -59, -58, -74, -25, -50, -49, -53, -52, -79, -78, -86, -85

Offset: 1

Amiram Eldar, Mar 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A057521, A370902.
Cf. A002117, A013661, A078434.
Similar sequences: A068762, A068773, A307704, A357817, A362028.

f[p_, e_] := If[e == 1, 1, p^e]; pfp[n_] := Times @@ f @@@ FactorInteger[n]; pfp[1] = 1; Accumulate[Array[(-1)^(# + 1) * pfp[#] &, 100]]

pfp(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, f[i, 1]^f[i, 2]));}
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * pfp(k); print1(s, ", "))};

a(n) = c_1 * n^(3/2) + c_2 * n^(4/3) + O(n^(6/5)), where c_1 = (zeta(3/2)/(3*zeta(3))) * ((9-12*sqrt(2))/23) * Product_{p prime} (1 + (sqrt(p)-1)/(p*(p-sqrt(p)+1))) = -0.40656281796860400941..., and c_2 = (zeta(4/3)/(4*zeta(2))) * ((2^(5/3)-3*2^(1/3)-1)/(2^(5/3)-2^(1/3)+1)) * Product_{p prime} (1 + (p^(1/3)-1)/(p*(p^(2/3)-p^(1/3)+1))) = -0.52513876339565998938... (Tóth, 2017).

A300853 L.g.f.: log(Product_{k>=1} (1 + x^(k^2))) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, -1, 1, 3, 1, -1, 1, -5, 10, -1, 1, 3, 1, -1, 1, 11, 1, -10, 1, 3, 1, -1, 1, -5, 26, -1, 10, 3, 1, -1, 1, -21, 1, -1, 1, 30, 1, -1, 1, -5, 1, -1, 1, 3, 10, -1, 1, 11, 50, -26, 1, 3, 1, -10, 1, -5, 1, -1, 1, 3, 1, -1, 10, 43, 1, -1, 1, 3, 1, -1, 1, -50, 1, -1, 26, 3, 1, -1, 1, 11, 91, -1, 1, 3, 1

Offset: 1

Ilya Gutkovskiy, Mar 13 2018

			L.g.f.: L(x) = x - x^2/2 + x^3/3 + 3*x^4/4 + x^5/5 - x^6/6 + x^7/7 - 5*x^8/8 + 10*x^9/9 - x^10/10 + ...
exp(L(x)) = 1 + x + x^4 + x^5 + x^9 + x^10 + x^13 + x^14 + ... + A033461(n)*x^n + ...

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000290, A033461, A035316, A039956, A056911, A304876, A304906.

Cf. A078434, A268682.

nmax = 85; Rest[CoefficientList[Series[Log[Product[(1 + x^k^2), {k, 1, Floor[nmax^(1/2) + 1]}]], {x, 0, nmax}],x] Range[0, nmax]]
nmax = 85; Rest[CoefficientList[Series[Sum[k^2 x^k^2/(1 + x^k^2), {k, 1,Floor[nmax^(1/2) + 1]}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, IntegerQ[#^(1/2)] &], {n, 85}]
f[p_, e_] := If[p == 2, (1 - (-2)^(e + 1))/3, (p^(2*Floor[e/2 + 1]) - 1)/(p^2 - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

seq(n)={Vec(deriv(log(prod(k=1, n, (1 + x^(k^2) + O(x*x^n))))))} \\ Andrew Howroyd, Jul 20 2018

a(n)={sumdiv(n, d, if(n%d^2, 0, (-1)^(n/d^2 + 1) * d^2))} \\ Andrew Howroyd, Jul 20 2018

G.f.: Sum_{k>=1} k^2*x^(k^2)/(1 + x^(k^2)).
a(n) = 1 if n is an odd squarefree (A056911).
a(n) = -1 if n is an even squarefree (A039956).
a(n) = Sum_{d^2|n} (-1)^(n/d^2 + 1) * d^2. - Andrew Howroyd, Jul 20 2018
Multiplicative with a(2^e) = (1 - (-2)^(e + 1))/3, and a(p^e) = (p^(2*floor(e/2 + 1)) - 1)/(p^2 - 1) for an odd prime p. - Amiram Eldar, Oct 25 2020
From Amiram Eldar, Dec 18 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * (1 - 1/2^(s-1)).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (1 - 1/sqrt(2)) * zeta(3/2)/3 = A268682 * A078434 / 3 = 0.255049... . (End)

Keyword:mult added by Andrew Howroyd, Jul 20 2018

A370329 a(n) is the number of coreful divisors of the n-th powerful number that are also powerful numbers.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 15 2024

A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n (see A307958).

The positive terms of A361430.

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

Cf. A001694, A002117, A062503, A078434, A307958, A360908 (analogous with squares), A361430, A370328.

f[p_, e_] := e - 1; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{max = 10^4}, s /@ Union@ Flatten@ Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]

lista(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[,2]; if(k == 1 || vecmin(e) > 1, print1(prod(i = 1, #e, e[i]-1), ", ")));}

a(n) = A361430(A001694(n)).
a(n) = 1 if and only if n is the square of a squarefree number (A062503).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3/2) * zeta(3) * Product_{p prime} (1 + 2/p^2 + 2/p^(5/2) - 1/p^3 - 2/p^(7/2) - 2/p^4) = 6.91748056612108993003... . (The infinite product of primes is the value of f(1/2) in A361430).

A058266 An approximation to sigma_{1/2}(n): floor( sum_{ d divides n } sqrt(d) ).

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 3, 7, 5, 7, 4, 12, 4, 8, 8, 11, 5, 13, 5, 14, 9, 10, 5, 19, 8, 11, 10, 16, 6, 21, 6, 16, 11, 12, 11, 25, 7, 12, 12, 23, 7, 24, 7, 19, 18, 13, 7, 30, 10, 19, 13, 20, 8, 26, 13, 26, 14, 15, 8, 39, 8, 15, 20, 24, 14, 28, 9, 22, 15, 28, 9, 41, 9

Offset: 1

N. J. A. Sloane, Dec 08 2000

nonn

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

Cf. A000203, A001157, A058267, A058268, A078434, A086671.

with(numtheory); f := proc(n) local d, t1, t2; t2 := 0; t1 := divisors(n); for d in t1 do t2 := t2 + sqrt(d) end do; t2 end proc; # exact value of sigma_{1/2}(n)
with(numtheory):seq(floor(sigma[1/2](n)),n=1..80);

f[n_] := Floor@DivisorSigma[1/2, n]; Array[f, 73] (* Robert G. Wilson v, Aug 17 2017*)

a(n) = floor(sumdiv(n, d, sqrt(d))); \\ Michel Marcus, Aug 17 2017

Sum_{k=1..n} a(k) ~ (2/3)*zeta(3/2) * n^(3/2). - Amiram Eldar, Jan 14 2023

A058267 An approximation to sigma_{1/2}(n): round( Sum_{ d divides n } sqrt(d) ).

Original entry on oeis.org

1, 2, 3, 4, 3, 7, 4, 7, 6, 8, 4, 12, 5, 9, 9, 11, 5, 14, 5, 14, 10, 10, 6, 20, 8, 11, 11, 16, 6, 21, 7, 17, 12, 12, 12, 25, 7, 13, 13, 23, 7, 24, 8, 19, 19, 14, 8, 31, 11, 20, 14, 20, 8, 26, 14, 26, 15, 15, 9, 39, 9, 16, 21, 25, 15, 28, 9, 23, 16, 28, 9, 42, 10

Offset: 1

N. J. A. Sloane, Dec 08 2000

nonn

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

Cf. A000203, A001157, A058266, A058268, A078434.

map(round @ numtheory:-sigma[1/2], [$1..100]); # Robert Israel, Aug 18 2017

f[n_] := Round@ DivisorSigma[1/2, n]; Array[f, 70] (* Robert G. Wilson v, Aug 17 2017 *)

a(n) = round(sumdiv(n, d, sqrt(d))); \\ Michel Marcus, Aug 17 2017

Sum_{k=1..n} a(k) ~ (2/3)*zeta(3/2) * n^(3/2). - Amiram Eldar, Jan 14 2023

A058268 An approximation to sigma_{1/2}(n): ceiling( sum_{d|n} sqrt(d) ).

Original entry on oeis.org

1, 3, 3, 5, 4, 7, 4, 8, 6, 8, 5, 13, 5, 9, 9, 12, 6, 14, 6, 15, 10, 11, 6, 20, 9, 12, 11, 17, 7, 22, 7, 17, 12, 13, 12, 26, 8, 13, 13, 24, 8, 25, 8, 20, 19, 14, 8, 31, 11, 20, 14, 21, 9, 27, 14, 27, 15, 16, 9, 40, 9, 16, 21, 25, 15, 29, 10, 23, 16, 29, 10, 42

Offset: 1

N. J. A. Sloane, Dec 08 2000

nonn

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

Cf. A000203, A001157, A058266, A058267, A078434.

with(numtheory); f := proc(n) local d, t1, t2; t2 := 0; t1 := divisors(n); for d in t1 do t2 := t2 + sqrt(d) end do; t2 end proc; # exact value of sigma_{1/2}(n)

a[n_] := Ceiling[DivisorSigma[1/2, n]]; Array[a, 70] (* Amiram Eldar, Jan 14 2023 *)

Sum_{k=1..n} a(k) ~ (2/3)*zeta(3/2) * n^(3/2). - Amiram Eldar, Jan 14 2023

A293235 a(n) is the sum of proper divisors of n that are square.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 5, 1, 1, 1, 21, 1, 1, 1, 14, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 21, 1, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 21, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 26, 5, 1, 1, 1, 21, 10, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1, 1, 1, 21, 1, 50, 10, 30, 1, 1, 1, 5, 1

Offset: 1

Antti Karttunen, Oct 08 2017

nonn

a(n) = 1 if and only if n > 1 is squarefree or the square of a prime. - Robert Israel, Oct 08 2017

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

Cf. A010052, A035316, A078434, A293228, A293234.

A035316:= n -> mul((p[1]^(p[2]+2-(p[2] mod 2))-1)/(p[1]^2-1), p = ifactors(n)[2]):
f:= n -> A035316(n) - `if`(issqr(n),n,0):
map(f, [$1..100]); # Robert Israel, Oct 08 2017

Table[Total[Select[Most[Divisors[n]],IntegerQ[Sqrt[#]]&]],{n,120}] (* Harvey P. Dale, Dec 29 2023 *)

PARI
```
A293235(n) = sumdiv(n,d,(d
				
```

a(n) = Sum_{d|n, dA010052(d)*d.
a(n) = A035316(n) - (A010052(n)*n).
G.f.: Sum_{k>=1} k^2 * x^(2*k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Apr 13 2021
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (zeta(3/2)-1)/3 = 0.537458449561... . - Amiram Eldar, Dec 01 2023

A370786 Powerful numbers with an odd number of prime factors (counted with multiplicity).

Original entry on oeis.org

8, 27, 32, 72, 108, 125, 128, 200, 243, 288, 343, 392, 432, 500, 512, 648, 675, 800, 968, 972, 1125, 1152, 1323, 1331, 1352, 1372, 1568, 1728, 1800, 2000, 2048, 2187, 2197, 2312, 2592, 2700, 2888, 3087, 3125, 3200, 3267, 3528, 3872, 3888, 4232, 4500, 4563, 4608

Offset: 1

Amiram Eldar, Mar 02 2024

Jakimczuk (2024) proved:
The number of terms that do not exceed x is N(x) = c * sqrt(x) + o(sqrt(x)) where c = (zeta(3/2)/zeta(3) - 1/zeta(3/2))/2 = 0.895230... .
The relative asymptotic density of this sequence within the powerful numbers is (1 - zeta(3)/(zeta(3/2)^2))/2 = 0.411930... .
In general, the relative asymptotic density of the s-full numbers (numbers whose exponents in their prime factorization are all >= s) with an odd number of prime factors (counted with multiplicity) within the s-full numbers is smaller than 1/2 when s is odd.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Arithmetical Functions over the Powerful Part of an Integer, ResearchGate, 2024.

Intersection of A001694 and A026424.
Complement of A370785 within A001694.
A370788 is a subsequence.
Cf. A002117, A078434, A090699.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, AllTrue[e, # > 1 &] && OddQ[Total[e]]]; Select[Range[2500], q]

is(n) = {my(e = factor(n)[, 2]); n > 1 && vecmin(e) > 1 && vecsum(e)%2;}

A211113 Decimal expansion of -zeta(-1/2).

Original entry on oeis.org

2, 0, 7, 8, 8, 6, 2, 2, 4, 9, 7, 7, 3, 5, 4, 5, 6, 6, 0, 1, 7, 3, 0, 6, 7, 2, 5, 3, 9, 7, 0, 4, 9, 3, 0, 2, 2, 2, 6, 2, 6, 8, 5, 3, 1, 2, 8, 7, 6, 7, 2, 5, 3, 7, 6, 1, 0, 1, 1, 3, 5, 5, 7, 1, 0, 6, 1, 4, 7, 2, 9, 1, 9, 3, 2, 2, 9, 2, 3, 4, 0, 4, 8, 7, 5, 4, 3, 2, 6, 6, 9, 4, 0, 7, 3, 3, 2, 1, 5, 6, 4, 3, 1, 0, 9, 9, 7, 5, 6

Offset: 0

Stanislav Sykora, May 17 2012

			0.207886224977354566017306725397049302226...

Wikipedia, Riemann zeta function

Cf. A059750, A078434.

Maple
```
-Zeta(-1/2) ; evalf(%) ;
```

RealDigits[-Zeta[-1/2], 10, 100][[1]] (* Amiram Eldar, Mar 24 2021 *)

-zeta(-1/2) \\ Charles R Greathouse IV, May 18 2012

Equals -zeta(-1/2) = zeta(3/2)/(4*Pi) = A078434/ (10*A019694).

cp's OEIS Frontend

A335324 Square part of 4th-power-free part of n.

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A370903 Partial alternating sums of the powerful part function (A057521).

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A300853 L.g.f.: log(Product_{k>=1} (1 + x^(k^2))) = Sum_{n>=1} a(n)*x^n/n.

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A370329 a(n) is the number of coreful divisors of the n-th powerful number that are also powerful numbers.

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A058266 An approximation to sigma_{1/2}(n): floor( sum_{ d divides n } sqrt(d) ).

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A058267 An approximation to sigma_{1/2}(n): round( Sum_{ d divides n } sqrt(d) ).

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A058268 An approximation to sigma_{1/2}(n): ceiling( sum_{d|n} sqrt(d) ).

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