A306633 -id:A306633 - cp's OEIS frontend

A371188 Indices of the squarefree numbers in the sequence of cubefree numbers.

1, 2, 3, 5, 6, 7, 9, 10, 12, 13, 14, 15, 17, 19, 20, 21, 23, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 40, 41, 44, 46, 47, 48, 49, 50, 52, 53, 55, 56, 57, 59, 60, 61, 62, 63, 66, 67, 68, 69, 70, 72, 73, 74, 75, 77, 79, 80, 81, 82, 86, 87, 88, 89, 90, 91

Offset: 1

Amiram Eldar, Mar 14 2024

The asymptotic density of this sequence is zeta(3)/zeta(2) = 0.730762... (A253905).

			The first 5 cubefree numbers are 1, 2, 3, 4, and 5. The 1st, 2nd, 3rd, and 5th, 1, 2, 3, and 5, are squarefree. Therefore, the first 4 terms of this sequence are 1, 2, 3, and 5.

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

Cf. A004709, A005117.

Cf. A253905, A306633.

freeQ[n_, k_] := AllTrue[FactorInteger[n], Last[#] < k &]; Position[Select[Range[200], freeQ[#, 3] &], _?(freeQ[#1, 2] &), Heads -> False] // Flatten

isfree(n, k) = n == 1 || vecmax(factor(n)[, 2]) < k;
lista(kmax) = {my(c = 0); for(k = 1, kmax, if(isfree(k, 3), c++; if(isfree(k, 2), print1(c, ", "))));}

A004709(a(n)) = A005117(n).

a(n) ~ c * n, where c = zeta(2)/zeta(3) = 1.368432... (A306633).

A374783 Numerator of the mean unitary abundancy index of the unitary divisors of n.

Original entry on oeis.org

1, 5, 7, 9, 11, 35, 15, 17, 19, 11, 23, 21, 27, 75, 77, 33, 35, 95, 39, 99, 5, 115, 47, 119, 51, 135, 55, 135, 59, 77, 63, 65, 161, 175, 33, 19, 75, 195, 63, 187, 83, 25, 87, 207, 209, 235, 95, 77, 99, 51, 245, 243, 107, 275, 23, 255, 91, 295, 119, 231, 123, 315

Offset: 1

Amiram Eldar, Jul 20 2024

The unitary abundancy index of a number k is A034448(k)/k = A332882(k)/A332883(k).
The record values of a(n)/A374784(n) are attained at the primorial numbers (A002110).
The least number k such that a(k)/A374784(k) is larger than 2, 3, 4, ..., is A002110(9) = 223092870, A002110(314) = 7.488... * 10^878, A002110(65599) = 5.373... * 10^356774, ... .

			For n = 4, 4 has 2 unitary divisors, 1 and 4. Their unitary abundancy indices are usigma(1)/1 = 1 and usigma(4)/4 = 5/4, and their mean unitary abundancy index is (1 + 5/4)/2 = 9/8. Therefore a(4) = numerator(9/8) = 9.

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

Cf. A002110, A005117, A034444, A034448, A077610, A306633, A332882, A332883, A343525, A374784 (denominators).

Similar sequences: A374777/A374778, A374786/A374787.

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

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

Let f(n) = a(n)/A374784(n). Then:
f(n) = (Sum_{d|n, gcd(d, n/d) = 1} usigma(d)/d) / ud(n), where usigma(n) is the sum of unitary divisors of n (A034448), and ud(n) is their number (A034444).
f(n) is multiplicative with f(p^e) = 1 + 1/(2*p^e).
f(n) = (Sum_{d|n, gcd(d, n/d) = 1} d*ud(d))/(n*ud(n)) = A343525(n)/(n*A034444(n)).
Dirichlet g.f. of f(n): zeta(s) * zeta(s+1) * Product_{p prime} (1 - 1/(2*p^(s+1)) - 1/(2*p^(2*s+1))).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} f(k) = Product_{p prime} (1 + 1/(2*p*(p+1))) = 1.17443669198552182119... . For comparison, the asymptotic mean of the unitary abundancy index over all the positive integers is zeta(2)/zeta(3) = 1.368432... (A306633).
Lim sup_{n->oo} f(n) = oo (i.e., f(n) is unbounded).
f(n) <= A374777(n)/A374778(n) with equality if and only if n is squarefree (A005117).

A328258 a(n) = Sum_{d|n, gcd(d,n/d) = 1} (-1)^(d + 1) * d.

Original entry on oeis.org

1, -1, 4, -3, 6, -4, 8, -7, 10, -6, 12, -12, 14, -8, 24, -15, 18, -10, 20, -18, 32, -12, 24, -28, 26, -14, 28, -24, 30, -24, 32, -31, 48, -18, 48, -30, 38, -20, 56, -42, 42, -32, 44, -36, 60, -24, 48, -60, 50, -26, 72, -42, 54, -28, 72, -56, 80, -30, 60, -72, 62, -32, 80, -63, 84

Offset: 1

Ilya Gutkovskiy, Oct 09 2019

Excess of sum of odd unitary divisors of n over sum of even unitary divisors of n.

a(n) = n+1 iff n is in A061345 \ {1}. - Bernard Schott, Mar 05 2023

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor.

Cf. A002129, A034448, A077610, A109712, A192066, A254503, A306633, A360156.

[&+[(-1)^(d+1)*d:d in Divisors(n)|Gcd(d, n div d) eq 1]:n in [1..70]]; // Marius A. Burtea, Oct 10 2019

f:= proc(n) local t;
  mul(1 - (-1)^t[1] * t[1]^t[2], t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Oct 10 2019

a[n_] := Sum[Boole[GCD[d, n/d] == 1] (-1)^(d + 1) d, {d, Divisors[n]}]; Table[a[n], {n, 1, 65}]
a[1] = 1; a[n_] := Times @@ (1 - (-1)^First[#] First[#]^Last[#] & /@ FactorInteger[n]); Table[a[n], {n, 1, 65}]

a(n) = sumdiv(n, d, if (gcd(d,n/d) == 1, (-1)^(d + 1) * d)); \\ Michel Marcus, Oct 10 2019

If n = Product (p_j^k_j) then a(n) = Product (1 - (-1)^p_j * p_j^k_j).
If n odd, a(n) = usigma(n), where usigma = A034448.
Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2)/(14*zeta(3)) = A306633 / 14 = 0.0977451... . - Amiram Eldar, Nov 17 2022
From Amiram Eldar, Jan 28 2023: (Start)
a(n) = 2 * A192066(n) - A034448(n).
a(n) = A192066(n) - A360156(n/2) if n is even, and A192066(n) otherwise.
Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s-1))*(2^(2*s)-2^(s+2)+2)/(2^(2*s)-2). (End)

A332882 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j^k_j).

Original entry on oeis.org

1, 3, 4, 5, 6, 2, 8, 9, 10, 9, 12, 5, 14, 12, 8, 17, 18, 5, 20, 3, 32, 18, 24, 3, 26, 21, 28, 10, 30, 12, 32, 33, 16, 27, 48, 25, 38, 30, 56, 27, 42, 16, 44, 15, 4, 36, 48, 17, 50, 39, 24, 35, 54, 14, 72, 9, 80, 45, 60, 2, 62, 48, 80, 65, 84, 24, 68, 45, 32, 72

Offset: 1

Ilya Gutkovskiy, Feb 28 2020

Numerator of sum of reciprocals of unitary divisors of n.

			1, 3/2, 4/3, 5/4, 6/5, 2, 8/7, 9/8, 10/9, 9/5, 12/11, 5/3, 14/13, 12/7, 8/5, 17/16, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Unitary Divisor.

Cf. A017665, A028235, A028236, A034448, A077610, A306633, A319676, A323166, A332880, A332883 (denominators).

Cf. also A348734, A348735.

a:= n-> numer(mul(1+i[1]^i[2], i=ifactors(n)[2])/n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020

Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]]^#[[2]] & /@ FactorInteger[n])], {n, 1, 70}] // Numerator
Table[Sum[If[GCD[d, n/d] == 1,  1/d, 0], {d, Divisors[n]}], {n, 1, 70}] // Numerator

a(n) = numerator(sumdiv(n, d, if (gcd(d, n/d)==1, 1/d))); \\ Michel Marcus, Feb 28 2020

a(n) = numerator of Sum_{d|n, gcd(d, n/d) = 1} 1/d.
a(n) = numerator of usigma(n)/n.
a(p) = p + 1, where p is prime.
a(n) = A034448(n) / A323166(n). - Antti Karttunen, Nov 13 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A332883(k) = zeta(2)/zeta(3) = 1.368432... (A306633). - Amiram Eldar, Nov 21 2022

A367171 The sum of divisors of the largest unitary divisor of n that is a term of A138302.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 1, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 4, 31, 42, 1, 56, 30, 72, 32, 1, 48, 54, 48, 91, 38, 60, 56, 6, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 3, 72, 8, 80, 90, 60, 168, 62, 96, 104, 1, 84, 144, 68, 126

Offset: 1

Amiram Eldar, Nov 07 2023

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

Cf. A000203, A048298, A138302, A306633, A367168, A367169, A367170.

Similar sequences: A351568, A351569.

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], (p^(e+1)-1)/(p-1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1 << valuation(f[i, 2], 2), (f[i, 1]^(f[i, 2]+1)-1)/(f[i, 1]-1), 1));}

Multiplicative with a(p^e) = (p^(A048298(e)+1)-1)/(p-1).
a(n) = A000203(A367168(n)).
a(n) <= A000203(n), with equality if and only if n is in A138302.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2)/zeta(3) = 1.368432... (A306633).

A342534 a(n) = Sum_{k=1..n} phi(gcd(k, n))^2.

Original entry on oeis.org

1, 2, 6, 7, 20, 12, 42, 26, 50, 40, 110, 42, 156, 84, 120, 100, 272, 100, 342, 140, 252, 220, 506, 156, 484, 312, 438, 294, 812, 240, 930, 392, 660, 544, 840, 350, 1332, 684, 936, 520, 1640, 504, 1806, 770, 1000, 1012, 2162, 600, 2022, 968, 1632, 1092, 2756, 876, 2200, 1092, 2052

Offset: 1

Seiichi Manyama, Mar 15 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000010, A029935, A029939, A065464, A306633, A338997, A342535.

a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^2 &]; Array[a, 50] (* Amiram Eldar, Mar 15 2021 *)

a(n) = sum(k=1, n, eulerphi(gcd(k, n))^2);

a(n) = sumdiv(n, d, eulerphi(n/d)*eulerphi(d)^2);

a(n) = Sum_{d|n} phi(n/d) * phi(d)^2.
a(n) = Sum_{k=1..n} phi(gcd(k,n))*phi(n/gcd(k,n)). - Richard L. Ollerton, May 10 2021
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = (p-1)*(p^(e-2) - p^(2*e-3) + p^(2*e-1)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(2)/(3*zeta(3)) * Product_{p prime} (1 - (2*p-1)/p^3) = A306633 * A065464 / 3 = 0.195343... . (End)

A033457 GCD-convolution of squares A000290 with themselves.

Original entry on oeis.org

1, 2, 6, 4, 19, 6, 28, 24, 45, 10, 98, 12, 79, 94, 120, 16, 201, 18, 238, 164, 171, 22, 436, 120, 229, 234, 426, 28, 695, 30, 496, 352, 369, 370, 1014, 36, 451, 470, 1068, 40, 1261, 42, 946, 1020, 639, 46, 1832, 336, 1225, 754, 1278, 52, 1899, 774, 1924, 920, 981

Offset: 0

N. J. A. Sloane

Danny Rorabaugh, Table of n, a(n) for n = 0..10000

Cf. A000010 (phi), A000290, A069097, A306633.

Table[Sum[d^2*EulerPhi[(n + 2)/d], {d, Most@ Divisors[n + 2]}], {n, 0, 47}] (* Michael De Vlieger, Mar 20 2015 *)
f[p_, e_] := p^(e - 1)*(p^e*(p + 1) - 1); a[n_] := Times @@ f @@@ FactorInteger[n + 2] - (n + 2)^2; Array[a, 100, 0] (* Amiram Eldar, Dec 06 2024 *)

a(n) = {my(f = factor(n+2)); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; p^(e-1)*(p^e*(p+1) - 1)) - (n+2)^2;} \\ Amiram Eldar, Dec 06 2024

sum([d^2*euler_phi(int((n+2)/d)) for d in range(1,n+2) if (n+2)%d==0]) # Danny Rorabaugh, Mar 20 2015

a(n-2) = Sum_{d|n, dVladeta Jovovic, Aug 27 2003
From Amiram Eldar, Dec 06 2024: (Start)
a(n) = A069097(n+2) - (n+2)^2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)/zeta(3) - 1)/3 = (A306633 - 1)/3 = 0.122810925... . (End)

A072158 Numerator of Sum_{k=1..n} phi(k)/k^3.

Original entry on oeis.org

1, 9, 259, 1063, 136331, 15259, 5305837, 21351973, 1740485813, 1745820149, 2337022458319, 2341131255319, 5164765371583843, 5173292359195843, 5182536034853059, 20760610355567611, 102246457919648504843, 3789825999242633809, 26045507479622115279931, 26064975970269506857723

Offset: 1

N. J. A. Sloane, Jun 28 2002

			1, 9/8, 259/216, 1063/864, 136331/108000, 15259/12000, ...

G. C. Greubel, Table of n, a(n) for n = 1..770

Cf. A000010, A072159, A306633.

List([1..25], n-> NumeratorRat( Sum([1..n], k-> Phi(k)/k^3) ) ); # G. C. Greubel, Aug 26 2019

[Numerator( &+[EulerPhi(k)/k^3: k in [1..n]] ): n in [1..25]]; // G. C. Greubel, Aug 26 2019

with(numtheory); seq(numer(add(phi(k)/k^3, k = 1..n)), n = 1..25); # G. C. Greubel, Aug 26 2019

Numerator[Table[Sum[EulerPhi[k]/k^3,{k,n}],{n,20}]] (* Harvey P. Dale, May 27 2012 *)
Numerator[Accumulate[Table[EulerPhi[k]/k^3, {k, 1, 30}]]] (* Amiram Eldar, Dec 28 2024 *)

a(n) = numerator( sum(k=1, n, eulerphi(k)/k^3)); \\ G. C. Greubel, Aug 26 2019

[numerator( sum(euler_phi(k)/k^3 for k in (1..n)) ) for n in (1..25)] # G. C. Greubel, Aug 26 2019

Limit_{n->oo} a(n)/A072159(n) = zeta(2)/zeta(3) = 1.368432... (A306633). - Amiram Eldar, Dec 28 2024

A073245 Sum of all cubefree numbers with the same squarefree kernel as the n-th squarefree number.

Original entry on oeis.org

1, 6, 12, 30, 72, 56, 180, 132, 182, 336, 360, 306, 380, 672, 792, 552, 1092, 870, 2160, 992, 1584, 1836, 1680, 1406, 2280, 2184, 1722, 4032, 1892, 3312, 2256, 3672, 2862, 3960, 4560, 5220, 3540, 3782, 5952, 5460, 9504, 4556, 6624, 10080, 5112, 5402

Offset: 1

Reinhard Zumkeller, Jul 21 2002

nonn

			14 is the 10th squarefree number: A005117(10)=14=2*7, the cubefree numbers with squarefree kernel =14 are 14, 28=2*2*7, 98=2*7*7 and 196=2*2*7*7; therefore a(10)=14+28+98+196=336; a(10)=A062822(10)*A005117(10)=24*14=336.

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

Cf. A004709, A005117, A007947, A062822, A064987, A072048, A306633.

Cf. A002117, A013661.

Map[# * DivisorSigma[1, #] &, Select[Range[200], SquareFreeQ]] (* Amiram Eldar, Oct 14 2020 *)

apply(x->(x*sigma(x)), select(issquarefree, [1..100])) \\ Michel Marcus, Oct 18 2020

from math import isqrt
from sympy import mobius, divisor_sigma
def A073245(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m*divisor_sigma(m) # Chai Wah Wu, Aug 12 2024

a(n) = A062822(n)*A005117(n).
Sum_{n>=1} 1/a(n) = A306633. - Amiram Eldar, Oct 14 2020
a(n) = A064987(A005117(n)). - Michel Marcus, Oct 18 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(2)^3/(3*zeta(3)) = 1.23423882415851340020... . - Amiram Eldar, Oct 09 2023

A327574 Decimal expansion of the constant that appears in the asymptotic formula for average order of the infinitary divisors sum function (A049417).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Sep 17 2019

The asymptotic mean of the infinitary abundancy index lim_{n->oo} (1/n) * Sum_{k=1..n} A049417(k)/k = 1.461436... is twice this constant. - Amiram Eldar, Jun 13 2020

			0.730718242127384225838975468173530161957256433861727...

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, section 1.7.5, pp. 53-54.

Graeme L. Cohen and Peter Hagis, Jr., Arithmetic functions associated with infinitary divisors of an integer, International Journal of Mathematics and Mathematical Sciences, Vol. 16, No. 2 (1993), pp. 373-383.

Cf. A049417, A050376, A327566.

Cf. A013661 (corresponding constant for all divisors), A275480 (exponential), A306633 (unitary), A307160 (bi-unitary).

$MaxExtraPrecision = 1000; m = 1000; em = 10; f[x_] := Sum[Log[1 + x^(2^e)/(1 + 1/x^(2^e))], {e, 0, em}]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[(1/2) * Exp[NSum[Indexed[c, k]*PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals Limit_{k->oo} A327566(k)/k^2.

Equals (1/2) * Product_{P} (1 + 1/(P*(P+1))), where P are numbers of the form p^(2^k) where p is prime and k >= 0 (A050376).

cp's OEIS Frontend

A371188 Indices of the squarefree numbers in the sequence of cubefree numbers.

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A374783 Numerator of the mean unitary abundancy index of the unitary divisors of n.

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A328258 a(n) = Sum_{d|n, gcd(d,n/d) = 1} (-1)^(d + 1) * d.

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A332882 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j^k_j).

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A367171 The sum of divisors of the largest unitary divisor of n that is a term of A138302.

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A342534 a(n) = Sum_{k=1..n} phi(gcd(k, n))^2.

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A033457 GCD-convolution of squares A000290 with themselves.

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