A013665 -id:A013665 - cp's OEIS frontend

A069091 Jordan function J_6(n).

1, 63, 728, 4032, 15624, 45864, 117648, 258048, 530712, 984312, 1771560, 2935296, 4826808, 7411824, 11374272, 16515072, 24137568, 33434856, 47045880, 62995968, 85647744, 111608280, 148035888, 187858944, 244125000, 304088904, 386889048

Offset: 1

Benoit Cloitre, Apr 05 2002

Moebius transform of n^6. - Enrique Pérez Herrero, Sep 14 2010

a(n) is divisible by 504 = (2^3)*(3^3)*7 = A006863(3) except for n = 1, 2, 3 and 7. See Lugo. - Peter Bala, Jan 13 2024

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

Enrique Pérez Herrero, Table of n, a(n) for n=1..2000
Michael Lugo, A little number theory problem (2008)
Wikipedia, Jordan's totient function.

Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A007434 (J_2), A059376 (J_3), A059377 (J_4), A059378 (J_5), A069092 - A069095 (J_7 through J_10).
Cf. A065959.
Cf. A013665.

with(numtheory): seq(add(d^6 * mobius(n/d), d in divisors(n)), n = 1..100); # Peter Bala, Jan 13 2024

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/# ]&]/;(n>0)&&IntegerQ[n]
A069091[n_IntegerQ]:=JordanTotient[n,6]; (* Enrique Pérez Herrero, Sep 14 2010 *)
f[p_, e_] := p^(6*e) - p^(6*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

for(n=1,100,print1(sumdiv(n,d,d^6*moebius(n/d)),","))

a(n) = Sum_{d|n} d^6*mu(n/d).
Multiplicative with a(p^e) = p^(6e)-p^(6(e-1)).
Dirichlet generating function: zeta(s-6)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^6*Product_{distinct primes p dividing n} (1-1/p^6). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ n^7 / (7*zeta(7)). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^6 = 1/zeta(7).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^6/(p^6-1)^2) = 1.0175973008... (End)
O.g.f.: Sum_{n >= 1} mu(n)*A(x^n)/(1 - x^n)^7 = x + 63*x^2 + 728*x^3 + 4032*x^4 + 15624*x^5 + ..., where A(x) = x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6 is the 6th Eulerian polynomial. See A008292. - Peter Bala, Jan 31 2022

A053166 Smallest positive integer for which n divides a(n)^4.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 4, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 4, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38

Offset: 1

Henry Bottomley, Feb 29 2000

According to Broughan (2002, 2003, 2006), a(n) is the "upper 4th root of n". The "lower 4th root of n" is sequence A053164. - Petros Hadjicostas, Sep 15 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.
Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, 101(2) (2002), 105-114.
Kevin A. Broughan, Relationship between the integer conductor and k-th root functions, Int. J. Pure Appl. Math. 5(3) (2003), 253-275.
Kevin A. Broughan, Relaxations of the ABC Conjecture using integer k'th roots, New Zealand J. Math. 35(2) (2006), 121-136.
Eric Weisstein's World of Mathematics, Smarandache Ceil Function.

Cf. A000188 (inner square root), A019554 (outer square root), A053150 (inner 3rd root), A019555 (outer 3rd root), A053164 (inner 4th root), A015052 (outer 5th root), A015053 (outer 6th root).

Cf. A000189, A000190, A008835, A013665, A015051.

f[p_, e_] := p^Ceiling[e/4]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 08 2020 *)

a(n) = my(f=factor(n)); for (i=1, #f~, f[i,2] = ceil(f[i,2]/4)); factorback(f); \\ Michel Marcus, Jun 09 2014

a(n) = n/A000190(n) = A019554(n)/(A008835(A019554(n)^2))^(1/4).
If n is 5th-power-free (i.e., not 32, 64, 128, 243, ...) then a(n) = A007947(n).
Multiplicative with a(p^e) = p^(ceiling(e/4)). - Christian G. Bower, May 16 2005
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(7)/2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5 - 1/p^6) = 0.3528057925... . - Amiram Eldar, Oct 27 2022

A009641 a(n) = Product_{i=0..6} floor((n+i)/7).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 4, 8, 16, 32, 64, 128, 192, 288, 432, 648, 972, 1458, 2187, 2916, 3888, 5184, 6912, 9216, 12288, 16384, 20480, 25600, 32000, 40000, 50000, 62500, 78125, 93750, 112500, 135000, 162000, 194400, 233280, 279936, 326592, 381024

Offset: 0

N. J. A. Sloane

For n >= 7, a(n) is the maximal product of seven positive integers with sum n. - Wesley Ivan Hurt, Jun 29 2022

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. El-Mikkawy and T. Sogabe, A new family of k-Fibonacci numbers, Appl. Math. Comput. 215 (2010) 4456-4461, Table 1 k=7.
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 6, -12, 6, 0, 0, 0, 0, -15, 30, -15, 0, 0, 0, 0, 20, -40, 20, 0, 0, 0, 0, -15, 30, -15, 0, 0, 0, 0, 6, -12, 6, 0, 0, 0, 0, -1, 2, -1).

Maximal product of k positive integers with sum n, for k = 2..10: A002620 (k=2), A006501 (k=3), A008233 (k=4), A008382 (k=5), A008881 (k=6), this sequence (k=7), A009694 (k=8), A009714 (k=9), A354600 (k=10).

Cf. A001015, A013665.

A009641[n_] := Product[Floor[(n + i)/7], {i, 0, 6}]; Array[A009641, 50, 0] (* Paolo Xausa, Jan 17 2025 *)

a(n) = prod(k=0, 6, (n+k)\7); \\ Georg Fischer, Nov 07 2019

a(n) = 2*a(n-1) - a(n-2) + 6*a(n-7) - 12*a(n-8) + 6*a(n-9) - 15*a(n-14) + 30*a(n-15) - 15*a(n-16) + 20*a(n-21) - 40*a(n-22) + 20*a(n-23) - 15*a(n-28) + 30*a(n-29) - 15*a(n-30) + 6*a(n-35) - 12*a(n-36) + 6*a(n-37) - a(n-42) + 2*a(n-43) - a(n-44). - Wesley Ivan Hurt, Jun 29 2022
a(7*n) = n^7 (A001015). - Bernard Schott, Nov 04 2022
Sum_{n>=7} 1/a(n) = 1 + zeta(7). - Amiram Eldar, Jan 10 2023

a(40)-a(44) from Georg Fischer, Nov 07 2019

A084220 a(n) = sigma_6(n^2)/sigma_3(n^2).

Original entry on oeis.org

1, 57, 703, 3641, 15501, 40071, 117307, 233017, 512461, 883557, 1770231, 2559623, 4824613, 6686499, 10897203, 14913081, 24132657, 29210277, 47039023, 56439141, 82466821, 100903167, 148023723, 163810951, 242203001, 275002941, 373584043

Offset: 1

Benoit Cloitre, Jun 21 2003

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001158, A013665, A013954.

Cf. A057660, A084218.

with(numtheory): a:=n->sigma[6](n^2)/sigma[3](n^2): seq(a(n),n=1..30); # Muniru A Asiru, Oct 09 2018

Table[DivisorSigma[6,n^2]/DivisorSigma[3,n^2],{n,30}] (* Harvey P. Dale, May 02 2012 *)
f[p_, e_] := (p^(6*e + 3) + 1)/(p^3 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Sep 13 2020 *)

a(n)=sumdiv(n^2,d,d^6)/sumdiv(n^2,d,d^3)

a(n) = sigma(n^2, 6)/sigma(n^2, 3); \\ Michel Marcus, Oct 09 2018

Multiplicative with a(p^e) = (p^(6*e + 3) + 1)/(p^3 + 1). - Amiram Eldar, Sep 13 2020
Sum_{k>=1} 1/a(k) = 1.019347996519986873084210965032965644185467985307512751244884310846924559959... - Vaclav Kotesovec, Sep 24 2020
Sum_{k=1..n} a(k) ~ c * n^7, where c = 90*zeta(7)/(7*Pi^4) = 0.133093... . - Amiram Eldar, Oct 30 2022
From Seiichi Manyama, May 18 2024: (Start)
a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( n/gcd(x_1, x_2, x_3, n) )^3.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_6(d). (End)

A321810 Sum of 6th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 730, 1, 15626, 730, 117650, 1, 532171, 15626, 1771562, 730, 4826810, 117650, 11406980, 1, 24137570, 532171, 47045882, 15626, 85884500, 1771562, 148035890, 730, 244156251, 4826810, 387952660, 117650, 594823322, 11406980, 887503682

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Eric Weisstein's World of Mathematics, Odd Divisor Function.
Index entries for sequences mentioned by Glaisher.

Column k=6 of A285425.
Cf. A050999, A051000, A051001, A051002, A321811 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013665, A013954.

f[2, e_] := 1; f[p_, e_] := (p^(6*e + 6) - 1)/(p^6 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 02 2022 *)

apply( A321810(n)=sigma(n>>valuation(n,2),6), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321810(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),6)) # Chai Wah Wu, Jul 16 2022

a(n) = A013954(A000265(n)) = sigma_6(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (2*k - 1)^6*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(6*e+6)-1)/(p^6-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^7, where c = zeta(7)/14 = 0.0720249... . (End)
a(n) + a(n/2)*2^6 = A013954(n) where a(.)=0 for non-integer arguments. - R. J. Mathar, Aug 15 2023

A351269 Sum of the 6th powers of the squarefree divisors of n.

Original entry on oeis.org

1, 65, 730, 65, 15626, 47450, 117650, 65, 730, 1015690, 1771562, 47450, 4826810, 7647250, 11406980, 65, 24137570, 47450, 47045882, 1015690, 85884500, 115151530, 148035890, 47450, 15626, 313742650, 730, 7647250, 594823322, 741453700, 887503682, 65, 1293240260

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Inverse Möbius transform of n^6 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

			a(4) = 65; a(4) = Sum_{d|4} d^6 * mu(d)^2 = 1^6*1 + 2^6*1 + 4^6*0 = 65.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A008683 (mu), A013661, A013665.

Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), A351265 (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), this sequence (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^6); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)

a(n) = Sum_{d|n} d^6 * mu(d)^2.
Multiplicative with a(p^e) = 1 + p^6. - Amiram Eldar, Feb 06 2022
G.f.: Sum_{k>=1} mu(k)^2 * k^6 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^7, where c = zeta(7)/(7*zeta(2)) = 0.0875718... . - Amiram Eldar, Nov 10 2022

A351603 a(n) = n^5 * Sum_{d^2|n} 1 / d^5.

Original entry on oeis.org

1, 32, 243, 1056, 3125, 7776, 16807, 33792, 59292, 100000, 161051, 256608, 371293, 537824, 759375, 1082368, 1419857, 1897344, 2476099, 3300000, 4084101, 5153632, 6436343, 8211456, 9768750, 11881376, 14407956, 17748192, 20511149, 24300000, 28629151, 34635776, 39135393

Offset: 1

Wesley Ivan Hurt, Feb 14 2022

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

Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: A046951 (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), this sequence (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

Cf. A013665.

f[p_, e_] := p^5*(p^(5*e) - p^(5*Floor[(e - 1)/2]))/(p^5 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 13 2022 *)

a(n) = n^5*sumdiv(n, d, if (issquare(d), 1/sqrtint(d^5))); \\ Michel Marcus, Feb 15 2022

Multiplicative with a(p^e) = p^5*(p^(5*e) - p^(5*floor((e-1)/2)))/(p^5 - 1). - Sebastian Karlsson, Feb 25 2022

Sum_{k=1..n} a(k) ~ c * n^6, where c = zeta(7)/6 = 0.168058... . - Amiram Eldar, Nov 13 2022

A352034 Sum of the 6th powers of the odd proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 730, 1, 1, 730, 15626, 1, 730, 1, 117650, 16355, 1, 1, 532171, 1, 15626, 118379, 1771562, 1, 730, 15626, 4826810, 532171, 117650, 1, 11406980, 1, 1, 1772291, 24137570, 133275, 532171, 1, 47045882, 4827539, 15626, 1, 85884500, 1, 1771562, 11938421, 148035890

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 15626; a(10) = Sum_{d|10, d<10, d odd} d^6 = 1^6 + 5^6 = 15626.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors.

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), this sequence (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).

Cf. A013665, A321810.

f[2, e_] := 1; f[p_, e_] := (p^(6*e+6) - 1)/(p^6 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^6, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)
Table[Total[Select[Most[Divisors[n]],OddQ]^6],{n,50}] (* Harvey P. Dale, Sep 15 2024 *)

a(n) = Sum_{d|n, d

G.f.: Sum_{k>=1} (2*k-1)^6 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

For odd n >1, a(n) = A321810(n)-n^6; for even n, a(n) = A321810(n). - R. J. Mathar, Aug 15 2023

Sum_{k=1..n} a(k) ~ c * n^7, where c = (zeta(7)-1)/14 = 0.0005963769... . - Amiram Eldar, Oct 11 2023

A352052 Sum of the 6th powers of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 64, 729, 4096, 15625, 46720, 117649, 262144, 532170, 1000064, 1771561, 2990080, 4826809, 7529600, 11406979, 16777216, 24137569, 34058944, 47045881, 64004096, 85884499, 113379968, 148035889, 191365120, 244156250, 308915840, 387952659, 481894400, 594823321

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^6 * Sum_{d|10, d<10, d odd} 1 / d^6 = 10^6 * (1/1^6 + 1/5^6) = 1000064.

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

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), A352047 (k=1), A352048 (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), this sequence (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A000035, A006519, A013665, A321810.

f:= proc(n) local m,d;
      m:= n/2^padic:-ordp(n,2);
      add((n/d)^6, d = select(`<`,numtheory:-divisors(m),n))
end proc:
map(f, [$1..30]); # Robert Israel, Apr 03 2023

Table[n^6*DivisorSum[n, 1/#^6 &, And[# < n, OddQ[#]] &], {n, 29}] (* Michael De Vlieger, Apr 04 2023 *)
a[n_] := DivisorSigma[-6, n/2^IntegerExponent[n, 2]] * n^6 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^6*sumdiv(n, d, if ((dMichel Marcus, Apr 04 2023

a(n) = n^6 * sigma(n >> valuation(n, 2), -6) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^6 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^6 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 18 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A321810(n) * A006519(n)^6 - A000035(n).

Sum_{k=1..n} a(k) = c * n^7 / 7, where c = 127*zeta(7)/128 = 1.000471548... . (End)

A293904 Decimal expansion of zeta(21).

Original entry on oeis.org

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

Offset: 1

Frank Ellermann, Oct 19 2017

Web searches find 1.0000004769329867878 in Python tools. Simon Plouffe published 1000 digits for zeta(9) up to zeta(2051) many years ago.

			1.000000476932986787806...

Simon Plouffe, Zeta(9) to Zeta(2051) in reverse order to 1000 digits each.
Simon Plouffe, Zeta(9) to Zeta(2051) in reverse order to 1000 digits each. [Cached copy, with permission]
Simon Plouffe, Zeta(2) to Zeta(4096) to 2048 digits each. (gzipped file)
Simon Plouffe, Identities inspired from Ramanujan Notebooks II, July 21, 1998.

Cf. A013661, A002117, A013662, A013663, A013664, A013665, A013666, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678.

RealDigits[Zeta[21], 10, 100][[1]] (* Amiram Eldar, May 31 2021 *)

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cp's OEIS Frontend

A069091 Jordan function J_6(n).

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A053166 Smallest positive integer for which n divides a(n)^4.

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A009641 a(n) = Product_{i=0..6} floor((n+i)/7).

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A084220 a(n) = sigma_6(n^2)/sigma_3(n^2).

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A321810 Sum of 6th powers of odd divisors of n.

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A351269 Sum of the 6th powers of the squarefree divisors of n.

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A351603 a(n) = n^5 * Sum_{d^2|n} 1 / d^5.

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