A072691 -id:A072691 - cp's OEIS frontend

A275710 Decimal expansion of the Dirichlet eta function at 7.

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

Offset: 0

Terry D. Grant, Aug 06 2016

			0.99259381992283028267...

Cf. A002162 (value at 1), A013665, A072691 (value at 2), A197070 (value at 3), A267315 (value at 4), A267316 (value at 5), A275703 (value at 6), A334668, A334669, A347150, A347059.

RealDigits[63 Zeta[7]/64, 10, 100] [[1]]

-polylog(7, -1) \\ Michel Marcus, Aug 20 2021

s = RLF(0); s
RealField(110)(s)
for i in range(1, 10000): s -= (-1)^i / i^7
print(s) # Terry D. Grant, Aug 06 2016

eta(7) = 63*zeta(7)/64 = (63*A013665)/64.
eta(7) = Lim_{n -> infinity} A334668(n)/A334669(n). - Petros Hadjicostas, May 07 2020
Equals Sum_{k>=1} (-1)^(k+1) / k^7. - Sean A. Irvine, Aug 19 2021

A164108 Decimal expansion of Pi^4/24.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 10 2009

Volume of the 8-dimensional unit sphere.

			4.0587121264167682181850138620293796354053160696952259038...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Cornel Ioan Vălean, Problem 11993, The American Mathematical Monthly, Vol. 124, No. 7 (2017), p. 659; An Integral Related to Euler Sums, Solution to Problem 11993 by Abdelhak Berkane, ibid., Vol. 126, No. 4 (2019), pp. 373-374.
Eric Weisstein's World of Mathematics, Hypersphere.
Wikipedia, n-sphere.
Index entries for transcendental numbers

Cf. A000796, A019699, A102753, A164103, A164105, A164106.

RealDigits[Pi^4/24, 10, 100][[1]] (* G. C. Greubel, Apr 09 2017 *)

PARI
```
Pi^4/24 \\ G. C. Greubel, Apr 09 2017
```

Equals A164109/8 = A092425/24 = A072691*A102753.

Pi^4/240 = -Integral_{x=0..1} log(1-x)*log(1+x)^2/x dx (Vălean, 2017). - Amiram Eldar, Mar 26 2022

A304411 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*k_j).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 9, 8, 18, 12, 24, 14, 24, 24, 12, 18, 24, 20, 36, 32, 36, 24, 36, 12, 42, 12, 48, 30, 72, 32, 15, 48, 54, 48, 48, 38, 60, 56, 54, 42, 96, 44, 72, 48, 72, 48, 48, 16, 36, 72, 84, 54, 36, 72, 72, 80, 90, 60, 144, 62, 96, 64, 18, 84, 144, 68, 108, 96, 144, 72, 72

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(24) = a(2^3*3) = (2 + 1)*3 * (3 + 1)*1 = 36.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000 .
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A000005, A000026, A000040, A000203, A003959, A001615, A003961, A005117, A005361, A007947, A008864, A045965, A048250, A064478, A081294 (numbers n such that a(n) is odd), A181797, A304407, A304408, A304409, A304412.

Cf. A072691, A330596.

a[n_] := Times @@ ((#[[1]] + 1) #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 72}]
Table[Total[Select[Divisors[n], SquareFreeQ]] DivisorSigma[0, n/Last[Select[Divisors[n], SquareFreeQ]]], {n, 72}]

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p+1)*e)} \\ Andrew Howroyd, Jul 24 2018

a(n) = A005361(n)*A048250(n) = A000005(n/A007947(n))*A000203(A007947(n)).
a(p^k) = (p + 1)*k where p is a prime and k > 0.
a(n) = Product_{p|n} (p + 1) if n is a squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = A072691 * A330596 = 0.6156455744... . - Amiram Eldar, Nov 30 2022

A324706 The sum of the tri-unitary divisors of n.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 15, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 60, 26, 42, 40, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 80, 85, 84, 144

Offset: 1

Amiram Eldar, Mar 11 2019

A divisor d of n is tri-unitary if the greatest common bi-unitary divisor of d and n/d is 1.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Graeme L. Cohen, On an integer's infinitary divisors, Mathematics of Computation, Vol. 54, No. 189 (1990), pp. 395-411.
Pentti Haukkanen, On the k-ary convolution of arithmetical functions, The Fibonacci Quarterly, Vol. 38, No. 5 (2000) pp. 440-445.

Cf. A000203, A034448, A049417, A072691, A188999, A322485, A324707, A324708, A324709.

f[p_, e_] := If[e == 3, (p^4-1)/(p-1), If[e==6, (p^8-1)/(p^2-1), p^e+1]]; a[1]=1; a[n_]:= Times @@ f @@@ FactorInteger[n]; Array[a, 100]

A324706(n) = { my(f = factor(n)); prod(i=1, #f~, if(3==f[i,2], sigma(f[i,1]^f[i,2]), if(6==f[i,2], ((f[i,1]^8)-1)/((f[i,1]^2)-1), 1+(f[i,1]^f[i,2])))); }; \\ Antti Karttunen, Mar 12 2019

Multiplicative with a(p^3) = 1 + p + p^2 + p^3, a(p^6) = 1 + p^2 + p^4 + p^6, and a(p^e) = 1 + p^e otherwise.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^3 + 1/p^4 - 2/p^6 + 2/p^8 - 1/p^9 - 1/p^12 + 1/p^13) = 0.72189237802... . - Amiram Eldar, Nov 24 2022

A344221 a(n) = Sum_{k=1..n} tau(gcd(k,n)^3), where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 5, 6, 13, 8, 30, 10, 29, 21, 40, 14, 78, 16, 50, 48, 61, 20, 105, 22, 104, 60, 70, 26, 174, 43, 80, 66, 130, 32, 240, 34, 125, 84, 100, 80, 273, 40, 110, 96, 232, 44, 300, 46, 182, 168, 130, 50, 366, 73, 215, 120, 208, 56, 330, 112, 290, 132, 160, 62, 624, 64, 170, 210, 253, 128, 420

Offset: 1

Seiichi Manyama, May 12 2021

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

Cf. A060648, A072691, A344222, A344223, A344224, A344225.

Table[Sum[DivisorSigma[0,GCD[k,n]^3],{k,n}],{n,100}] (* Giorgos Kalogeropoulos, May 13 2021 *)
f[p_, e_] := (p^e*(p + 2) - 3)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sum(k=1, n, numdiv(gcd(k, n)^3));

a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d^3));

PARI
```
a(n) = n*sumdiv(n, d, 3^omega(d)/d);
```

a(n) = Sum_{d|n} phi(n/d) * tau(d^3).
a(n) = n * Sum_{d|n} 3^omega(d) / d.
If p is prime, a(p) = 3 + p.
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = (p^e*(p + 2) - 3)/(p - 1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 2/p^2) = 1.8019184198... . (End)

A360522 a(n) = Sum_{d|n} Max({d'; d'|n, gcd(d, d') = 1}).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 11, 11, 18, 12, 24, 14, 24, 24, 20, 18, 33, 20, 36, 32, 36, 24, 44, 27, 42, 30, 48, 30, 72, 32, 37, 48, 54, 48, 66, 38, 60, 56, 66, 42, 96, 44, 72, 66, 72, 48, 80, 51, 81, 72, 84, 54, 90, 72, 88, 80, 90, 60, 144, 62, 96, 88, 70, 84, 144, 68

Offset: 1

Amiram Eldar, Feb 10 2023

a(n) is the sum of delta_d(n) over the divisors d of n, where delta_d(n) is the greatest divisor of n that is relatively prime to n.
Denoted by Sur(n) in Khan (2005).
Related sequences: A048691(n) = Sum_{d|n} #{d'; d' | n, gcd(d, d') = 1}, and A328485(n) = Sum_{d|n} Sum_{d' | n, gcd(d, d') = 1} d' (number and sum of divisors instead of maximal divisor, respectively).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Mizan R. Khan, Problem 10922, The American Mathematical Monthly, Vol. 109, No. 2 (2002), p. 201; Michael R. Avidon, The Sum of Divisors Won't Die, solution, ibid., Vol. 110, No. 10 (2003), p. 959.
Mizan R. Khan, A variant of the divisor functions sigma_a(n), JP Journal of Algebra, Number Theory and Applications, Vol. 5, No. 3 (2005), pp. 561-574.

Cf. A000203, A005117, A034448, A360523.
Cf. A065465, A072691, A152649, A330523.
Cf. A048691, A328485.

f[p_, e_] := p^e + e; 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]^f[i,2] + f[i,2]);}

Multiplicative with a(p^e) = p^e + e.
Dirichlet g.f.: zeta(s-1)*zeta(s)^2 * Product_{p prime} (1 - 1/p^s - 1/p^(2*s-1) + 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = A072691 * A065465 = A152649 * A330523 = 0.7250160726810604158... .
a(n) <= A000203(n) with equality if and only if n is squarefree (A005117).
limsup_{n->oo} sigma(n)/a(n) = oo, where sigma(n) is the sum of divisors of n (A000203) (Khan, 2002).
liminf_{n->oo} a(n)/usigma(n) = 1, where usigma(n) is the sum of unitary divisors of n (A034448) (Khan, 2005).
limsup_{n->oo} a(n)/usigma(n) = (55/54) * Product_{p prime} (1 + 1/(p^2+1)) = 1.4682298236... (Khan, 2005).

A097448 If n is square, replace it with sqrt(n).

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 12, 13, 14, 15, 4, 17, 18, 19, 20, 21, 22, 23, 24, 5, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 7, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 8, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 0

Cino Hilliard, Aug 23 2004

			Among the first five integers 0, 1, 2, 3, and 4 the squares are 0, 1, and 4. Thus the first five terms in the sequence are 0, 1, 2, 3, and 2.

Cf. A016627, A072691, A097449.

Table[If[IntegerQ[Sqrt[n]],Sqrt[n],n],{n,0,100}] (* Harvey P. Dale, Jul 09 2017 *)

g(n) = for(x=0,n,if(issquare(x),y=sqrt(x),y=x);print1(floor(y)","))

for(n=0,74,print1(if(issquare(n,&m),m,n)", ")) \\ Zak Seidov, Feb 21 2013

Sum_{n>=1} (-1)^(n+1)/n = 2*log(2) - Pi^2/12 = A016627 - A072691. - Amiram Eldar, Jul 07 2024

A175475 Decimal expansion of the Dickman function evaluated at 1/3.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, May 25 2010

Density of the cube root-smooth numbers, see A090081. - Charles R Greathouse IV, Jul 14 2014

			F(1/3) = 0.04860838829113156690718...

David Broadhurst, Dickman polylogarithms and their constants arXiv:1004.0519 [math-ph], 2010.
K. Soundararajan, An asymptotic expansion related to the Dickman function, arXiv:1005.3494 [math.NT], 2010.

Cf. A002391, A072691, A175478.

N[1 - Log[3] + Log[3]^2/2 - Pi^2/12 + PolyLog[2, 1/3], 105] // RealDigits // First // Prepend[#, 0]& (* Jean-François Alcover, Feb 05 2013 *)

1-log(3)+log(3)^2/2-Pi^2/12+polylog(2,1/3) \\ Charles R Greathouse IV, Jul 14 2014

Equals 1 - log(3) + log^2(3)/2 - Pi^2/12 + Sum_{n>=1} 1/(n^2*3^n), where Sum_{n>=1} 1/(n^2*3^n) = 0.3662132299770634876167462976642627638...

A299692 a(n) is the total area that is visible in the perspective view of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

3, 10, 20, 35, 51, 75, 97, 128, 159, 197, 231, 283, 323, 375, 429, 492, 544, 619, 677, 759, 833, 913, 983, 1091, 1172, 1266, 1360, 1472, 1560, 1692, 1786, 1913, 2027, 2149, 2267, 2430, 2542, 2678, 2812, 2982, 3106, 3286, 3416, 3588, 3756, 3920, 4062, 4282, 4437, 4630, 4804, 5006, 5166, 5394, 5576, 5808, 6002

Offset: 1

Omar E. Pol, Mar 06 2018

nonn

a(n) is also the sum of all divisors of all positive integers <= n, plus the n-th oblong number, since A024916(n) equals the total area of the horizontal terraces of the stepped pyramid with n levels, and A002378(n) equals the total area of the vertical sides that are visible (see link).

a(n) is also the sum of all aliquot divisors of all positive integers <= n, plus the n-th triangular matchstick number.

			For n = 3 the areas of the terraces of the first three levels starting from the top of the stepped pyramid are 1, 3 and 4 respectively. On the other hand the areas of the vertical sides that are visible are [1, 1], [2, 2], [2, 1, 1, 2], or in successive levels 2, 4, 6 respectively. Hence the total area that is visible is equal to 1 + 3 + 4 + 2 + 4 + 6 = 8 + 12 = 20, so a(3) = 20.
For n = 16 the total number of horizontal and vertical cells that are visible are 220 and 272 respectively. So a(16) = 220 + 272 = 492 (see the link).

Omar E. Pol, Perspective view of the pyramid with 16 levels, contains 492 cells.

Partial sums of A224880.

Cf. A002378, A024916, A045943, A072691, A153485, A196020, A236104, A237048, A237270, A237271, A237591, A237593, A244050, A245092, A262626, A328366.

Accumulate[Table[DivisorSigma[1, n] + 2*n, {n, 1, 50}]] (* Amiram Eldar, Mar 21 2024 *)

a(n) = sum(k=1, n, n\k*k) + n*(n+1); \\ Michel Marcus, Jun 21 2018

from math import isqrt
def A299692(n): return n*(n+1)+(-(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) # Chai Wah Wu, Oct 22 2023

a(n) = A024916(n) + A002378(n).
a(n) = A153485(n) + A045943(n).
a(n) = A328366(n)/2. - Omar E. Pol, Apr 22 2020
a(n) = c * n^2 + O(n*log(n)), where c = zeta(2)/2 + 1 = A072691 + 1 = 1.822467... . - Amiram Eldar, Mar 21 2024

A323309 The sum of exponential semiproper divisors of n.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 18, 13, 14, 15, 18, 17, 24, 19, 30, 21, 22, 23, 30, 30, 26, 30, 42, 29, 30, 31, 34, 33, 34, 35, 72, 37, 38, 39, 50, 41, 42, 43, 66, 60, 46, 47, 54, 56, 60, 51, 78, 53, 60, 55, 70, 57, 58, 59, 90, 61, 62, 84, 66, 65, 66, 67

Offset: 1

Amiram Eldar, Jan 10 2019

An exponential semiproper divisor of n is a divisor d such that rad(d) = rad(n) and GCD(d/rad(n), n/d) = 1, were rad(n) is the largest squarefree divisor of n (A007947).

Ivan Neretin, Table of n, a(n) for n = 1..10000
Nicusor Minculete, A new class of divisors: the exponential semiproper divisors, Bulletin of the Transilvania University of Brasov, Mathematics, Informatics, Physics, Series III, Vol. 7 No. 1 (2014), pp. 37-46.

Cf. A007947, A034448, A072691, A323308, A323310.

f[p_, e_] := If[e==1, p, p^e + p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = my(f=factor(n)); for (k=1, #f~, if (f[k,2] > 1, f[k,1] += f[k,1]^f[k,2]); f[k,2] = 1); factorback(f); \\ Michel Marcus, Jan 10 2019

a(n) = A007947(n) * A034448(n/A007947(n)).
Multiplicative with a(p^e) = p for e = 1 and p^e + p otherwise.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.5628034365... . - Amiram Eldar, Dec 01 2022

cp's OEIS Frontend

A275710 Decimal expansion of the Dirichlet eta function at 7.

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A164108 Decimal expansion of Pi^4/24.

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A304411 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*k_j).

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A324706 The sum of the tri-unitary divisors of n.

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A344221 a(n) = Sum_{k=1..n} tau(gcd(k,n)^3), where tau(n) is the number of divisors of n.

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A360522 a(n) = Sum_{d|n} Max({d'; d'|n, gcd(d, d') = 1}).

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A097448 If n is square, replace it with sqrt(n).

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