A082020 -id:A082020 - cp's OEIS frontend

A088245 Decimal expansion of 9/(2*Pi^2).

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

Offset: 0

Eric W. Weisstein, Sep 25 2003

The asymptotic density of squarefree numbers not divisible by 3 (A261034). - Amiram Eldar, May 22 2020

			0.455945326390519971497457584443774375...

See the Hardy reference given under A030059, eq. (4.9.4), p. 64, from the corrected formula on p. 65 for s=2. - Wolfdieter Lang, Oct 18 2016

Eric Weisstein's World of Mathematics, Moebius Function
Index entries for transcendental numbers

Cf. A082020, A088246, A030059, A261034.

RealDigits[N[9/(2 Pi^2), 120]] // First (* Michael De Vlieger, Oct 23 2015 *)

PARI
```
9/(2*Pi^2) \\ Altug Alkan, Oct 23 2015
```

Equals Sum_{n > 0} 1/A030059(n)^2 (the sum of reciprocals of squarefree numbers with an odd number of prime factors). Convergence is very slow. - Michel Lagneau, Oct 23 2015

A376936 Powerful numbers divisible by cubes of 2 distinct primes.

Original entry on oeis.org

216, 432, 648, 864, 1000, 1296, 1728, 1944, 2000, 2592, 2744, 3375, 3456, 3888, 4000, 5000, 5184, 5400, 5488, 5832, 6912, 7776, 8000, 9000, 9261, 10000, 10125, 10368, 10584, 10648, 10800, 10976, 11664, 13500, 13824, 15552, 16000, 16200, 16875, 17496, 17576, 18000

Offset: 1

Michael De Vlieger, Oct 16 2024

Numbers m with coreful divisors d, m/d such that neither d | m/d nor m/d | d, i.e., numbers m such that there exists a divisor pair (d, m/d) such that rad(d) = rad(m/d) but gcd(d, m/d) > 1 is neither d nor m/d, where rad = A007947. Divisors in each pair must be dissimilar and each in A126706.
Proper subset of A320966.
Contains A372695, A177493, and A162142. Does not contain A085986.

			216 is in the sequence since rad(12) | rad(18), but 12 does not divide 18 and 18 does not divide 12.
432 is a term since rad(18) | rad(24), but 18 does not divide 24 and 24 does not divide 18.
Table of coreful divisors d, a(n)/d such that neither d | a(n)/d nor a(n)/d | d for select a(n)
   n |   a(n)   divisor pairs d X a(n)/d
  ---------------------------------------------------------------------------
   1 |   216:   12 X 18;
   2 |   432:   18 X 24;
   3 |   648:   12 X 54;
   4 |   864:   24 X 36, 18 X 48;
   5 |  1000:   20 X 50;
   6 |  1296:   24 X 54;
   7 |  1728:   18 X 96, 36 X 48;
   8 |  1944:   12 X 162, 36 X 54;
   9 |  2000:   40 X 50;
  10 |  2592:   24 X 108, 48 X 54;
  11 |  2744:   28 X 98;
  12 |  3375:   45 X 75;
  13 |  3456:   18 X 192, 36 X 96, 48 X 72;
  22 |  7776:   24 X 324, 48 X 162, 54 X 144, 72 X 108;
  58 | 31104:   48 X 648, 54 X 576, 96 X 324, 108 X 288, 144 X 216, 162 X 192

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Notes on this sequence

Cf. A007947, A030078, A085986, A126706, A162142, A177493, A286708, A307958, A320966, A361430, A370329, A372695, A376514.

Cf. A082020, A082695.

Union@ Select[
  Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[20000],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - (15/Pi^2) * (1 + Sum_{prime} 1/((p-1)*(p^2+1))) = 0.021194288968234037106579437374641326044... . - Amiram Eldar, Nov 08 2024

A344753 a(n) = sigma(n) + psi(n) - 2n = Sum_{d|n, d

Original entry on oeis.org

0, 2, 2, 5, 2, 12, 2, 11, 7, 16, 2, 28, 2, 20, 18, 23, 2, 39, 2, 38, 22, 28, 2, 60, 11, 32, 22, 48, 2, 84, 2, 47, 30, 40, 26, 91, 2, 44, 34, 82, 2, 108, 2, 68, 60, 52, 2, 124, 15, 83, 42, 78, 2, 120, 34, 104, 46, 64, 2, 192, 2, 68, 74, 95, 38, 156, 2, 98, 54, 148, 2, 195, 2, 80, 94, 108, 38, 180, 2, 170, 67, 88, 2

Offset: 1

Antti Karttunen, May 28 2021

Sigma is the sum of divisors (A000203), and psi is Dedekind psi-function (A001615). Coincides with the latter only on perfect numbers (A000396).

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001065, A001615, A008683, A008966, A033879, A173557, A306927, A344705, A342001, A344705, A344754, A344755, A344997, A344998.
Cf. A000396 [where a(n) = A001615(n)], A005100 [a(n) < A001615(n)], A005101 [a(n) > A001615(n)].
Cf. also A051709, A345001.
Cf. A013661, A082020.

a[n_] := Sum[d + If[SquareFreeQ[n/d], d, 0], {d, Most[Divisors[n]]}];
Array[a, 100] (* Jean-François Alcover, Jun 12 2021 *)

PARI
```
A344753(n) = sumdiv(n,d,(d
				
```

a(n) = Sum_{d|n, dA008966(n/d) * d).
a(n) = A001065(n) + A306927(n).
a(n) = A001615(n) - A033879(n).
a(n) = A344705(n) + 2*A001065(n) - n.
For squarefree n, a(n) = 2*A001065(n).
a(n) = A344997(n) / A173557(n) = A344998(n) / A342001(n). - Antti Karttunen, Jun 06 2021
Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = Pi^2/6 + 15/Pi^2 - 2 = 1.164751... . - Amiram Eldar, Dec 08 2023

New primary definition added by Antti Karttunen, Jun 06 2021

A173290 Partial sums of A001615.

Original entry on oeis.org

1, 4, 8, 14, 20, 32, 40, 52, 64, 82, 94, 118, 132, 156, 180, 204, 222, 258, 278, 314, 346, 382, 406, 454, 484, 526, 562, 610, 640, 712, 744, 792, 840, 894, 942, 1014, 1052, 1112, 1168, 1240, 1282, 1378, 1422, 1494, 1566, 1638, 1686, 1782, 1838, 1928, 2000, 2084

Offset: 1

Jonathan Vos Post, Feb 15 2010

nonn

a(n) is even for n >= 2. - Jianing Song, Nov 24 2018

W. Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications, Volume 14, Number 2, 2015, Pages 73-88; http://scientificadvances.co.in; DOI: http://dx.doi.org/10.18642/jantaa_7100121599

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
W. Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications, Volume 14, Number 2, 2015, Pages 73-88.

Cf. A001615, A063659.
Cf. A082020.
Cf. A175836 (partial products of the Dedekind psi function).

[(&+[MoebiusMu(k)^2*Floor(n/k)*Floor(1 + n/k): k in [1..n]])/2: n in [1..60]]; // G. C. Greubel, Nov 23 2018

with(numtheory): a:=n->(1/2)*add(mobius(k)^2*floor(n/k)*floor(1+n/k),k=1..n); seq(a(n),n=1..55); # Muniru A Asiru, Nov 24 2018

Table[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k], {k,1,n}], {n,60}] (* G. C. Greubel, Nov 23 2018 *)
psi[n_] := If[n==1, 1, n*Times@@(1 + 1/FactorInteger[n][[;;,1]])]; Accumulate[Array[psi, 50]] (* Amiram Eldar, Nov 23 2018 *)

S(n) = sum(k=1, sqrtint(n), moebius(k)*(n\(k*k))); \\ see: A013928
a(n) = sum(k=1, sqrtint(n), k*(k+1) * (S(n\k) - S(n\(k+1))))/2 + sum(k=1, n\(1+sqrtint(n)), moebius(k)^2*(n\k)*(1+n\k))/2; \\ Daniel Suteu, Nov 23 2018

def A173290(n) :
    return add(k*mul(1+1/p for p in prime_divisors(k)) for k in (1..n))
[A173290(n) for n in (1..52)]  # Peter Luschny, Jun 10 2012

a(n) = Sum_{i=1..n} A001615(i) = Sum_{i=1..n} (n * Product_{p|n, p prime} (1 + 1/p)).
a(n) = 15*n^2/(2*Pi^2) + O(n*log(n)). - Enrique Pérez Herrero, Jan 14 2012
a(n) = Sum_{i=1..n} A063659(i) * floor(n/i). - Enrique Pérez Herrero, Feb 23 2013
a(n) = (1/2)*Sum_{k=1..n} mu(k)^2 * floor(n/k) * floor(1+n/k), where mu(k) is the Moebius function. - Daniel Suteu, Nov 19 2018
a(n) = (Sum_{k=1..floor(sqrt(n))} k*(k+1) * (A013928(1+floor(n/k)) - A013928(1+floor(n/(k+1)))) + Sum_{k=1..floor(n/(1+floor(sqrt(n))))} mu(k)^2 * floor(n/k) * floor(1+n/k))/2. - Daniel Suteu, Nov 23 2018

A074135 Triangle read by rows: for 1 <= k < n, a(n, k) is the least positive integer not already used. a(n, n) is the least positive integer not already used that makes the row sum a multiple of n.

Original entry on oeis.org

1, 2, 4, 3, 5, 7, 6, 8, 9, 13, 10, 11, 12, 14, 18, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 40, 36, 37, 38, 39, 41, 42, 43, 44, 49, 45, 46, 47, 48, 50, 51, 52, 53, 54, 64, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 75, 66, 67, 68, 69, 70, 71

Offset: 1

Amarnath Murthy, Aug 27 2002

Alternative (but equivalent) definition: In the following square array, numbers (not occurring earlier) are entered like this a(1, 1), a(1, 2), a(2, 1), a(3, 1), a(2, 2), a(1, 3), a(1, 4), a(2, 3), a(3, 2), a(4, 1), a(5, 1), a(4, 2), ... such that the n-th diagonal sum is a multiple of n. 1 2 7 6 18... 4 5 8 14... 3 9 12... 13 11... 10... ... sequence contains terms as they are entered.

Cf. A074132, A074133, A074134, A074136.

Cf. A082017, A082018, A082019, A082020.

Edited and extended by David Wasserman, Oct 31 2006

Further edited by N. J. A. Sloane Jan 17 2009 at the suggestion of R. J. Mathar

A306927 a(n) = A001615(n) - n.

Original entry on oeis.org

0, 1, 1, 2, 1, 6, 1, 4, 3, 8, 1, 12, 1, 10, 9, 8, 1, 18, 1, 16, 11, 14, 1, 24, 5, 16, 9, 20, 1, 42, 1, 16, 15, 20, 13, 36, 1, 22, 17, 32, 1, 54, 1, 28, 27, 26, 1, 48, 7, 40, 21, 32, 1, 54, 17, 40, 23, 32, 1, 84, 1, 34, 33, 32, 19, 78, 1, 40, 27, 74, 1, 72

Offset: 1

Torlach Rush, Mar 16 2019

Analogous to A051953.
a(n) = A051953(n) if n is an element of A000961.
a(n) > A051953(n) if n is an element of A024619.
The sum of the proper divisors d of n such that n/d is squarefree. - Amiram Eldar, Sep 06 2019

			0 is a term because A001615(1) - 1 = 0.
1 is a term because A001615(2) - 2 = 1.
3 is a term because A001615(9) - 9 = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Dedekind psi function.

Cf. A000961, A001615, A024619, A051953, A082020.

a[1] = 0; a[n_] := n * (Times @@ (1 + 1/FactorInteger[n][[;; , 1]]) - 1); Array[a, 100] (* Amiram Eldar, Sep 06 2019 *)

a(n) = n*(sumdivmult(n, d, issquarefree(d)/d) - 1); \\ Michel Marcus, Mar 18 2019

a(n) = A001615(n) - n.
a(n) = Sum_{d|n, dAmiram Eldar, Sep 06 2019
Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = 15/Pi^2 - 1 = 0.519817... . - Amiram Eldar, Dec 08 2023

A157292 Decimal expansion of 315/(2*Pi^4).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Feb 26 2009

Equals the asymptotic mean of the abundancy index of the 5-free numbers (numbers that are not divisible by a 5th power other than 1) (Jakimczuk and Lalín, 2022). - Amiram Eldar, May 12 2023

			1.61689220511... = (1+1/2^2+1/2^4)*(1+1/3^2+1/3^4)*(1+1/5^2+1/5^4)*(1+1/7^2+1/7^4)*...

Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (1).
Index entries for transcendental numbers.

Cf. A001248, A004709, A030514, A092744, A013661, A013664, A082020.

Maple
```
evalf(315/2/Pi^4) ;
```

RealDigits[N[Zeta[2]/Zeta[6], 150]][[1]] (* Geoffrey Critzer, Feb 16 2015 *)

315/2/Pi^4 \\ Charles R Greathouse IV, Oct 01 2022

Equals Product_{p = primes} (1 + 1/p^2 + 1/p^4), whereas, the product over (1 + 2/p^2 + 1/p^4) equals A082020^2.
Equals A013661/A013664 = Product_{i>=1} (1+1/A001248(i)+1/A030514(i)).
Equals 315*A092744/2.
Equals Sum_{n>=1} 1/A004709(n)^2. - Geoffrey Critzer, Feb 16 2015

A366440 The sum of divisors of the cubefree numbers (A004709).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 13, 18, 12, 28, 14, 24, 24, 18, 39, 20, 42, 32, 36, 24, 31, 42, 56, 30, 72, 32, 48, 54, 48, 91, 38, 60, 56, 42, 96, 44, 84, 78, 72, 48, 57, 93, 72, 98, 54, 72, 80, 90, 60, 168, 62, 96, 104, 84, 144, 68, 126, 96, 144, 72, 74, 114, 124, 140

Offset: 1

Amiram Eldar, Oct 10 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634; p. 619, eq. (2).

Cf. A000203, A004709, A358040.
Cf. A002117, A082020.
Similar sequences: A062822, A065764, A180114, A362986, A366439.

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

lista(max) = for(k = 1, max, my(f = factor(k), iscubefree = 1); for(i = 1, #f~, if(f[i, 2] > 2, iscubefree = 0; break)); if(iscubefree, print1(sigma(f), ", ")));

from sympy import mobius, integer_nthroot, divisor_sigma
def A366440(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return divisor_sigma(m) # Chai Wah Wu, Aug 06 2024

a(n) = A000203(A004709(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 15*zeta(3)/(2*Pi^2) = A082020 * A002117 / 2 = 0.913453711751... .
The asymptotic mean of the abundancy index of the cubefree numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A004709(k) = 15/Pi^2 = 1.519817... (A082020).

A332880 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j).

Original entry on oeis.org

1, 3, 4, 3, 6, 2, 8, 3, 4, 9, 12, 2, 14, 12, 8, 3, 18, 2, 20, 9, 32, 18, 24, 2, 6, 21, 4, 12, 30, 12, 32, 3, 16, 27, 48, 2, 38, 30, 56, 9, 42, 16, 44, 18, 8, 36, 48, 2, 8, 9, 24, 21, 54, 2, 72, 12, 80, 45, 60, 12, 62, 48, 32, 3, 84, 24, 68, 27, 32, 72

Offset: 1

Ilya Gutkovskiy, Feb 28 2020

Numerator of sum of reciprocals of squarefree divisors of n.

(6/Pi^2) * A332881(n)/a(n) is the asymptotic density of numbers that are coprime to their digital sum in base n+1 (see A094387 and A339076 for bases 2 and 10). - Amiram Eldar, Nov 24 2022

			1, 3/2, 4/3, 3/2, 6/5, 2, 8/7, 3/2, 4/3, 9/5, 12/11, 2, 14/13, 12/7, 8/5, 3/2, 18/17, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001615, A008683, A017665, A028235, A028236, A048250, A076512, A306695, A308443, A308462, A332881 (denominators), A332882.
Cf. also A348734, A348735.
Cf. A059956, A065463, A082020, A094387, A339076.

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

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

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A332880(n) = numerator(A001615(n)/n);

Numerators of coefficients in expansion of Sum_{k>=1} mu(k)^2*x^k/(k*(1 - x^k)).
a(n) = numerator of Sum_{d|n} mu(d)^2/d.
a(n) = numerator of psi(n)/n.
a(p) = p + 1, where p is prime.
a(n) = A001615(n) / A306695(n) = A001615(n) / gcd(n, A001615(n)). - Antti Karttunen, Nov 15 2021
From Amiram Eldar, Nov 24 2022: (Start)
Asymptotic means:
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A332881(k) = 15/Pi^2 = 1.519817... (A082020).
Limit_{m->oo} (1/m) * Sum_{k=1..m} A332881(k)/a(k) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463). (End)

A088246 Decimal expansion of 21/(2*Pi^2).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Sep 25 2003

			1.06387242824454660016073436370214020849576713405858...

Eric Weisstein's World of Mathematics, Moebius Function.
Index entries for transcendental numbers.

Cf. A007434, A030229, A082020, A088245.

Cf. A104141.

RealDigits[21/2/Pi^2, 10, 100][[1]] (* Amiram Eldar, May 23 2020 *)

21/2/Pi^2 \\ Charles R Greathouse IV, Oct 01 2022

Equals Sum_{k>=1} 1/A030229(k)^2. - Amiram Eldar, May 23 2020
Equals zeta(4)/zeta(6). - Terry D. Grant, Nov 07 2020
Equals Sum_{k>=1} A007434(k)/k^6. - Amiram Eldar, Jan 25 2024

cp's OEIS Frontend

A088245 Decimal expansion of 9/(2*Pi^2).

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A376936 Powerful numbers divisible by cubes of 2 distinct primes.

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A344753 a(n) = sigma(n) + psi(n) - 2n = Sum_{d|n, d

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A173290 Partial sums of A001615.

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A074135 Triangle read by rows: for 1 <= k < n, a(n, k) is the least positive integer not already used. a(n, n) is the least positive integer not already used that makes the row sum a multiple of n.

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A306927 a(n) = A001615(n) - n.

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A157292 Decimal expansion of 315/(2*Pi^4).

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