A192265 -id:A192265 - cp's OEIS frontend

A100879 a(n) = n^sigma(n).

1, 8, 81, 16384, 15625, 2176782336, 5764801, 35184372088832, 2541865828329, 1000000000000000000, 3138428376721, 1648446623609512543951043690496, 3937376385699289, 3214199700417740936751087616

Offset: 1

Odimar Fabeny, Jan 09 2005

			a(1) = 1^1 = 1;
a(2) = 2^(1+2) = 8;
a(3) = 3^(1+3) = 81;
a(4) = 4^(1+2+4) = 16384.

Cf. A000203, A192265.

with(numtheory): seq(n^add(d,d=divisors(n)),n=1..17); # C. Ronaldo, Jan 19 2005

Table[n^DivisorSigma[1,n],{n,20}] (* Harvey P. Dale, Oct 05 2011 *)

a(n)=n^sigma(n) \\ Charles R Greathouse IV, Aug 25 2014

a(n) = n^A000203(n). - Michel Marcus, Mar 17 2018

Sum_{n>=1} 1/a(n) = A192265. - Amiram Eldar, Nov 15 2020

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005

A192266 Decimal expansion of Sum_{k >= 1} 1/k^sigma_(k) where sigma_(n) is the sum of the anti-divisors of n.

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava, Jun 27 2011

Continued fraction (2,7,1,4,1,1,1,6,4,1,11,1,2...).

			1/1^sigma*(1)+ 1/2^sigma*(2) + 1/3^sigma*(3) + 1/4^sigma*(4) + 1/5^sigma*(5) + 1/6^sigma*(6) + ... = 1/1^0 + 1/2^0 + 1/3^2 + 1/4^3 + 1/5^5 + 1/6^4 + ... = 2.12782780242507..

Cf. A066417, A192265.

with(numtheory): P:=proc(i) local a,j,k,n,s; d:=2; for n from 3 to i do k:=0; j:=n;
while j mod 2 <> 1 do k:=k+1; j:=j/2; od; a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
d:=d+1/n^a; od; print(evalf(d, 300)); end: P(100);

f[n_] := Total@ Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]; First@ RealDigits@ N[Sum[1/k^f@ k, {k, 120}], 86] (* _Michael De Vlieger, Oct 08 2015 *)

Corrected and edited by R. J. Mathar, Jun 27 2011

A239725 Decimal expansion of sum(1/k^(phi(k)), k=1..infinity), where phi(n) is the Euler totient function.

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava, Mar 25 2014

Rational approximation: 29329/17218.

Continued fraction: (1, 1, 2, 2, 1, 2, 4, 36, 3, 1, 11, …).

			1.70339179928153054530520483941958492888306265194505541291458253107862176...

Cf. A000010, A192265, A192266.

with(numtheory); P:=proc(q) local k; print(evalf(add(1/k^phi(k),k=1..q),100)); end: P(1000);

cp's OEIS Frontend

A100879 a(n) = n^sigma(n).

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A192266 Decimal expansion of Sum_{k >= 1} 1/k^sigma_(k) where sigma_(n) is the sum of the anti-divisors of n.

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A239725 Decimal expansion of sum(1/k^(phi(k)), k=1..infinity), where phi(n) is the Euler totient function.

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cp's OEIS Frontend

A100879 a(n) = n^sigma(n).

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A192266 Decimal expansion of Sum_{k >= 1} 1/k^sigma_*(k) where sigma_*(n) is the sum of the anti-divisors of n.

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A239725 Decimal expansion of sum(1/k^(phi(k)), k=1..infinity), where phi(n) is the Euler totient function.

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Maple

A192266 Decimal expansion of Sum_{k >= 1} 1/k^sigma_(k) where sigma_(n) is the sum of the anti-divisors of n.