A217872 -id:A217872 - cp's OEIS frontend

A023887 a(n) = sigma_n(n): sum of n-th powers of divisors of n.

1, 5, 28, 273, 3126, 47450, 823544, 16843009, 387440173, 10009766650, 285311670612, 8918294543346, 302875106592254, 11112685048647250, 437893920912786408, 18447025552981295105, 827240261886336764178, 39346558271492178925595, 1978419655660313589123980

Offset: 1

Olivier Gérard

Logarithmic derivative of A023881.

Compare to A217872(n) = sigma(n)^n.

			The divisors of 6 are 1, 2, 3 and 6, so a(6) = 1^6 + 2^6 + 3^6 + 6^6 = 47450.

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.

Nick Hobson, Table of n, a(n) for n = 1..100

Cf. A000203, A001157-A001160, A013954-A013972, A023881, A199858, subdiagonal in A109974.

A023887 := proc(n)
    numtheory[sigma][n](n) ;
end proc:
seq(A023887(n),n=1..10) ; # R. J. Mathar, Apr 06 2022

Table[DivisorSigma[n,n],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)

makelist(divsum(n,n),n,1,20); /* Emanuele Munarini, Mar 26 2011 */

a(n) = sigma(n,n); \\ Nick Hobson, Nov 25 2006

from sympy import divisor_sigma
def A023887(n): return divisor_sigma(n,n) # Chai Wah Wu, Jun 19 2022

G.f.: Sum_{n>0} (n*x)^n/(1-(n*x)^n). - Vladeta Jovovic, Oct 27 2002
From Nick Hobson, Nov 25 2006: (Start)
If the canonical prime factorization of n > 1 is the product of p^e(p) then sigma_n(n) = Product_p ((p^(n*(e(p)+1)))-1)/(p^n-1).
sigma_n(n) is odd if and only if n is a square or twice a square. (End)
Conjecture: sigma_m(n) = sigma(n^m * rad(n)^(m-1))/sigma(rad(n)^(m-1)) for n > 0 and m > 0, where sigma = A000203 and rad = A007947. - Velin Yanev, Aug 24 2017
a(n) ~ n^n. - Vaclav Kotesovec, Nov 02 2018
Sum_{n>=1} 1/a(n) = A199858. - Amiram Eldar, Nov 19 2020

Edited by N. J. A. Sloane, Nov 25 2006

A156217 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^n*x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 1, 5, 26, 634, 2273, 502568, 821149, 323391480, 1514316108, 360153555251, 440146271717, 19353735645240631, 19423959923754863, 1599762850473552085, 35664862808194240282, 45517618403498070780338, 45844669861151626268272

Offset: 0

Paul D. Hanna, Feb 06 2009

nonn

Logarithmic derivative yields A217872.

Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

			G.f.: A(x) = 1 + x + 5*x^2 + 26*x^3 + 634*x^4 + 2273*x^5 + 502568*x^6 + ...
log(A(x)) = x + 3^2*x^2/2 + 4^3*x^3/3 + 7^4*x^4/4 + 6^5*x^5/5 + 12^6*x^6/6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..335

Cf. A217872, A224439, A000041, A000203.

A156217 := proc(n) option remember ; if n = 0 then 1; else add( (numtheory[sigma](k))^k*procname(n-k),k=1..n)/n ; fi; end: seq(A156217(n),n=0..10) ; # R. J. Mathar, Apr 02 2009

{a(n)=polcoeff(exp(sum(m=1,n,sigma(m)^m*x^m/m)+x*O(x^n)),n)}

{a(n)=if(n==0,1,(1/n)*sum(k=1,n,sigma(k)^k*a(n-k)))}

a(n) = (1/n)*Sum_{k=1..n} sigma(k)^k * a(n-k) for n>0, with a(0)=1.

A236285 a(n) = tau(n)^sigma(n), where tau(n) = A000005(n) = the number of divisors of n and sigma(n) = A000203(n) = the sum of divisors of n.

Original entry on oeis.org

1, 8, 16, 2187, 64, 16777216, 256, 1073741824, 1594323, 68719476736, 4096, 6140942214464815497216, 16384, 281474976710656, 281474976710656, 4656612873077392578125, 262144, 2227915756473955677973140996096, 1048576, 481229803398374426442198455156736

Offset: 1

Jaroslav Krizek, Jan 21 2014

nonn

			a(4) = tau(4)^sigma(4) = 3^7 = 2187.

Jaroslav Krizek, Table of n, a(n) for n = 1..100

Cf. A000005 (tau(n)), A000203 (sigma(n)), A062758 (n^tau(n)), A217872 (sigma(n)^n), A236284 (tau(n)^n), A236286.

Table[DivisorSigma[0, n]^DivisorSigma[1, n], {n, 1000}]

s=[]; for(n=1, 20, s=concat(s, sigma(n, 0)^sigma(n))); s \\ Colin Barker, Jan 22 2014

a(n) = A000005(n)^A000203(n).

A236284 a(n) = tau(n)^n, where tau(n) = A000005(n) = the number of divisors of n.

Original entry on oeis.org

1, 4, 8, 81, 32, 4096, 128, 65536, 19683, 1048576, 2048, 2176782336, 8192, 268435456, 1073741824, 152587890625, 131072, 101559956668416, 524288, 3656158440062976, 4398046511104, 17592186044416, 8388608, 4722366482869645213696, 847288609443, 4503599627370496

Offset: 1

Jaroslav Krizek, Jan 21 2014

nonn

			a(4) = tau(4)^4 = 3^4 = 81.

Jaroslav Krizek, Table of n, a(n) for n = 1..100

Cf. A000005 (tau(n)), A062758 (n^tau(n)), A217872 (sigma(n)^n), A236285 (tau(n)^sigma(n)), A236286.

Mathematica
```
Table[DivisorSigma[0, n]^n, {n, 1000}]
```

s=[]; for(n=1, 30, s=concat(s, sigma(n, 0)^n)); s \\ Colin Barker, Jan 22 2014

a(n) = A000005(n)^n.

A236286 a(n) = tau(n)^sigma(n) / tau(n)^n, where tau(n) = A000005(n) = the number of divisors of n and sigma(n) = A000203(n) = the sum of divisors of n.

Original entry on oeis.org

1, 2, 2, 27, 2, 4096, 2, 16384, 81, 65536, 2, 2821109907456, 2, 1048576, 262144, 30517578125, 2, 21936950640377856, 2, 131621703842267136, 4194304, 268435456, 2, 324518553658426726783156020576256, 729, 4294967296, 67108864, 6140942214464815497216, 2

Offset: 1

Jaroslav Krizek, Jan 21 2014

nonn

a(n) = tau(n)^sigma_p(n), where sigma_p(n) = A001065(n) = the sum of proper divisors of n.

			a(4) = tau(4)^sigma(4) / tau(4)^4 = 3^7 /3^4 = 27.

Jaroslav Krizek, Table of n, a(n) for n = 1..50

Cf. A000005 (tau(n)), A000203 (sigma(n)), A001065 (sigma_p(n)), A062758 (n^tau(n)), A217872 (sigma(n)^n), A236284 (tau(n)^n), A236285 (tau(n)^sigma(n)).

Table[DivisorSigma[0, n]^[DivisorSigma[1, n] - n], {n, 1000}]

s=[]; for(n=1, 30, s=concat(s, sigma(n, 0)^sigma(n)/sigma(n, 0)^n)); s \\ Colin Barker, Jan 22 2014

a(n) = A236285(n) / A236284(n) = A000005(n)^A000203(n) / A000005(n)^n = A000005(n)^A001065(n).

a(p) = 2 for p = primes (A000040).

A300906 Numbers k such that sigma(k)^k divides k^sigma(k).

Original entry on oeis.org

1, 6, 28, 84, 120, 364, 420, 496, 672, 840, 1080, 1320, 1488, 1782, 2280, 2760, 3276, 3360, 3472, 3480, 3720, 3780, 5640, 7080, 7392, 7440, 7560, 8128, 8736, 9240, 9480, 10416, 10920, 11880, 12400, 15456, 15960, 16368, 16380, 17880, 18360, 18600, 19320, 20520

Offset: 1

Jaroslav Krizek, Mar 20 2018

nonn

Numbers k such that A217872(k) divides A100879(k).
Numbers k such that A300905(k) = 0.
Corresponding quotients: 1, 729, 123476695691247935826229781856256, ...
m-perfect numbers k (A007691) are terms iff m divides k.

			6 is a term because 6^sigma(6) / sigma(6)^6 = 6^12 / 12^6 = 2176782336 / 2985984 = 729 (integer).

Cf. A000203, A100879, A217872, A300905.

Filtered([1..30000],n->PowerModInt(n,Sigma(n),Sigma(n)^n)=0); # Muniru A Asiru, Mar 20 2018

[n: n in[1..20000]  | n^SumOfDivisors(n) mod SumOfDivisors(n)^n eq 0];

with(numtheory):
select(n->n &^ sigma(n) mod sigma(n)^n =0, [`$`(1..30000)]); # Muniru A Asiru, Mar 20 2018

isok(n) = my(s = sigma(n)); Mod(n, s^n)^s == 0; \\ Michel Marcus, Mar 23 2018

A158947 Inverse Euler transform of A156217.

Original entry on oeis.org

1, 4, 21, 598, 1555, 497652, 299593, 320361028, 1178277701, 357046721884, 67546215517, 19351522090518670, 61054982558011, 1502551437369035044, 33657152197069739919, 45463945109198918808616, 128583032925805678351

Offset: 1

Vladeta Jovovic, Mar 31 2009

Seiichi Manyama, Table of n, a(n) for n = 1..335
N. J. A. Sloane, Transforms

Cf. A008683, A156217, A217872.

A158947 := proc(n) add(numtheory[sigma](d)^d*numtheory[mobius](n/d),d=numtheory[divisors](n))/n ; end: seq( A158947(n),n=1..40) ; # R. J. Mathar, Apr 02 2009
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(A156217):
seq(a(n), n = 1..17); # Peter Luschny, Nov 21 2022

f[n_] := Block[{d = Divisors@n}, Plus @@ (DivisorSigma[1, d]^d*MoebiusMu[n/d])/n]; Array[f, 17] (* Robert G. Wilson v, May 04 2009 *)

a(n) = (1/n)*Sum_{d|n} sigma(d)^d*moebius(n/d).

Extended by R. J. Mathar, Apr 02 2009

A224440 a(n) = sigma(n)^(n-1).

Original entry on oeis.org

1, 3, 16, 343, 1296, 248832, 262144, 170859375, 815730721, 198359290368, 61917364224, 8293509467471872, 56693912375296, 876488338465357824, 21035720123168587776, 23465261991844685929951, 121439531096594251776, 1117116121846700839825703079, 262144000000000000000000

Offset: 1

Paul D. Hanna, Apr 06 2013

nonn

Here sigma(n) = A000203(n) is the sum of the divisors of n.

			L.g.f.: L(x) = x + 3^1*x^2/2 + 4^2*x^3/3 + 7^3*x^4/4 + 6^4*x^5/5 + 12^5*x^6/6 +...
where exponentiation yields the g.f. of A224439:
exp(L(x)) = 1 + x + 2*x^2 + 7*x^3 + 93*x^4 + 357*x^5 + 41927*x^6 +...

Seiichi Manyama, Table of n, a(n) for n = 1..335

Cf. A224439, A217872.

{a(n)=sigma(n)^(n-1)}
for(n=1, 20, print1(a(n), ", "))

Logarithmic derivative of A224439.

Offset corrected by Seiichi Manyama, Nov 10 2017

A236287 a(n) = sigma(n)^tau(n), where tau(n) = A000005(n) = the number of divisors of n and sigma(n) = A000203(n) = the sum of divisors of n.

Original entry on oeis.org

1, 9, 16, 343, 36, 20736, 64, 50625, 2197, 104976, 144, 481890304, 196, 331776, 331776, 28629151, 324, 3518743761, 400, 5489031744, 1048576, 1679616, 576, 167961600000000, 29791, 3111696, 2560000, 30840979456, 900, 722204136308736, 1024, 62523502209, 5308416

Offset: 1

Jaroslav Krizek, Jan 23 2014

nonn

			a(4) = sigma(4)^tau(4) = 7^3 = 343.

Jaroslav Krizek, Table of n, a(n) for n = 1..100

Cf. A000005 (tau(n)), A000203 (sigma(n)), A062758 (n^tau(n)), A217872 (sigma(n)^n), A236285 (tau(n)^sigma(n)), A236286.

Table[DivisorSigma[1, n]^DivisorSigma[0, n], {n, 1000}]

s=[]; for(n=1, 40, s=concat(s, sigma(n, 1)^sigma(n, 0))); s \\ Colin Barker, Jan 24 2014

a(n) = A000203(n)^A000005(n).

A236288 a(n) = sigma(n)^n / sigma(n)^tau(n), where tau(n) = A000005(n) = the number of divisors of n and sigma(n) = A000203(n) = the sum of divisors of n.

Original entry on oeis.org

1, 1, 4, 7, 216, 144, 32768, 50625, 4826809, 34012224, 5159780352, 481890304, 4049565169664, 63403380965376, 1521681143169024, 25408476896404831, 6746640616477458432, 12381557655576425121, 13107200000000000000000, 53148384174432398229504, 38685626227668133590597632

Offset: 1

Jaroslav Krizek, Jan 23 2014

nonn

Conjecture: number 1 is the only number n such that sigma(n)^(n - tau(n)) = sigma(n+1)^(n + 1 - tau(n+1)).

Conjecture: number 1 is the only number n such that sigma(n)^(n - tau(n)) = sigma(k)^(k - tau(k)) has solution for distinct numbers n and k.

			a(4) = sigma(4)^(4 - tau(4)) = 7^(4 - 3) = 7.

Jaroslav Krizek, Table of n, a(n) for n = 1..50

Cf. A000005 (tau(n)), A000203 (sigma(n)), A062758 (n^tau(n)), A217872 (sigma(n)^n), A236285 (tau(n)^sigma(n)), A236287 (sigma(n)^tau(n)).

Table[DivisorSigma[1, n]^[n - DivisorSigma[0, n]], {n, 50}]

s=[]; for(n=1, 30, s=concat(s, sigma(n, 1)^(n-sigma(n, 0)))); s \\ Colin Barker, Jan 24 2014

a(n) = sigma(n)^(n - tau(n)).

a(n) = A217872(n) / A236287(n) = A000203(n)^n / A000203(n)^A000005(n) = A000203(n)^A049820(n).

cp's OEIS Frontend

A023887 a(n) = sigma_n(n): sum of n-th powers of divisors of n.

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A156217 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^n*x^n/n ), a power series in x with integer coefficients.

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A236285 a(n) = tau(n)^sigma(n), where tau(n) = A000005(n) = the number of divisors of n and sigma(n) = A000203(n) = the sum of divisors of n.

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A236284 a(n) = tau(n)^n, where tau(n) = A000005(n) = the number of divisors of n.

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A236286 a(n) = tau(n)^sigma(n) / tau(n)^n, where tau(n) = A000005(n) = the number of divisors of n and sigma(n) = A000203(n) = the sum of divisors of n.

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A300906 Numbers k such that sigma(k)^k divides k^sigma(k).

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A158947 Inverse Euler transform of A156217.

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