A055225 -id:A055225 - cp's OEIS frontend

A073705 a(n) = Sum_{ d divides n } (n/d)^(2d).

1, 5, 10, 33, 26, 182, 50, 577, 811, 1750, 122, 16194, 170, 18982, 74900, 135425, 290, 847127, 362, 2498178, 4901060, 4209430, 530, 78564226, 9766251, 67138102, 387952660, 542674914, 842, 4866184552, 962, 8606778369, 31382832260, 17179953862, 6385992100, 422091411267, 1370, 274878038710

Offset: 1

Paul D. Hanna, Aug 04 2002

a(n) is the number of linear partitions of the linearly ordered set [n] = {1,2,...,n} with blocks of the same size, where each block has two element marked (possibly equal). For instance, for n = 3, we have the following 10 linear partitions (where the marked elements are denoted by a and b, or by X when they coincide):

(X)(X)(X), (ab3), (a2b), (1ab), (ba3), (b2a), (1ba), (X23), (1X3), (12X). - Emanuele Munarini, Feb 03 2014

			a(10) = (10/1)^(2*1) +(10/2)^(2*2) +(10/5)^(2*5) +(10/10)^(2*10) = 1750 because positive divisors of 10 are 1, 2, 5, 10.

Seiichi Manyama, Table of n, a(n) for n = 1..3143

Cf. A038041, A055225, A151954, A236696.

Table[Total[Quotient[n, x = Divisors[n]]^(2*x)], {n, 34}] (* Jayanta Basu, Jul 08 2013 *)

a(n):= lsum(d^(2*n/d),d,listify(divisors(n)));
makelist(a(n),n,1,40); /* Emanuele Munarini , Feb 03 2014 */

a(n)=sumdiv(n, d, (d)^(2*n/d) );  /* Joerg Arndt, Oct 07 2012 */

G.f.: Sum_{n>=1} -log(1 - (n^2)*x^n)/n = Sum_{n>=1} a(n) x^n/n.

G.f.: Sum_{k>=1} k^2*x^k/(1-k^2*x^k). - Benoit Cloitre, Apr 21 2003

Corrected a(14) and inserted missing a(16) by Jayanta Basu, Jul 08 2013

A076717 a(n) = -Sum_{d|n} (-n/d)^d.

Original entry on oeis.org

1, 1, 4, -1, 6, 4, 8, -25, 37, 16, 12, -106, 14, 92, 384, -561, 18, -65, 20, -706, 2552, 1948, 24, -15658, 3151, 8048, 20440, -2570, 30, -33326, 32, -135393, 178512, 130816, 94968, -583219, 38, 523964, 1596560, -2465370, 42, -2521186, 44, -15082, 16364502, 8388124, 48, -78560082, 823593, 23888231

Offset: 1

Vladeta Jovovic, Oct 27 2002

sign

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A055225, A308367, A321438.

a(n) = -sumdiv(n, d, (-n/d)^d); \\ Michel Marcus, Mar 22 2021

G.f.: Sum_{n>0} n*x^n/(1+n*x^n).

A294956 a(n) = Sum_{d|n} d^(d + n/d).

Original entry on oeis.org

1, 9, 82, 1041, 15626, 280212, 5764802, 134221889, 3486785131, 100000078254, 3138428376722, 106993207077516, 3937376385699290, 155568095598166344, 6568408355713287812, 295147905180426634241, 14063084452067724991010

Offset: 1

Seiichi Manyama, Nov 12 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..385

Cf. A055225, A062796, A294957.

sd[n_]:=Total[#^(#+n/#)&/@Divisors[n]]; Array[sd,20] (* Harvey P. Dale, Mar 28 2021 *)

PARI
```
a(n) = sumdiv(n, d, d^(d+n/d));
```

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-k*x^k)^(k^(k-1)))))) \\ Seiichi Manyama, Jun 09 2019

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^(k+1)*x^k/(1-k*x^k))) \\ Seiichi Manyama, Jan 11 2023

L.g.f.: -log(Product_{k>=1} (1 - k*x^k)^(k^(k-1))) = Sum_{k>=1} a(k)*x^k/k. - Seiichi Manyama, Jun 09 2019

G.f.: Sum_{k>0} k^(k+1) * x^k / (1 - k * x^k). - Seiichi Manyama, Jan 11 2023

A309369 a(n) = Sum_{d|n} phi(n/d)^d, where phi = Euler totient function (A000010).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 10, 15, 22, 11, 34, 13, 44, 105, 42, 17, 116, 19, 314, 357, 112, 23, 426, 1045, 158, 747, 1474, 29, 5290, 31, 594, 3069, 274, 24185, 6082, 37, 344, 9945, 67922, 41, 63542, 43, 12170, 303225, 508, 47, 74834, 279979, 1050022, 135201, 29098, 53, 309872, 4294345

Offset: 1

Ilya Gutkovskiy, Jul 25 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A000010, A055225, A164941, A264782, A279789.

Table[Sum[EulerPhi[n/d]^d, {d, Divisors[n]}], {n, 1, 55}]
nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 - EulerPhi[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 55; CoefficientList[Series[-Log[Product[(1 - EulerPhi[k] x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest

a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^(gcd(k, n)-1)); \\ Seiichi Manyama, Mar 13 2021

G.f.: Sum_{k>=1} phi(k)*x^k/(1 - phi(k)*x^k).
L.g.f.: -log(Product_{k>=1} (1 - phi(k)*x^k)^(1/k)).
a(p) = p for p prime.
a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^(gcd(k, n) - 1). - Seiichi Manyama, Mar 13 2021

A327238 Expansion of Sum_{k>=1} ((1 + k * x^k)^k - 1).

Original entry on oeis.org

1, 4, 9, 20, 25, 63, 49, 160, 108, 350, 121, 940, 169, 1225, 1475, 2304, 289, 7560, 361, 8025, 12446, 7139, 529, 58192, 3750, 13858, 61965, 102655, 841, 191181, 961, 318464, 220704, 40460, 354172, 1304370, 1369, 63175, 629863, 4012608, 1681, 1916733, 1849

Offset: 1

Ilya Gutkovskiy, Sep 14 2019

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

Cf. A055225, A056045, A318636, A327124.

nmax = 43; CoefficientList[Series[Sum[((1 + k x^k)^k - 1), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (n/#)^# Binomial[n/#, #] &], {n, 1, 43}]

a(n)={sumdiv(n, d, (n/d)^d * binomial(n/d,d))} \\ Andrew Howroyd, Sep 14 2019

a(n) = Sum_{d|n} (n/d)^d * binomial(n/d,d).

a(p) = p^2, where p is prime.

A308509 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*n/d).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 4, 3, 1, 9, 10, 9, 2, 1, 17, 28, 33, 6, 4, 1, 33, 82, 129, 26, 24, 2, 1, 65, 244, 513, 126, 182, 8, 4, 1, 129, 730, 2049, 626, 1458, 50, 41, 3, 1, 257, 2188, 8193, 3126, 11954, 344, 577, 37, 4, 1, 513, 6562, 32769, 15626, 99594, 2402, 8705, 811, 68, 2

Offset: 1

Seiichi Manyama, Jun 02 2019

			Square array begins:
   1,  1,   1,    1,     1,     1,      1, ...
   2,  3,   5,    9,    17,    33,     65, ...
   2,  4,  10,   28,    82,   244,    730, ...
   3,  9,  33,  129,   513,  2049,   8193, ...
   2,  6,  26,  126,   626,  3126,  15626, ...
   4, 24, 182, 1458, 11954, 99594, 840242, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=0..3 give A000005, A055225, A073705, A073706.

Cf. A294579.

T[n_, k_] := DivisorSum[n, #^(k*n/#) &]; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 11 2021 *)

T(n,k) = sumdiv(n, d, (n/d)^(k*d));
matrix(9, 9, n, k, T(n,k-1)) \\ Michel Marcus, Jun 02 2019

L.g.f. of column k: -log(Product_{j>=1} (1 - j^k*x^j)^(1/j)).

A(n,k) = Sum_{d|n} (n/d)^(k*d).

A362683 Expansion of Sum_{k>0} (1/(1 - k*x^k)^2 - 1).

Original entry on oeis.org

2, 7, 10, 25, 16, 78, 22, 153, 136, 298, 34, 1254, 40, 1214, 2004, 3825, 52, 11385, 58, 20894, 18932, 25006, 70, 150002, 18826, 115274, 199828, 389510, 88, 1334624, 94, 1725281, 2131188, 2360266, 725948, 14878299, 112, 10486958, 22329428, 37317986, 124, 120957336, 130

Offset: 1

Seiichi Manyama, Jun 13 2023

nonn

Cf. A338662, A363639, A363640.

Cf. A007503, A055225, A359103, A363646.

a[n_] := DivisorSum[n, (n/#)^# * (# + 1) &]; Array[a, 50] (* Amiram Eldar, Jul 17 2023 *)

PARI
```
a(n) = sumdiv(n, d, (n/d)^d*(d+1));
```

a(n) = Sum_{d|n} (n/d)^d * (d+1) = A055225(n) + A359103(n).

If p is prime, a(p) = 1 + 3*p.

A180851 Sum of increasing powers of divisors: a(n) = Sum_{i=1..q} d(i)^i where d(1) < d(2) < ... < d(q) are the divisors of n.

Original entry on oeis.org

1, 5, 10, 69, 26, 1328, 50, 4165, 739, 10130, 122, 2994048, 170, 38764, 50760, 1052741, 290, 34072601, 362, 64100694, 194834, 235592, 530, 110111416192, 15651, 459178, 532180, 482430598, 842, 656271867808, 962, 1074794565, 1187262

Offset: 1

Jason Earls, Sep 21 2010

			For n=4, the divisors of 4 are [1, 2, 4] and summing them as increasing powers yields: 1^1+2^2+4^3 = 69.
For n=12, the divisors of 12 are [1, 2, 3, 4, 6, 12] and summing them as increasing powers yields: 1^1+2^2+3^3+4^4+6^5+12^6 = 2994048.

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

Cf. A027750.
Positions of primes: A180852.
Comparable sequences: A055225, A264786, A344459.

f:= proc(n) local D,k;
  D:=sort(convert(numtheory:-divisors(n),list));
  add(D[k]^k,k=1..nops(D))
end proc:
map(f, [$1..100]); # Robert Israel, Sep 11 2020

Total[Divisors[#]^Range[DivisorSigma[0,#]]]&/@Range[40] (* Harvey P. Dale, Aug 16 2011 *)

a(n) = my(d = divisors(n)); sum(k=1, #d, d[k]^k); \\ Michel Marcus, Jan 01 2016

Name made precise by Peter Munn, Sep 19 2024

A327579 a(n) = n! * Sum_{d|n} d^(n/d) / d!.

Original entry on oeis.org

1, 4, 9, 76, 125, 4686, 5047, 389768, 1995849, 62445610, 39916811, 23574862092, 6227020813, 5667436494734, 55630647072015, 2922249531801616, 355687428096017, 2425220588831040018, 121645100408832019, 1364553980880330240020, 18677216386213152768021, 1152100749379237026969622

Offset: 1

Ilya Gutkovskiy, Sep 17 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..425

Cf. A055225, A057625, A327578, A354843.

a[n_] := n! Sum[d^(n/d)/d!, {d, Divisors[n]}]; Table[a[n], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[x^k/((k - 1)! (1 - k x^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

a(n) = n! * sumdiv(n, d, d^(n/d) / d!); \\ Michel Marcus, Sep 17 2019

E.g.f.: Sum_{k>=1} x^k / ((k - 1)! * (1 - k * x^k)).

A294463 E.g.f.: Product_{k>0} (1-k*x^k)^(1/k).

Original entry on oeis.org

1, -1, -2, 0, -12, 180, -1080, 15120, -45360, -15120, 6501600, 166320000, -6017457600, 73297224000, 724669545600, -32528399904000, 169180371360000, 6794185638240000, -119705492402496000, 2601008778880512000, -119160456995099520000

Offset: 0

Seiichi Manyama, Oct 31 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425

Cf. A028343, A055225, A294462.

N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1-k*x^k)^(1/k))))

a(0) = 1 and a(n) = -(n-1)! * Sum_{k=1..n} A055225(k)*a(n-k)/(n-k)! for n > 0.

E.g.f.: exp(-Sum_{k>=1} Sum_{j>=1} j^(k-1)*x^(j*k)/k). - Ilya Gutkovskiy, May 28 2018

cp's OEIS Frontend

A073705 a(n) = Sum_{ d divides n } (n/d)^(2d).

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A076717 a(n) = -Sum_{d|n} (-n/d)^d.

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A294956 a(n) = Sum_{d|n} d^(d + n/d).

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A309369 a(n) = Sum_{d|n} phi(n/d)^d, where phi = Euler totient function (A000010).

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A327238 Expansion of Sum_{k>=1} ((1 + k * x^k)^k - 1).

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A308509 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*n/d).

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A362683 Expansion of Sum_{k>0} (1/(1 - k*x^k)^2 - 1).

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A180851 Sum of increasing powers of divisors: a(n) = Sum_{i=1..q} d(i)^i where d(1) < d(2) < ... < d(q) are the divisors of n.

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