A087909 -id:A087909 - cp's OEIS frontend

A208239 Triangle read by rows: T(n,m) = n + k - n/k, where k is the m-th divisor of n; 1 <= m <= tau(n).

1, 1, 3, 1, 5, 1, 4, 7, 1, 9, 1, 5, 7, 11, 1, 13, 1, 6, 10, 15, 1, 9, 17, 1, 7, 13, 19, 1, 21, 1, 8, 11, 13, 16, 23, 1, 25, 1, 9, 19, 27, 1, 13, 17, 29, 1, 10, 16, 22, 31, 1, 33, 1, 11, 15, 21, 25, 35, 1, 37, 1, 12, 19, 21, 28, 39, 1, 17, 25, 41, 1, 13, 31, 43, 1, 45, 1, 14, 19, 22, 26

Offset: 1

Gerasimov Sergey, Jan 11 2013

n-th row sum is equal to A038040(n) = d(n)*n, where d = A000005.
Numbers n such that n + k - n/k is noncomposite number for all divisors k of n: 1, 2, 3, 6, 7, 10, 15, 19, 22, 30, 31, 37, 42, 57, 70, 79, 87, 97,...
Numbers n such that n + k - n/k is nonprime number for all divisor k of n: 1, 5, 8, 11, 13, 17, 23, 25, 29, 32, 38, 41, 43, 47, 53, 56, 59, 61, 62, 67, 68, 71, 73, 80, 81, 83, 88, 89, 93, 98, 101, 103, 107, 109, 111, 113, 121, 123, 125, 127,...
Smallest m such that n = m + k - m/k for all k is divisor of n, or 0 if no such m exists : 1, 0, 2, 4, 3, 8, 4, 12, 5, 8, 6, 20, 7, 24, 8, 12, 9, 32, 10, 36, 11, 16, 12, 44, 13, 24, 14, 20, 15, 56, 16, 60, 17, 24,..
Number of ways to write n as (p - q)/(1 - 1/q), where p is prime and q is a prime divisor of n: 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 0, 1, 0, 1, 2, 0, 0, 1, 1, 0, 1, 2, 0, 1, 0, 1, 0, 1, 0, 3, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 3, 0, 0, 1, 1, 0,...
Numbers n of the form (s - r)(1/s - 1) where s is divisor of n and r is anti-divisor of n:  10, 12, 14,...
The n-th row starts with 1 and ends with 2n-1; the first differences are symmetric w.r.t. reversal of the row (which corresponds to exchange of k and n/k). The second term in even lines is n/2+2. - M. F. Hasler, Jan 26 2013
If n is prime then n-th row is 1, 2n-1. - Zak Seidov, Feb 22 2013
T(n,A000005(n)) = A005408(n-1). - Reinhard Zumkeller, Feb 25 2013

			Triangle begins:
1,
1, 3,
1, 5,
1, 4, 7,
1, 9,
1, 5, 7, 11,
1, 13,
1, 6, 10, 15,
1, 9, 17,
1, 7, 13, 19,
1, 21,
1, 8, 11, 13, 16, 23.
In this last, 12th line (ending with 2*12-1), the first differences are (7,3,2,3,7).

Zak Seidov, Rows n = 1..200 of irregular triangle, flattened

Row lengths are A000005.

Cf. A027750, A038040, A087909.

a208239 n k = a208239_row n !! k
a208239_row n = map (+ n) $ zipWith (-) divs $ reverse divs
                where divs = a027750_row n
a208239_tabl = map a208239_row [1..]
-- Reinhard Zumkeller, Feb 25 2013

row[n_] := Table[n + k - n/k, {k, Divisors[n]}]; Table[row[n], {n, 1, 24}] // Flatten (* Jean-François Alcover, Jan 21 2013 *)

T(n,k) = n + A027750(n,k) + A027750(n,A000005(n)+1-k), 1<=k<=A000005(n). - Reinhard Zumkeller, Feb 25 2013

A309633 G.f.: x * Sum_{k>=1} x^k / (1 - a(k)*x^k).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 5, 6, 4, 1, 7, 1, 13, 11, 6, 1, 7, 17, 7, 11, 31, 1, 45, 1, 10, 18, 6, 146, 34, 1, 9, 27, 141, 1, 261, 1, 78, 364, 8, 1, 44, 730, 537, 18, 145, 1, 255, 1281, 2203, 51, 33, 1, 2213, 1, 47, 7461, 221, 4722, 1159, 1, 85, 38, 27948, 1, 2342, 1, 36, 17060, 347, 63146, 3427, 1

Offset: 1

Ilya Gutkovskiy, Aug 10 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..14665

Cf. A028815 (positions of 1's), A087909, A127525, A309634.

a[n_] := a[n] = SeriesCoefficient[x Sum[x^k/(1 - a[k] x^k), {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 80}]
a[n_] := a[n] = Sum[a[d]^((n - 1)/d - 1) , {d, Divisors[n - 1]}]; a[1] = 0; a[2] = 1; Table[a[n], {n, 1, 80}]

seq(n)={my(v=vector(n)); for(n=1, #v-1, v[n+1]=sumdiv(n, d, v[d]^(n/d-1))); v} \\ Andrew Howroyd, Aug 10 2019

a(1) = 0; a(n+1) = Sum_{d|n} a(d)^(n/d-1).

A356543 a(n) = Sum_{d|n} (d!)^(n/d-1).

Original entry on oeis.org

1, 2, 2, 4, 2, 12, 2, 34, 38, 138, 2, 1546, 2, 5106, 15698, 54274, 2, 889314, 2, 5689090, 25448258, 39917826, 2, 2486196610, 207360002, 6227024898, 131683574018, 215393466370, 2, 14769495662082, 2, 86475697160194, 1593350982706178, 355687428161538, 648227266560002

Offset: 1

Seiichi Manyama, Aug 11 2022

nonn

Cf. A087909, A342628, A356541.

a[n_] := DivisorSum[n, (#)!^(n/# - 1) &]; Array[a, 35] (* Amiram Eldar, Aug 30 2023 *)

PARI
```
a(n) = sumdiv(n, d, d!^(n/d-1));
```

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-k!*x^k)))

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

If p is prime, a(p) = 2.

A356661 a(n) = n! * Sum_{d|n} 1/d^(n/d - 1).

Original entry on oeis.org

1, 4, 12, 60, 240, 1860, 10080, 95760, 766080, 8210160, 79833600, 1100484000, 12454041600, 188172784800, 2683799838720, 44951306400000, 711374856192000, 13745322470880000, 243290200817664000, 5142812718440517120, 103294640229580800000, 2351280996859354560000

Offset: 1

Seiichi Manyama, Aug 21 2022

nonn

Cf. A087905, A087909, A098558, A356662.

a[n_] := n! * DivisorSum[n, 1/#^(n/# - 1) &]; Array[a, 22] (* Amiram Eldar, Aug 21 2022 *)

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

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(1-x^k/k))))

a(p) = 2 * p! for prime p.

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

A359134 a(n) = Sum_{d|n} (2*d)^(n/d - 1).

Original entry on oeis.org

1, 3, 5, 13, 17, 55, 65, 201, 293, 779, 1025, 3365, 4097, 12303, 17781, 49681, 65537, 204547, 262145, 791549, 1095429, 3145751, 4194305, 12897625, 16787217, 50331675, 68788805, 201591509, 268435457, 815505231, 1073741825, 3223326753, 4355433957, 12884901923

Offset: 1

Seiichi Manyama, Jan 13 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A087909, A278741, A349970, A359733, A359796.

a[n_] := DivisorSum[n, (2*#)^(n/# - 1) &]; Array[a, 30] (* Amiram Eldar, Aug 14 2023 *)

PARI
```
a(n) = sumdiv(n, d, (2*d)^(n/d-1));
```

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-2*k*x^k)))

G.f.: Sum_{k>0} x^k / (1 - 2 * k * x^k).

If p is prime, a(p) = 1 + 2^(p-1).

A359811 a(n) = Sum_{d|n} 2^(d-1) * d^(n/d-1).

Original entry on oeis.org

1, 3, 5, 13, 17, 53, 65, 177, 293, 625, 1025, 2541, 4097, 8769, 17109, 34561, 65537, 136013, 262145, 534481, 1054629, 2110465, 4194305, 8449325, 16787217, 33615873, 67155845, 134403521, 268435457, 537370845, 1073741825, 2148270081, 4295327397, 8591179777

Offset: 1

Seiichi Manyama, Jan 14 2023

Cf. A087909, A359134, A359730, A359796.

Table[Sum[2^(d-1) * d^(n/d - 1), {d, Divisors[n]}], {n, 1, 40}] (* Vaclav Kotesovec, Jan 14 2023 *)

a(n) = sumdiv(n, d, 2^(d-1)*d^(n/d-1));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, 2^(k-1)*x^k/(1-k*x^k)))

G.f.: Sum_{k>0} 2^(k-1) * x^k / (1 - k * x^k).
If p is prime, a(p) = 1 + 2^(p-1).
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Jan 14 2023

A213919 Triangle read by rows: T(n,m) = (n/k)^(k-1), where k is the m-th divisor of n, 1 <= m <= tau(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 4, 8, 1, 1, 9, 1, 1, 5, 16, 1, 1, 1, 1, 6, 16, 27, 32, 1, 1, 1, 1, 7, 64, 1, 1, 25, 81, 1, 1, 8, 64, 128, 1, 1, 1, 1, 9, 36, 243, 256, 1, 1, 1, 1, 10, 125, 256, 512, 1, 1, 49, 729, 1, 1, 11, 1024, 1, 1, 1, 1, 12, 64, 216, 1024, 2187, 2048, 1, 1, 625, 1, 1, 13, 4096, 1, 1, 81, 6561, 1, 1, 14, 343, 4096

Offset: 1

Gerasimov Sergey, Mar 05 2013

Divisor k of composite number n with maximal value (n/k)^(k-1): 2, 3, 4, 3, 5, 6, 7, 5, 8, 9, 10, 7, 11, 8, 5, 13, 9, 14,...

			Triangle begins:
  1;
  1, 1;
  1, 1;
  1, 2, 1;
  1, 1;
  1, 3, 4, 1;
  1, 1;
  1, 4, 8, 1;
  1, 9, 1;
  1, 5, 16, 1;
  1, 1;
  1, 6, 16, 27, 32, 1.

Cf. A000005 (row lengths), A027750, A087909 (row sums), A167401, A208239.

T(n,k) = A027750(n, A000005(n) + 1 - k)/(A027750(n,k) - 1), 1 <= k <= A000005(n).

a(83) corrected by Jason Yuen, Oct 27 2024

A214845 Triangle read by rows: T(n,m) =(n/k)^(k-1) mod k, where k is the m-th divisor of n, 1 <= m <= tau(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 3, 2, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 3, 4, 1, 0, 1, 0, 0, 1, 1, 2, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 4, 3, 8, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 3, 1, 2, 1, 0, 1, 0, 1, 1, 1, 5, 3, 4, 1

Offset: 1

Gerasimov Sergey, Mar 08 2013

Row lengths are tau(n) = A000005(n).

The sequence of row sums starts: 0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 7, 1, 3, 3, 1, 1, 9, 1, 5, 3, 3, 1, 17, 1, 3, 1, 7, 1, 16, 1, 1, 3, 3, 3, 26, 1, 3, 3, 19, 1, 12, 1, 7, 18, 3, 1, 27, 1, 23...

			Triangle begins:
0;
0,1;
0,1;
0,0,1;
0,1;
0,1,1,1;
0,1;
0,0,0,1;
0,0,1;
0,1,1,1;
0,1;
0,0,1,3,2,1;
0,1;
0,1,1,1;
0,1,1,1;
0,0,0,0,1;
0,1;
0,1,0,3,4,1;

Cf. A000005, A027750, A055225, A087909, A213919.

A214845 := proc(n,m)
    sort(convert(numtheory[divisors](n),list)) ;
    k := op(m,%) ;
    modp((n/k)^(k-1),k) ;
end proc:
for n from 1 to 30 do
    for m from 1 to numtheory[tau](n) do
        printf("%d,",A214845(n,m)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Apr 17 2013

A359812 a(n) = Sum_{d|n} (-1)^(d-1) * d^(n/d-1).

Original entry on oeis.org

1, 0, 2, -2, 2, -1, 2, -12, 11, -11, 2, -27, 2, -57, 108, -200, 2, -40, 2, -653, 780, -1013, 2, -1177, 627, -4083, 6644, -11959, 2, 5043, 2, -49680, 59172, -65519, 18028, -26670, 2, -262125, 531612, -713423, 2, 515723, 2, -3144419, 5180382, -4194281, 2

Offset: 1

Seiichi Manyama, Jan 14 2023

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

Cf. A087909, A359811.

a[n_] := DivisorSum[n, (-1)^(#-1) * #^(n/# - 1) &]; Array[a, 50] (* Amiram Eldar, Aug 09 2023 *)

a(n) = sumdiv(n, d, (-1)^(d-1)*d^(n/d-1));

my(N=50, x='x+O('x^N)); Vec(-sum(k=1, N, (-x)^k/(1-k*x^k)))

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

If p is an odd prime, a(p) = 2.

A326121 Expansion of Sum_{k>=1} k^2 * x^(2k) / (1 - k x^k).

Original entry on oeis.org

0, 1, 1, 5, 1, 18, 1, 33, 28, 58, 1, 246, 1, 178, 369, 577, 1, 1539, 1, 2774, 2531, 2170, 1, 16706, 3126, 8362, 20413, 35366, 1, 116444, 1, 135425, 178479, 131362, 94933, 1110999, 1, 524650, 1596521, 2530946, 1, 7280892, 1, 8403734, 16364457, 8389138, 1, 78568322, 823544, 43420683

Offset: 1

Ilya Gutkovskiy, Sep 10 2019

nonn

Cf. A055225, A078308, A087909.

nmax = 50; CoefficientList[Series[Sum[k^2 x^(2 k)/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (n/#)^# &, # > 1 &], {n, 1, 50}]

a(n)={sumdiv(n, d, if(d > 1, (n/d)^d))} \\ Andrew Howroyd, Sep 10 2019

a(n) = Sum_{d|n, d>1} (n/d)^d = Sum_{d|n, d

a(p) = 1, where p is prime.

a(n) = A055225(n) - n.

Previous Showing 11-20 of 20 results.

cp's OEIS Frontend

A208239 Triangle read by rows: T(n,m) = n + k - n/k, where k is the m-th divisor of n; 1 <= m <= tau(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A309633 G.f.: x * Sum_{k>=1} x^k / (1 - a(k)*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A356543 a(n) = Sum_{d|n} (d!)^(n/d-1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A356661 a(n) = n! * Sum_{d|n} 1/d^(n/d - 1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A359134 a(n) = Sum_{d|n} (2*d)^(n/d - 1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A359811 a(n) = Sum_{d|n} 2^(d-1) * d^(n/d-1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A213919 Triangle read by rows: T(n,m) = (n/k)^(k-1), where k is the m-th divisor of n, 1 <= m <= tau(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A214845 Triangle read by rows: T(n,m) =(n/k)^(k-1) mod k, where k is the m-th divisor of n, 1 <= m <= tau(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

A326121 Expansion of Sum_{k>=1} k^2 * x^(2k) / (1 - k x^k).