cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A343544 a(n) = n * Sum_{d|n} binomial(d+2,3)/d.

Original entry on oeis.org

1, 6, 13, 32, 40, 94, 91, 184, 204, 320, 297, 612, 468, 770, 850, 1184, 986, 1752, 1349, 2280, 2114, 2662, 2323, 4184, 3125, 4264, 4266, 5740, 4524, 7660, 5487, 8352, 7546, 9180, 8470, 13212, 9176, 12654, 12194, 16640, 12382, 19628, 14233, 20724, 19590, 22034, 18471, 30416, 21462
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Links

Crossrefs

Cf. A000203, A038040, A059358, A309731, A343545, A343546, A343547.

Programs

  • Maple
    f:= n -> n/6*add((d+1)*(d+2),d=numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Apr 26 2021
  • Mathematica
    a[n_] := n * DivisorSum[n, Binomial[# + 2, 3]/# &]; Array[a, 50] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, binomial(d+2, 3)/d);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+2, 3)*x^k/(1-x^k)^2))

Formula

G.f.: Sum_{k>=1} k * x^k/(1 - x^k)^4 = Sum_{k>=1} binomial(k+2,3) * x^k/(1 - x^k)^2.

A343549 a(n) = n * Sum_{d|n} binomial(d+n-1,n)/d.

Original entry on oeis.org

1, 5, 13, 49, 131, 545, 1723, 6809, 24484, 94445, 352727, 1366273, 5200313, 20135939, 77571083, 301034537, 1166803127, 4540794476, 17672631919, 68943346009, 269129827042, 1052178506615, 4116715363823, 16124644677569, 63205303337656, 247964681424725, 973469783435197
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Links

Crossrefs

Cf. A000203, A038040, A309731, A343544, A343545, A343546, A343547, A343548.

Programs

  • Mathematica
    a[n_] := n * DivisorSum[n, Binomial[# + n - 1, n]/# &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, binomial(d+n-1, n)/d);

Formula

a(n) = [x^n] Sum_{k>=1} k * x^k/(1 - x^k)^(n+1).
a(n) = [x^n] Sum_{k>=1} binomial(k+n-1,n) * x^k/(1 - x^k)^2.
From Seiichi Manyama, Jun 14 2023: (Start)
a(n) = Sum_{d|n} binomial(d+n-1,d).
a(n) = [x^n] Sum_{k>=1} (1/(1 - x^k)^n - 1). (End)

A348980 a(n) = Sum_{d|n} d * A322582(n/d), where A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

Original entry on oeis.org

0, 1, 1, 5, 1, 9, 1, 17, 8, 13, 1, 37, 1, 17, 15, 49, 1, 51, 1, 57, 19, 25, 1, 117, 14, 29, 43, 77, 1, 105, 1, 129, 27, 37, 23, 191, 1, 41, 31, 185, 1, 141, 1, 117, 99, 49, 1, 325, 20, 117, 39, 137, 1, 237, 31, 253, 43, 61, 1, 405, 1, 65, 131, 321, 35, 213, 1, 177, 51, 209, 1, 579, 1, 77, 145, 197, 35, 249, 1, 521
Offset: 1

Views

Author

Antti Karttunen, Nov 08 2021

Keywords

Comments

Dirichlet convolution of A322582 with the identity function, A000027.

Links

Crossrefs

Cf. A000027, A003958, A038040, A322582, A348981 (Möbius transform), A348982, A348983, A349130.
Cf. also A347130, A349140.

Programs

  • Mathematica
    f[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, #*(n/# - s[n/#]) &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)
  • PARI
    A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));
A348980(n) = sumdiv(n,d,d*A322582(n/d));

Formula

a(n) = Sum_{d|n} d * A322582(n/d).
For all n >= 1, a(n) <= A347130(n) <= A349140(n).
a(n) = A038040(n) - A349130(n). - Antti Karttunen, Nov 14 2021

A349130 a(n) = Sum_{d|n} d * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1).

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 13, 15, 19, 27, 21, 35, 25, 39, 45, 31, 33, 57, 37, 63, 65, 63, 45, 75, 61, 75, 65, 91, 57, 135, 61, 63, 105, 99, 117, 133, 73, 111, 125, 135, 81, 195, 85, 147, 171, 135, 93, 155, 127, 183, 165, 175, 105, 195, 189, 195, 185, 171, 117, 315, 121, 183, 247, 127, 225, 315, 133, 231, 225, 351, 141
Offset: 1

Views

Author

Antti Karttunen, Nov 09 2021

Keywords

Comments

Dirichlet convolution of A003958 with the identity function, A000027.
Dirichlet convolution of sigma (A000203) with A003966.

Links

Crossrefs

Cf. A000203, A003958, A003966, A038040, A348980, A349129, A349131 (Möbius transform), A349132, A349133, A349170.

Programs

  • Mathematica
    f[p_, e_] := p^(e + 1) - (p - 1)^(e + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)
  • PARI
    A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A349130(n) = sumdiv(n,d,d*A003958(n/d));

Formula

a(n) = Sum_{d|n} d * A003958(n/d).
a(n) = Sum_{d|n} A349131(d).
a(n) = Sum_{d|n} A000203(d) * A003966(n/d).
a(n) = A038040(n) - A348980(n).
For all n >= 1, a(n) <= A349129(n) <= A349170(n).
Multiplicative with a(p^e) = p^(e+1) - (p-1)^(e+1). - Amiram Eldar, Nov 09 2021

A349140 a(n) = Sum_{d|n} d * A348507(n/d), where A348507(n) = A003959(n) - n, where A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

0, 1, 1, 7, 1, 11, 1, 33, 10, 15, 1, 61, 1, 19, 17, 131, 1, 77, 1, 89, 21, 27, 1, 263, 16, 31, 67, 117, 1, 145, 1, 473, 29, 39, 25, 379, 1, 43, 33, 395, 1, 189, 1, 173, 137, 51, 1, 997, 22, 155, 41, 201, 1, 443, 33, 527, 45, 63, 1, 743, 1, 67, 177, 1611, 37, 277, 1, 257, 53, 265, 1, 1541, 1, 79, 187, 285, 37, 321
Offset: 1

Views

Author

Antti Karttunen, Nov 08 2021

Keywords

Comments

Dirichlet convolution of A348507 with the identity function, A000027.
Dirichlet convolution of sigma with A348971.

Links

Crossrefs

Cf. A000027, A000203, A003959, A038040, A348507, A348971, A349141 (Möbius transform), A349142, A349143, A349170.
Cf. also A347130, A348980.

Programs

  • Mathematica
    f[p_, e_] := (p + 1)^e; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; a[n_] := DivisorSum[n, #*s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)
  • PARI
    A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349140(n) = sumdiv(n,d,d*A348507(n/d));

Formula

a(n) = Sum_{d|n} d * A348507(n/d).
a(n) = Sum_{d|n} A000203(d) * A348971(n/d).
a(n) = Sum_{d|n} A349141(d).
For all n >= 1, a(n) >= A347130(n) >= A348980(n).
a(n) = A349170(n) - A038040(n). - Antti Karttunen, Nov 15 2021

A349170 a(n) = Sum_{d|n} d * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

1, 5, 7, 19, 11, 35, 15, 65, 37, 55, 23, 133, 27, 75, 77, 211, 35, 185, 39, 209, 105, 115, 47, 455, 91, 135, 175, 285, 59, 385, 63, 665, 161, 175, 165, 703, 75, 195, 189, 715, 83, 525, 87, 437, 407, 235, 95, 1477, 169, 455, 245, 513, 107, 875, 253, 975, 273, 295, 119, 1463, 123, 315, 555, 2059, 297, 805, 135, 665
Offset: 1

Views

Author

Antti Karttunen, Nov 09 2021

Keywords

Comments

Dirichlet convolution of A003959 with the identity function, A000027.
Dirichlet convolution of sigma (A000203) with A003968.

Links

Crossrefs

Cf. A000027, A000203, A003959, A003968, A038040, A349129, A349130, A349140, A349171 (Möbius transform), A349172, A349173.

Programs

  • Mathematica
    f[p_, e_] := (p + 1)^(e + 1) - p^(e + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)
  • PARI
    A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A349170(n) = sumdiv(n,d,d*A003959(n/d));

Formula

a(n) = Sum_{d|n} d * A003959(n/d).
a(n) = Sum_{d|n} A349171(d).
a(n) = Sum_{d|n} A000203(d) * A003968(n/d).
a(n) = A038040(n) + A349140(n).
For all n >= 1, a(n) >= A349129(n) >= A349130(n).
Multiplicative with a(p^e) = (p+1)^(e+1) - p^(e+1). - Amiram Eldar, Nov 09 2021

A036438 Integers which can be written as m*tau(m) for some m, where tau = A000005.

Original entry on oeis.org

1, 4, 6, 10, 12, 14, 22, 24, 26, 27, 32, 34, 38, 40, 46, 56, 58, 60, 62, 72, 74, 75, 80, 82, 84, 86, 88, 94, 104, 106, 108, 118, 120, 122, 132, 134, 136, 140, 142, 146, 147, 152, 156, 158, 166, 168, 178, 184, 192, 194, 202, 204, 206, 214, 218, 220, 226, 228, 232
Offset: 1

Views

Author

Simon Colton (simonco(AT)cs.york.ac.uk)

Keywords

Comments

Invented by the HR concept formation program.

Examples

			10 = 5 * tau(5).

Links

Crossrefs

Cf. A000005, A001747, A036434.
Range of A038040.

Programs

  • Mathematica
    q[k_] := AnyTrue[Divisors[k], # * DivisorSigma[0, #] == k &]; Select[Range[250], q] (* Amiram Eldar, Feb 01 2025 *)
  • PARI
    isok(n) = {for (k=1, n, if (k*numdiv(k) == n, return (1));); return (0);} \\ Michel Marcus, Dec 09 2014
  • PARI
    up_to = 65536;
A036438list(up_to) = { my(v=vector(up_to), m = Map()); for(n=1,#v,mapput(m,n*numdiv(n),n)); my(k=0,u=0); while((k<#v)&&(u<#v), u++; if(mapisdefined(m,u), k++; v[k] = u)); vector(k,i,v[i]); };
v036438 = A036438list(up_to);
A036438(n) = v036438[n]; \\ Antti Karttunen, Jul 18 2020

A143520 a(n) is n times number of divisors of n if n is odd, zero if n is twice odd, n times number of divisors of n/4 if n is divisible by 4.

Original entry on oeis.org

1, 0, 6, 4, 10, 0, 14, 16, 27, 0, 22, 24, 26, 0, 60, 48, 34, 0, 38, 40, 84, 0, 46, 96, 75, 0, 108, 56, 58, 0, 62, 128, 132, 0, 140, 108, 74, 0, 156, 160, 82, 0, 86, 88, 270, 0, 94, 288, 147, 0, 204, 104, 106, 0, 220, 224, 228, 0, 118, 240, 122, 0, 378, 320, 260, 0, 134, 136
Offset: 1

Views

Author

Michael Somos, Aug 22 2008

Keywords

Examples

			q + 6*q^3 + 4*q^4 + 10*q^5 + 14*q^7 + 16*q^8 + 27*q^9 + 22*q^11 + 24*q^12 + ...

Links

Crossrefs

Cf. A001620, A027748, A038040, A112329, A124010.

Programs

  • Haskell
    a143520 n = product $ zipWith (\p e -> (e + 2 * mod p 2 - 1) * p ^ e)
                              (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jan 21 2014
  • Mathematica
    Abs@Total[# (-1)^Divisors[#]] & /@ Range[68] (* George Beck, Oct 25 2014 *)
f[p_, e_] := (e + 1)*p^e; f[2, e_] := (e - 1)*2^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 29 2022 *)
  • PARI
    {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; (e - (-1)^p) * p^e)))}
  • PARI
    {a(n) = if( n<1, 0, polcoeff( sum(k=1, n, k * x^k / (1 - (-x)^k)^2, x*O(x^n)), n))}

Formula

a(n) is multiplicative with a(2^e) = (e-1) * 2^e if e>0, a(p^e) = (e+1) * p^e if p>2.
a(4*n + 2) = 0.
G.f.: Sum_{k>0} k * x^k / (1 - (-x)^k)^2.
A038040(2*n + 1) = a(2*n + 1); 4 * A038040(n) = a(4*n).
From Amiram Eldar, Nov 29 2022: (Start)
a(n) = n * A112329(n).
Dirichlet g.f.: zeta(s-1)^2*(1+2^(3-2*s)-2^(2-s)).
Sum_{k=1..n} a(k) ~ n^2*log(n)/4 + (4*gamma-1)*n^2/8, where gamma is Euler's constant (A001620). (End)

A304409 If n = Product (p_j^k_j) then a(n) = Product (p_j*(k_j + 1)).

Original entry on oeis.org

1, 4, 6, 6, 10, 24, 14, 8, 9, 40, 22, 36, 26, 56, 60, 10, 34, 36, 38, 60, 84, 88, 46, 48, 15, 104, 12, 84, 58, 240, 62, 12, 132, 136, 140, 54, 74, 152, 156, 80, 82, 336, 86, 132, 90, 184, 94, 60, 21, 60, 204, 156, 106, 48, 220, 112, 228, 232, 118, 360, 122, 248, 126, 14, 260
Offset: 1

Views

Author

Ilya Gutkovskiy, May 12 2018

Keywords

Examples

			a(12) = a(2^2*3) = 2*(2 + 1) * 3*(1 + 1) = 36.

Links

Crossrefs

Cf. A000005, A000026, A000040, A001221, A005117, A007947, A016754 (numbers n such that a(n) is odd), A034444, A038040, A064549, A299822, A304407, A304408, A304410 (fixed points), A304411, A304412.

Programs

  • Mathematica
    a[n_] := Times @@ (#[[1]] (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 65}]
Table[DivisorSigma[0, n] Last[Select[Divisors[n], SquareFreeQ]], {n, 65}]
  • PARI
    a(n)={numdiv(n)*factorback(factorint(n)[, 1])} \\ Andrew Howroyd, Jul 24 2018

Formula

a(n) = A000005(n)*A007947(n).
a(p^k) = p*(k + 1) where p is a prime and k > 0.
a(n) = 2^omega(n)*n if n is a squarefree (A005117), where omega() = A001221.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 2/p^(s-1) - 2/p^s - 1/p^(2*s-1) + 1/p^(2*s)). - Amiram Eldar, Sep 17 2023
From Vaclav Kotesovec, Jun 06 2025: (Start)
Let f(s) = Product_{p prime} (1 - 1/p^(2*s-1) + 2/p^(s-1) + 1/p^(2*s) - 2/p^s) * ((p^s - p)/(p^s - 1))^2.
Dirichlet g.f.: zeta(s-1)^2 * f(s).
Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where
f(2) = Product_{p prime} (1 - (3*p^2 + p - 1)/(p^2 * (p+1)^2)) = 0.40693068229776748114138817391056656864938379...,
f'(2) = f(2) * Sum_{p prime} 2*(3*p^4-3*p^2+1) * log(p) / ((p-1)*(p+1)*(p^4+2*p^3-2*p^2-p+1)) = f(2) * 2.2612432627709318567813765271568350301741329636853...
and gamma is the Euler-Mascheroni constant A001620. (End)

A328260 a(n) = n * omega(n).

Original entry on oeis.org

0, 2, 3, 4, 5, 12, 7, 8, 9, 20, 11, 24, 13, 28, 30, 16, 17, 36, 19, 40, 42, 44, 23, 48, 25, 52, 27, 56, 29, 90, 31, 32, 66, 68, 70, 72, 37, 76, 78, 80, 41, 126, 43, 88, 90, 92, 47, 96, 49, 100, 102, 104, 53, 108, 110, 112, 114, 116, 59, 180, 61, 124, 126, 64, 130, 198, 67, 136, 138, 210
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 09 2019

Keywords

Links

Crossrefs

Cf. A001221, A001222, A007947, A038040, A061397, A066959, A069359, A246655 (fixed points), A293548, A298473.

Programs

  • Magma
    [0] cat [n*(#PrimeDivisors(n)):n in [2..70]]; // Marius A. Burtea, Oct 10 2019
  • Mathematica
    Table[n PrimeNu[n], {n, 1, 70}]
nmax = 70; CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 - x^Prime[k])^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n)=n*omega(n) \\ Charles R Greathouse IV, Mar 16 2022

Formula

G.f.: Sum_{k>=1} prime(k) * x^prime(k) / (1 - x^prime(k))^2.
a(n) = bigomega(rad(n)^n).
a(n) = Sum_{d|n} A061397(n/d) * d.
Define f(x) = #{n <= x: a(n) <= x}. Gabdullin & Iudelevich show that f(x) ~ x/log log x. - Charles R Greathouse IV, Mar 16 2022
Previous Showing 41-50 of 114 results. Next

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