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.

Showing 1-6 of 6 results.

A007429 Inverse Moebius transform applied twice to natural numbers.

Original entry on oeis.org

1, 4, 5, 11, 7, 20, 9, 26, 18, 28, 13, 55, 15, 36, 35, 57, 19, 72, 21, 77, 45, 52, 25, 130, 38, 60, 58, 99, 31, 140, 33, 120, 65, 76, 63, 198, 39, 84, 75, 182, 43, 180, 45, 143, 126, 100, 49, 285, 66, 152, 95, 165, 55, 232, 91, 234, 105, 124, 61, 385, 63, 132, 162, 247, 105, 260
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Sum of the divisors d1 from the ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - Wesley Ivan Hurt, Mar 22 2022
a(n) is the sum of the sum-of-divisors of the divisors of n. - M. F. Hasler, Mar 29 2024

References

  • David M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston, MA, 1976, p. 120.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000203, A007430, A134699, A152649, A280077.

Programs

  • Magma
    [&+[SumOfDivisors(d): d in Divisors(n)]: n in [1..100]] // Jaroslav Krizek, Sep 24 2016
  • Maple
    A007429 := proc(n)
    add(numtheory[sigma](d),d=numtheory[divisors](n)) ;
end proc:
seq(A007429(n),n=1..100) ; # R. J. Mathar, Aug 28 2015
  • Mathematica
    f[n_] := Plus @@ DivisorSigma[1, Divisors@n]; Array[f, 52] (* Robert G. Wilson v, May 05 2010 *)
f[p_, e_] := (p*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 10 2022 *)
  • PARI
    A007429_upto(N)=vector(N,n, sumdiv(n,d, sigma(d))) \\ edited by M. F. Hasler, Mar 29 2024
  • PARI
    a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)^2/(1-p*X))[n]) \\ Ralf Stephan
  • PARI
    N=17; default(seriesprecision,N); x=z+O(z^(N+1))
c=sum(j=1,N,j*x^j);
t=1/prod(j=1,N, eta(x^(j))^(1/j))
t=log(t)
t=serconvol(t,c)
Vec(t)
/* Joerg Arndt, May 03 2008 */
  • PARI
    a(n)=sumdiv(n,d, sumdiv(d,t, t ) );  /* Joerg Arndt, Oct 07 2012 */
  • Python
    from math import prod
from sympy import factorint
def A007429(n): return prod((p*(p**(e+1)-1)-(p-1)*(e+1))//(p-1)**2 for p,e in factorint(n).items()) # Chai Wah Wu, Mar 28 2024
  • Sage
    def A(n): return sum(sigma(d) for d in n.divisors()) # Travis Scholl, Apr 14 2016

Formula

a(n) = Sum_{d|n} sigma(d), Dirichlet convolution of A000203 and A000012. - Jason Earls, Jul 07 2001
a(n) = Sum_{d|n} d*tau(n/d). - Vladeta Jovovic, Jul 31 2002
Multiplicative with a(p^e) = (p*(p^(e+1)-1)-(p-1)*(e+1))/(p-1)^2. - Vladeta Jovovic, Dec 25 2001
G.f.: Sum_{k>=1} sigma(k)*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Moebius transform of A007430. - Benoit Cloitre, Mar 03 2004
Dirichlet g.f.: zeta(s-1)*zeta^2(s).
Equals A051731^2 * [1, 2, 3, ...]. Equals row sums of triangle A134577. - Gary W. Adamson, Nov 02 2007
Row sums of triangle A134699. - Gary W. Adamson, Nov 06 2007
a(n) = n * (Sum_{d|n} tau(d)/d) = n * (A276736(n) / A276737(n)). - Jaroslav Krizek, Sep 24 2016
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(sigma(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 26 2018
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/72 = 1.352904... (A152649). - Amiram Eldar, Oct 22 2022

A318845 a(n) = Sum_{d|n} (-1)^(n/d+1) * Sum_{j|d} sigma(j), where sigma(j) = sum of divisors of j (A000203).

Original entry on oeis.org

1, 3, 6, 6, 8, 18, 10, 10, 24, 24, 14, 36, 16, 30, 48, 15, 20, 72, 22, 48, 60, 42, 26, 60, 46, 48, 82, 60, 32, 144, 34, 21, 84, 60, 80, 144, 40, 66, 96, 80, 44, 180, 46, 84, 192, 78, 50, 90, 76, 138, 120, 96, 56, 246, 112, 100, 132, 96, 62, 288, 64, 102, 240, 28, 128, 252, 70, 120, 156, 240
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 04 2018

Keywords

Links

Crossrefs

Cf. A000203, A007429, A007430, A288417, A318768.

Programs

  • Mathematica
    Table[Sum[(-1)^(n/d + 1) Sum[DivisorSigma[1, j], {j, Divisors[d]}], {d, Divisors[n]}], {n, 70}]
nmax = 70; Rest[CoefficientList[Series[Sum[DivisorSum[k, DivisorSigma[1, #] &] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 70; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^(DivisorSum[k, DivisorSigma[1, #] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
f[p_, e_] := (2*p^(e+3) - (e^2+5*e+6)*p^2 + (2*e^2+8*e+6)*p - e^2 - 3*e -2)/(2*(p-1)^3); f[2, e_] := (e+1)*(e+2)/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2025 *)
  • PARI
    a(n) = {my(f = factor(n), p , e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(p == 2, (e+1)*(e+2)/2, (2*p^(e+3) - (e^2+5*e+6)*p^2 + (2*e^2+8*e+6)*p - e^2 - 3*e -2)/(2*(p-1)^3)));} \\ Amiram Eldar, May 26 2025

Formula

G.f.: Sum_{k>=1} A007429(k)*x^k/(1 + x^k).
L.g.f.: log(Product_{k>=1} (1 + x^k)^(A007429(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
From Amiram Eldar, May 26 2025: (Start)
Multiplicative with a(2^e) = (e+1)*(e+2)/2, and a(p^e) = (2*p^(e+3) - (e^2+5*e+6)*p^2 + (2*e^2+8*e+6)*p - e^2 - 3*e -2)/(2*(p-1)^3) for an odd prime p.
Dirichlet g.f: zeta(s-1) * zeta(s)^2 * (1 - 1/2^(s-1)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^6/864 = 1.112718... . (End)

A341638 a(n) = Sum_{d|n} phi(d) * sigma(d) * tau(n/d).

Original entry on oeis.org

1, 5, 10, 23, 26, 50, 50, 101, 97, 130, 122, 230, 170, 250, 260, 427, 290, 485, 362, 598, 500, 610, 530, 1010, 671, 850, 904, 1150, 842, 1300, 962, 1761, 1220, 1450, 1300, 2231, 1370, 1810, 1700, 2626, 1682, 2500, 1850, 2806, 2522, 2650, 2210, 4270, 2493, 3355, 2900, 3910, 2810, 4520
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 16 2021

Keywords

Comments

Inverse Moebius transform of A062952.

Links

Crossrefs

Cf. A000005, A000010, A000203, A007430, A062354, A062952, A191831.
Cf. A002117, A013661, A330523.

Programs

  • Mathematica
    Table[Sum[EulerPhi[d] DivisorSigma[1, d] DivisorSigma[0, n/d], {d, Divisors[n]}], {n, 54}]
Table[Sum[DivisorSigma[0, GCD[n, k]] DivisorSigma[1, n/GCD[n, k]], {k, n}], {n, 54}]
f[p_, e_] := (p^(2*e + 4) - p^(e + 3) - 2*p^(e + 2) - p^(e + 1) + (e + 1)*p^3 - (e - 1)*p + 1)/(p^2 - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 26 2023 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)*numdiv(n/d)); \\ Michel Marcus, Feb 17 2021

Formula

a(n) = Sum_{k=1..n} tau(gcd(n,k)) * sigma(n/gcd(n,k)).
a(n) = Sum_{d|n} A062952(d).
a(n) = Sum_{k=1..n} tau(n/gcd(n,k))*sigma(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 09 2021
From Amiram Eldar, Jan 26 2023: (Start)
Multiplicative with a(p^e) = (p^(2*e+4) - p^(e+3) - 2*p^(e+2) - p^(e+1) + (e+1)*p^3 - (e-1)*p + 1)/(p^2-1)^2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)^2/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A013661 * A002117^2 * A330523 / 3 = 0.424578... . (End)

A343570 If n = Product (p_j^k_j) then a(n) = Product (p_j^k_j + 3), with a(1) = 1.

Original entry on oeis.org

1, 5, 6, 7, 8, 30, 10, 11, 12, 40, 14, 42, 16, 50, 48, 19, 20, 60, 22, 56, 60, 70, 26, 66, 28, 80, 30, 70, 32, 240, 34, 35, 84, 100, 80, 84, 40, 110, 96, 88, 44, 300, 46, 98, 96, 130, 50, 114, 52, 140, 120, 112, 56, 150, 112, 110, 132, 160, 62, 336, 64, 170, 120, 67, 128, 420, 70, 140, 156, 400, 74
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 20 2021

Keywords

Comments

The unitary analog of A007430.

Links

Crossrefs

Cf. A001221, A007430, A034444, A034448, A107759.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Times @@ ((#[[1]]^#[[2]] + 3) & /@ FactorInteger[n]); Table[a[n], {n, 71}]
  • PARI
    a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = f[k,1]^f[k,2] + 3; f[k,2] = 1); factorback(f); \\ Michel Marcus, Apr 20 2021

Formula

a(n) = Sum_{d|n, gcd(d, n/d) = 1} usigma(d) * 2^omega(n/d).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} A107759(d).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 2/p^2 - 3/p^3) = 1.1848008127... . - Amiram Eldar, Nov 13 2022

A134676 A127172 * A127648.

Original entry on oeis.org

1, 3, 2, 3, 0, 3, 6, 6, 0, 4, 3, 0, 0, 0, 5, 9, 6, 9, 0, 0, 6, 3, 0, 0, 0, 0, 0, 7, 10, 12, 0, 12, 0, 0, 0, 8, 6, 0, 9, 0, 0, 0, 0, 0, 9, 9, 6, 0, 0, 15, 0, 0, 0, 0, 10
Offset: 1

Views

Author

Gary W. Adamson, Nov 05 2007

Keywords

Comments

Left column = A007425: (1, 3, 3, 6, 3, 9, 3, 10, ...).
Row sums = A007430: (1, 5, 6, 16, 8, 30, 10, 42, ...).

Examples

			First few rows of the triangle:
  1;
  3, 2;
  3, 0, 3;
  6, 6, 0, 4;
  3, 0, 0, 0, 5;
  9, 6, 9, 0, 0, 6;
  3, 0, 0, 0, 0, 0, 7;
  ...

Crossrefs

Cf. A127172, A051731, A127648, A007430, A007425.

Formula

A127172 * A127648 = A051731^3 * A127648 as infinite lower triangular matrices.

A140704 A051731^3 * A000012.

Original entry on oeis.org

1, 4, 1, 4, 1, 1, 10, 4, 1, 1, 4, 1, 1, 1, 1, 16, 7, 4, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 20, 10, 4, 4, 1, 1, 1, 1, 10, 4, 4, 1, 1, 1, 1, 1, 1, 16, 7, 4, 4, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 40, 22, 13, 7, 4, 4, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Gary W. Adamson, May 24 2008

Keywords

Comments

Row sums = A007430: (1, 5, 6, 16, 8, 30, 10,...).
Left column = A007426: (1, 4, 4, 10, 4, 16, 4,...).

Examples

			First few rows of the triangle are:
1;
4, 1;
4, 1, 1;
10, 4, 1, 1;
4, 1, 1, 1, 1;
16, 7, 4, 1, 1, 1;
4, 1, 1, 1, 1, 1, 1;
20, 10, 4, 4, 1, 1, 1, 1;
10, 4, 4, 1, 1, 1, 1, 1, 1;
...

Crossrefs

Cf. A051731, A007430, A007426.

Formula

A051731^3 * A000012 as infinite lower triangular matrices, where A051731 = the inverse Mobius transform and A000012 an infinite lower triangular matrix with all 1's.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.