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 1-8 of 8 results.

A054862 (+2)-unitary sigma 3-multiple perfect number: numbers k such that (+2)usigma(k) = 3*k, where (+2)usigma(k) = A107759(k).

Original entry on oeis.org

105, 45045, 237405, 101846745, 121486365, 274680671265
Offset: 1

Views

Author

Yasutoshi Kohmoto

Keywords

Examples

			(+2)usigma(12) = (2+4)*(2+3) = 30.
Factorizations: 3*5*7, 3^2*5*7*11*13, 3*5*7^2*17*19, 3^2*5*7^2*11*13*17*19, 3^3*5*7*11*13*29*31, 3^3*5*7^2*11*13*17*19*29*31.

Crossrefs

Cf. A052396, A107759.

Programs

  • Mathematica
    s[n_] := Times @@ (2 + Power @@@ FactorInteger[n]); s[1] = 1; Select[Range[2.5*10^5], s[#] == 3*# &] (* Amiram Eldar, Aug 26 2022 *)

Extensions

Offset corrected by Amiram Eldar, Aug 26 2022

A051378 Sum of (1+e)-divisors of n. Let n = Product_i p(i)^r(i) then (1+e)-sigma(n) = Product_i (1 + Sum_{s|r(i)} p(i)^s).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 11, 13, 18, 12, 28, 14, 24, 24, 23, 18, 39, 20, 42, 32, 36, 24, 44, 31, 42, 31, 56, 30, 72, 32, 35, 48, 54, 48, 91, 38, 60, 56, 66, 42, 96, 44, 84, 78, 72, 48, 92, 57, 93, 72, 98, 54, 93, 72, 88, 80, 90, 60, 168, 62, 96, 104, 79, 84, 144, 68, 126, 96
Offset: 1

Views

Author

Yasutoshi Kohmoto

Keywords

Links

Crossrefs

Cf. A051377, A049599.
Cf. A027748, A124010, A027750, A069915, A107758, A107759.

Programs

  • Haskell
    a051378 n = product $ zipWith sum_1e (a027748_row n) (a124010_row n)
   where sum_1e p e = 1 + sum [p ^ d | d <- a027750_row e]
-- Reinhard Zumkeller, Mar 13 2012
  • Maple
    A051378 := proc(n)
    local a,d,p,e,sp;
    a := 1;
    for d in ifactors(n)[2] do
        p := op(1,d) ;
        e := op(2,d) ;
        sp := 1;
        for s in numtheory[divisors](e) do
            sp := sp+p^s ;
        end do:
        a := a*sp ;
    end do:
    a;
end proc: # R. J. Mathar, Oct 26 2015
  • Mathematica
    a[1] = 1; a[p_?PrimeQ] = p+1; a[n_] := Times @@ (1 + Sum[First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, May 04 2012 *)
  • PARI
    a(n)=my(f=factor(n));prod(i=1,#f[,1],sumdiv(f[i,2],d,f[i,1]^d)+1) \\ Charles R Greathouse IV, Nov 22 2011

Formula

Multiplicative with a(p^e) = 1 + Sum_{d|e} p^d. - Vladeta Jovovic, Apr 23 2002
a(n) = Sum_{d|n, gcd(d, n/d) = 1} A051377(d). - Daniel Suteu, Nov 01 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 + (1-1/p)*Sum_{k>=2} p^k/(p^(2*k)-1)) = 0.76636964336546210751... . - Amiram Eldar, Oct 31 2023

Extensions

Corrected and extended by Naohiro Nomoto, Apr 12 2001

A343443 If n = Product (p_j^k_j) then a(n) = Product (k_j + 2), with a(1) = 1.

Original entry on oeis.org

1, 3, 3, 4, 3, 9, 3, 5, 4, 9, 3, 12, 3, 9, 9, 6, 3, 12, 3, 12, 9, 9, 3, 15, 4, 9, 5, 12, 3, 27, 3, 7, 9, 9, 9, 16, 3, 9, 9, 15, 3, 27, 3, 12, 12, 9, 3, 18, 4, 12, 9, 12, 3, 15, 9, 15, 9, 9, 3, 36, 3, 9, 12, 8, 9, 27, 3, 12, 9, 27, 3, 20, 3, 9, 12, 12, 9, 27, 3, 18
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 15 2021

Keywords

Comments

Inverse Moebius transform of A056671.
a(n) depends only on the prime signature of n (see formulas). - Bernard Schott, May 03 2021

Links

Crossrefs

Cf. A000005, A001221, A005361, A007425, A025847, A034444, A056671, A064549, A074816, A107758, A107759, A348018, A363194.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Times @@ ((#[[2]] + 2) & /@ FactorInteger[n]); Table[a[n], {n, 80}]
a[n_] := Sum[If[GCD[d, n/d] == 1, DivisorSigma[0, d], 0], {d, Divisors[n]}]; Table[a[n], {n, 80}]
  • PARI
    a(n) = sumdiv(n, d, if(gcd(d, n/d)==1, numdiv(d))) \\ Andrew Howroyd, Apr 15 2021
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 + X - X^2)/(1-X)^2)[n], ", ")) \\ Vaclav Kotesovec, Feb 11 2023
  • Python
    from math import prod
from sympy import factorint
def A343443(n): return prod(e+2 for e in factorint(n).values()) # Chai Wah Wu, Feb 21 2025

Formula

a(n) = 2^omega(n) * tau_3(n) / tau(n), where omega = A001221, tau = A000005 and tau_3 = A007425.
a(n) = Sum_{d|n, gcd(d, n/d) = 1} tau(d).
From Bernard Schott, May 03 2021: (Start)
a(p^k) = k+2 for p prime, or signature [k].
a(A006881(n)) = 9 for signature [1, 1].
a(A054753(n)) = 12 for signature [2, 1].
a(A065036(n)) = 15 for signature [3, 1].
a(A085986(n)) = 16 for signature [2, 2].
a(A178739(n)) = 18 for signature [4, 1].
a(A143610(n)) = 20 for signature [3, 2].
a(A007304(n)) = 27 for signature [1, 1, 1]. (End)
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 1/p^s - 1/p^(2*s)). - Vaclav Kotesovec, Feb 11 2023
From Amiram Eldar, Sep 01 2023: (Start)
a(n) = A000005(A064549(n)).
a(n) = A363194(A348018(n)). (End)

A107758 (+2)Sigma(n): If n = Product p_i^r_i then a(n) = Product (2 + Sum p_i^s_i, s_i=1 to r_i) = Product (1 + (p_i^(r_i+1)-1)/(p_i-1)), a(1) = 1.

Original entry on oeis.org

1, 4, 5, 8, 7, 20, 9, 16, 14, 28, 13, 40, 15, 36, 35, 32, 19, 56, 21, 56, 45, 52, 25, 80, 32, 60, 41, 72, 31, 140, 33, 64, 65, 76, 63, 112, 39, 84, 75, 112, 43, 180, 45, 104, 98, 100, 49, 160, 58, 128, 95, 120, 55, 164
Offset: 1

Views

Author

Yasutoshi Kohmoto, May 25 2005

Keywords

Examples

			a(6) = (2+2)*(2+3) = 20.

Links

Crossrefs

Cf. A000203, A052396, A072691, A107759.

Programs

  • Maple
    A107758 := proc(n) local pf,p ; if n = 1 then 1; else pf := ifactors(n)[2] ; mul( 1+(op(1,p)^(op(2,p)+1)-1)/(op(1,p)-1), p=pf) ; end if; end proc:
seq(A107758(n),n=1..60) ; # R. J. Mathar, Jan 07 2011
  • Mathematica
    Table[DivisorSum[n, DivisorSigma[1, #] &, CoprimeQ[n/#, #] &], {n, 54}] (* Michael De Vlieger, Jun 27 2018 *)
f[p_, e_] := 1 + (p^(e + 1) - 1)/(p - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Aug 26 2022 *)
  • PARI
    a(n) = sumdiv(n, d, if(gcd(n/d, d) == 1, sigma(d))); \\ Daniel Suteu, Jun 27 2018

Formula

a(n) = Sum_{d|n, gcd(n/d, d) = 1} sigma(d), where sigma(d) is the sum of the divisors of d. - Daniel Suteu, Jun 27 2018
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 1.0741158... . - Amiram Eldar, Nov 01 2022

A343525 If n = Product (p_j^k_j) then a(n) = Product (2*p_j^k_j + 1), with a(1) = 1.

Original entry on oeis.org

1, 5, 7, 9, 11, 35, 15, 17, 19, 55, 23, 63, 27, 75, 77, 33, 35, 95, 39, 99, 105, 115, 47, 119, 51, 135, 55, 135, 59, 385, 63, 65, 161, 175, 165, 171, 75, 195, 189, 187, 83, 525, 87, 207, 209, 235, 95, 231, 99, 255, 245, 243, 107, 275, 253, 255, 273, 295, 119, 693, 123, 315, 285, 129, 297, 805
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 18 2021

Keywords

Comments

The unitary analog of A060640.

Links

Crossrefs

Cf. A001221, A034444, A034448, A060640, A107759, A145388, A298473.

Programs

  • Maple
    a:= n-> mul(2*i[1]^i[2]+1, i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Apr 18 2021
  • Mathematica
    a[1] = 1; a[n_] := Times @@ ((2 #[[1]]^#[[2]] + 1) & /@ FactorInteger[n]); Table[a[n], {n, 66}]
  • PARI
    a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = 2*f[k,1]^f[k,2]+1; f[k,2]=1); factorback(f); \\ Michel Marcus, Apr 18 2021

Formula

a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * usigma(n/d).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * 2^omega(d).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-1) - 2/p^(2*s-1)). - Amiram Eldar, Jul 24 2024

A343442 If n = Product (p_j^k_j) then a(n) = Product (p_j + 2), with a(1) = 1.

Original entry on oeis.org

1, 4, 5, 4, 7, 20, 9, 4, 5, 28, 13, 20, 15, 36, 35, 4, 19, 20, 21, 28, 45, 52, 25, 20, 7, 60, 5, 36, 31, 140, 33, 4, 65, 76, 63, 20, 39, 84, 75, 28, 43, 180, 45, 52, 35, 100, 49, 20, 9, 28, 95, 60, 55, 20, 91, 36, 105, 124, 61, 140, 63, 132, 45, 4, 105, 260, 69, 76, 125, 252
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 15 2021

Keywords

Links

Crossrefs

Cf. A000203, A001615, A007429, A007947, A008683, A048250, A062953, A063441, A074816, A092261, A107758, A107759, A319132, A335005.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Times @@ ((#[[1]] + 2) & /@ FactorInteger[n]); Table[a[n], {n, 70}]
nmax = 70; CoefficientList[Series[Sum[MoebiusMu[k]^2 DivisorSigma[1, k] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sumdiv(n, d, moebius(d)^2 * sigma(d)) \\ Andrew Howroyd, Apr 15 2021

Formula

G.f.: Sum_{k>=1} mu(k)^2 * sigma(k) * x^k / (1 - x^k), where mu = A008683 and sigma = A000203.
a(n) = Sum_{d|n} mu(d)^2 * sigma(d).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/(12*zeta(3)) = 0.684216... (A335005). - Amiram Eldar, Nov 13 2022
a(n) = Sum_{d|n} mu(d)^2*psi(d), where psi is A001615. - Ridouane Oudra, Jul 24 2025

A343569 If n = Product (p_j^k_j) then a(n) = Product (2*(p_j^k_j + 1)), with a(1) = 1.

Original entry on oeis.org

1, 6, 8, 10, 12, 48, 16, 18, 20, 72, 24, 80, 28, 96, 96, 34, 36, 120, 40, 120, 128, 144, 48, 144, 52, 168, 56, 160, 60, 576, 64, 66, 192, 216, 192, 200, 76, 240, 224, 216, 84, 768, 88, 240, 240, 288, 96, 272, 100, 312, 288, 280, 108, 336, 288, 288, 320, 360, 120, 960, 124, 384, 320, 130
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 20 2021

Keywords

Crossrefs

Cf. A001221, A034444, A034448, A034761, A064840, A107759, A333557, A343525.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Times @@ (2 (#[[1]]^#[[2]] + 1) & /@ FactorInteger[n]); Table[a[n], {n, 64}]
  • PARI
    a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = 2*f[k,1]^f[k,2] + 2; f[k,2] = 1); factorback(f); \\ Michel Marcus, Apr 20 2021

Formula

a(n) = usigma(n) * 2^omega(n).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} usigma(d) * usigma(n/d).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * 2^omega(d) * 2^omega(n/d).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} A343525(d).

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
Showing 1-8 of 8 results.

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