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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A324987 a(n) = Product_{d|n} (tau(d)*sigma(d)) where tau(k) = the number of divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 6, 8, 126, 12, 2304, 16, 7560, 312, 5184, 24, 8128512, 28, 9216, 9216, 1171800, 36, 21026304, 40, 27433728, 16384, 20736, 48, 234101145600, 1116, 28224, 49920, 65028096, 60, 110075314176, 64, 442940400, 36864, 46656, 36864, 60754075619328, 76, 57600, 50176
Offset: 1

Views

Author

Jaroslav Krizek, Mar 23 2019

Keywords

Comments

n divides a(n) for n: 1, 2, 6, 8, 12, 18, 24, 28, 36, 40, 48, 54, 56, 72, 80, 84, 96, 108, 112, 117, ...

Examples

			a(6) = (tau(1)*sigma(1)) * (tau(2)*sigma(2)) * (tau(3)*sigma(3)) * (tau(6)*sigma(6)) = (1*1) * (2*3) * (2*4) * (4*12) = 2304.

Crossrefs

Cf. A000005, A000203, A064840, A206032, A211776, A324986.

Programs

  • Magma
    [&*[NumberOfDivisors(d) * SumOfDivisors(d): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    Table[Product[DivisorSigma[0, k]*DivisorSigma[1, k], {k, Divisors[n]}], {n, 1, 50}] (* Vaclav Kotesovec, Mar 23 2019 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(k=1, #d, numdiv(d[k])*sigma(d[k])); \\ Michel Marcus, Mar 23 2019

Formula

a(n) = Product_{d|n} tau(d) * Product_{d|n} sigma(d) = A211776(n) * A206032(n).
a(p) = 2*(p + 1) for p = primes (A000040).
a(n) = Product_{d|n} A064840(d). - Antti Karttunen, Mar 28 2019

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).

A068354 Numbers n such that sigma(n)*tau(n) > prime(2*n) where sigma(n) is the sum of divisors of n and tau(n) the number of divisors of n (A000005).

Original entry on oeis.org

4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 128, 130, 132, 135, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 165, 168, 170
Offset: 1

Views

Author

Benoit Cloitre, Feb 28 2002

Keywords

Crossrefs

Cf. A000005, A000203, A031215, A064840.

Programs

  • Mathematica
    Select[Range[200],DivisorSigma[0,#]DivisorSigma[1,#]>Prime[2#]&] (* Harvey P. Dale, Aug 15 2011 *)
  • PARI
    for(n=1,1000, if(sigma(n)*numdiv(n)>prime(2*n),print1(n,",")))

A068578 Numbers k such that tau(k)*sigma(k) > k*phi(k).

Original entry on oeis.org

2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 30, 36, 40, 42, 48, 60, 72, 84, 90, 120, 180
Offset: 1

Views

Author

Benoit Cloitre, Mar 26 2002

Keywords

Comments

Numbers k such that A064840(k) > A002618(k).

Crossrefs

Cf. A000005, A000010, A000203, A002618, A064840.

Programs

  • Mathematica
    Select[Range[180], DivisorSigma[0, #] * DivisorSigma[1, #] > # * EulerPhi[#] &] (* Amiram Eldar, Jun 12 2022 *)
  • PARI
    is(k) = {my(f=factor(k)); numdiv(k)*sigma(k) > k*eulerphi(k); } \\ Jinyuan Wang, Apr 05 2020

A306655 Numbers n such that lcm(sigma(n), n) = tau(n) * sigma(n) where sigma(k) = the sum of the divisors of k (A000203) and tau(k) = the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 18, 468, 9360, 10880, 79360, 84480, 387072, 777216, 3801600, 7282688, 15037440, 17418240, 27067392, 65544192, 752903424, 1218032640, 4227842304, 4737761280, 6410638080, 11949932544, 19327057920, 26372530800, 37645171200, 224956569600, 243520929792, 876611248128
Offset: 1

Views

Author

Jaroslav Krizek, Mar 03 2019

Keywords

Comments

Numbers n such that A009242(n) = A000005(n) * A000203(n) = A064840(n).
Also numbers n such that A017666(n) = denominator(sigma(n)/n) = tau(n) = A000005(n).
a(29) > 10^12. - Giovanni Resta, Mar 04 2019

Examples

			18 is a term because lcm(sigma(18), 18) = lcm(39, 18) = 234 = tau(18) * sigma(18) = 6 * 39.

Crossrefs

Cf. A000005, A000203, A009242, A064840.
Cf. A069810 (gcd(sigma(n), n) = tau(n)).

Programs

  • Magma
    [n: n in [1..1000000] | LCM(SumOfDivisors(n), n) eq NumberOfDivisors(n)* SumOfDivisors(n)]
  • Mathematica
    Select[Range[1000000], LCM[DivisorSigma[1, #], #] == DivisorSigma[0, #] * DivisorSigma[1, #]&] (* Vaclav Kotesovec, Mar 04 2019 *)
  • PARI
    isok(n) = my(sn = sigma(n)); lcm(sn, n) == sn*numdiv(n); \\ Michel Marcus, Mar 04 2019

Extensions

a(13)-a(16) from Vaclav Kotesovec, Mar 04 2019
a(17) from Daniel Suteu, Mar 04 2019
a(18)-a(28) from Giovanni Resta, Mar 04 2019
Previous Showing 21-25 of 25 results.

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