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 1-4 of 4 results.

A082055 Product of common prime-divisors (without multiplicity) of sigma(n) and phi(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 6, 2, 1, 2, 3, 2, 2, 2, 2, 2, 2, 1, 6, 2, 2, 2, 2, 2, 1, 2, 2, 6, 1, 2, 6, 2, 2, 2, 6, 2, 2, 6, 2, 2, 2, 3, 1, 2, 2, 2, 6, 2, 6, 2, 2, 2, 2, 2, 6, 2, 1, 6, 2, 2, 2, 2, 6, 2, 3, 2, 6, 2, 2, 6, 6, 2, 2, 1, 2, 2, 2, 2, 6, 2, 10, 2, 6, 2, 2, 2, 2, 6, 2, 2, 3, 6, 1, 2, 2, 2, 6, 6
Offset: 1

Views

Author

Labos Elemer, Apr 03 2003

Keywords

Comments

The squarefree kernel of the greatest common divisor of sigma(n) and phi(n). - Antti Karttunen, Jan 22 2020

Links

Crossrefs

Cf. A000203, A000010, A007947, A009223, A081396, A082054.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] Table[Apply[Times, Intersection[ba[EulerPhi[w]], ba[DivisorSigma[1, w]]]], {w, 1, 256}]
  • PARI
    A082055(n) = factorback(factorint(gcd(sigma(n), eulerphi(n)))[, 1]); \\ Antti Karttunen, Jan 22 2020

Formula

a(n) = A007947(A009223(n)). - Antti Karttunen, Jan 22 2020

A081383 Least x = a(n) such that number of common prime factors (ignoring multiplicity) of sigma(x) = A000203(x) and phi(x) = A000010(x) equals n.

Original entry on oeis.org

3, 14, 209, 3596, 41624, 2003639, 24206049, 2562857198, 57721363052
Offset: 1

Views

Author

Labos Elemer, Mar 28 2003

Keywords

Comments

a(10) <= 6804704928496. - Donovan Johnson, Jun 15 2013

Examples

			x = 209: sigma(209) = 240 = 2^4*3*5, phi(209) = 180 = 2^2*3^2*5, common factor set = {2,3,5}, so a(3) = 209.

Crossrefs

Cf. A000203, A000010, A081396.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] t=Table[0, {10}]; Do[s=Length[Intersection[ba[EulerPhi[n]], ba[DivisorSigma[1, n]]]]; If[s<11&&t[[s]]==0, t[[s]]=n], {n, 1, 1000000}]; t
  • PARI
    a(n)=my(k=prod(i=1,n,prime(i))); while(omega(gcd(sigma(k),eulerphi(k)))!=n, k++); k \\ Charles R Greathouse IV, Feb 14 2013

Formula

a(n) = min{x: A081396(x) = n}.

Extensions

a(6)-a(8) from Donovan Johnson, May 24 2009
a(9) from Donovan Johnson, Jun 14 2013

A081397 a(n) = least k such that GCD of sigma(k) = A000203(k) and phi(k) = A000010(k) equals n-th primorial = A002110(n).

Original entry on oeis.org

3, 14, 488, 16072, 284229, 29221621, 25014312576
Offset: 1

Views

Author

Labos Elemer, Mar 28 2003

Keywords

Comments

a(8) <= 1524139588224. - Donovan Johnson, Dec 14 2009

Examples

			k=284229: sigma(284229) = 482790 = 2*3*5*7*11*19, phi(284229) = 166320 = 2*2*2*3*3*3*3*5*7*11, common factor set = {2,3,5,7,11}, gcd(sigma,phi) = 2310.

Crossrefs

Cf. A000010, A000203, A002110, A081383, A081396.

Formula

a(n) = Min_{k : A081396(k) = A002110(n)}.

Extensions

a(6) corrected and a(7) from Donovan Johnson, Dec 14 2009

A081398 Numbers k for which the number of common prime factors of sigma(k) and phi(k) is exactly six (ignoring multiplicity).

Original entry on oeis.org

2003639, 2179316, 2180057, 2382974, 2689561, 2720567, 2761873, 2933675, 3145572, 3435381, 3925463, 4007278, 4137111, 4212692, 4360114, 4947971, 5172881, 5379122, 5441134, 5458673, 5523746, 5675816, 5748831, 5867350, 5957435, 6010917, 6537948, 6540171, 6561511
Offset: 1

Views

Author

Labos Elemer, Mar 28 2003

Keywords

Comments

Numbers k such that A081396(k) = 6. - Amiram Eldar, Mar 25 2024

Examples

			k = 400: sigma(400) = 6846840 = 2*2*2*3*3*5*7*11*13*19, phi(400) = 1755600 = 2*2*2*2*3*5*5*7*11*19, the common prime set = {2,3,5,7,11,19} with six primes.

Links

Crossrefs

Cf. A000010, A000203, A001221, A009223, A002110, A081383, A081396, A081397.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[ffi[x][[2*w - 1]], {w, 1, lf[x]}] ; Do[s = Length[Intersection[ba[EulerPhi[n]], ba[DivisorSigma[1, n]]]]; If[Greater[s, 5], Print[{n, s}]], {n, 1, 10000000}]
  • PARI
    is(n) = {my(f = factor(n)); omega(gcd(sigma(f), eulerphi(f))) == 6;} \\ Amiram Eldar, Mar 25 2024

Extensions

3925463 inserted and more terms added by Amiram Eldar, Mar 25 2024
Showing 1-4 of 4 results.

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