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.

A066086 Greatest common divisor of product (p-1) and product (p+1), where p ranges over distinct prime divisors of n; a(n) = gcd(A048250(n), A173557(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 6, 8, 1, 2, 2, 2, 2, 4, 2, 2, 2, 2, 6, 2, 6, 2, 8, 2, 1, 4, 2, 24, 2, 2, 6, 8, 2, 2, 12, 2, 2, 8, 2, 2, 2, 2, 2, 8, 6, 2, 2, 8, 6, 4, 2, 2, 8, 2, 6, 4, 1, 12, 4, 2, 2, 4, 24, 2, 2, 2, 6, 8, 6, 12, 24, 2, 2, 2, 2, 2, 12, 4, 6, 8, 2, 2, 8, 8, 2, 4, 2, 24, 2, 2, 6, 4
Offset: 1

Views

Author

Labos Elemer, Dec 04 2001

Keywords

Comments

Frequently equal, but not identical, to A009223 (i.e. GCD of sigma and phi of n).

Links

Crossrefs

Cf. A048250, A173557, A023900, A000203, A007947, A000010, A009223, A066087, A322359, A322360, A322362.

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]}] cor[x_] := Apply[Times, ba[x]] g1[x_] := GCD[DivisorSigma[1, x], EulerPhi[x]] g2[x_] := GCD[DivisorSigma[1, cor[x]], EulerPhi[cor[x]]] Table[g2[w], {w, 1, 128}]
a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, GCD[Times@@((#-1)& @@@ f), Times@@((#+1)& @@@ f)]]]; Array[a, 100] (* Amiram Eldar, Dec 05 2018 *)
  • PARI
    a(n)=my(f=factor(n)[,1]);gcd(prod(i=1,#f,f[i]+1),prod(i=1,#f,f[i]-1)) \\ Charles R Greathouse IV, Feb 14 2013

Formula

a(n) = gcd(A048250(n), A023900(n)) = gcd(A000203(A007947(n)), A000010(A007947(n))).
a(n) = A322360(n) / A322359(n). - Antti Karttunen, Dec 04 2018

Extensions

Name edited, part of the old name transferred to the formula section by Antti Karttunen, Dec 04 2018

A322360 Multiplicative with a(p^e) = p^2 - 1.

Original entry on oeis.org

1, 3, 8, 3, 24, 24, 48, 3, 8, 72, 120, 24, 168, 144, 192, 3, 288, 24, 360, 72, 384, 360, 528, 24, 24, 504, 8, 144, 840, 576, 960, 3, 960, 864, 1152, 24, 1368, 1080, 1344, 72, 1680, 1152, 1848, 360, 192, 1584, 2208, 24, 48, 72, 2304, 504, 2808, 24, 2880, 144, 2880, 2520, 3480, 576, 3720, 2880, 384, 3, 4032, 2880
Offset: 1

Views

Author

Antti Karttunen, Dec 04 2018

Keywords

Comments

Absolute values of A046970, the Dirichlet inverse of the Jordan function J_2 (A007434).
Absolute values of the Möbius transform of A055491. (See Benoit Cloitre's May 31 2002 comment in A046970).

Links

Crossrefs

Absolute values of A046970.
Cf. A048250, A173557, A066086, A322359, A351265.

Programs

  • Maple
    a:= n-> mul(i[1]^2-1, i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 05 2021
  • Mathematica
    a[n_] := If[n==1, 1, Times @@ ((#^2-1)& @@@ FactorInteger[n])]; Array[a, 50]  (* Amiram Eldar, Dec 05 2018 *)
  • PARI
    A322360(n) = factorback(apply(p -> (p*p)-1, factor(n)[, 1]));
  • PARI
    A322360(n) = abs(sumdiv(n,d,moebius(n/d)*(core(d)^2)));

Formula

Multiplicative with a(p^e) = p^2 - 1.
a(n) = Product_{p prime divides n} (p^2 - 1).
a(n) = abs(A046970(n)).
a(n) = A048250(n) * A173557(n) = A066086(n) * A322359(n).
G.f. for a signed version of the sequence: Sum_{n >= 1} mu(n)*n^2*x^n/(1 - x^n) = Sum_{n >= 1} (-1)^omega(n)*a(n)*x^n = x - 3*x^2 - 8*x^3 - 3*x^4 - 24*x^5 + 24*x^6 - 48*x^7 - 3*x^8 - 8*x^9 + 72*x^10 - ..., where mu(n) is the Möbius function A008683(n) and omega(n) = A001221(n) is the number of distinct primes dividing n. - Peter Bala, Mar 05 2022
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} (p-1)*(p^2 + 2*p + 2)/(p*(p^2 + p + 1)) = 0.187556464... . - Amiram Eldar, Oct 22 2022
a(n) = A007434(A007947(n)). - Enrique Pérez Herrero, Oct 14 2024

A322321 a(n) = lcm(A003557(n), A173557(n)).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 2, 12, 6, 8, 8, 16, 6, 18, 4, 12, 10, 22, 4, 20, 12, 18, 6, 28, 8, 30, 16, 20, 16, 24, 6, 36, 18, 24, 4, 40, 12, 42, 10, 24, 22, 46, 8, 42, 20, 32, 12, 52, 18, 40, 12, 36, 28, 58, 8, 60, 30, 12, 32, 48, 20, 66, 16, 44, 24, 70, 12, 72, 36, 40, 18, 60, 24, 78, 8, 54, 40, 82, 12, 64, 42, 56, 20
Offset: 1

Views

Author

Antti Karttunen, Dec 05 2018

Keywords

Links

Crossrefs

Cf. A000010, A003557, A173557, A322320, A322351, A322352.
Cf. also A322319, A322359.

Programs

  • Mathematica
    a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, LCM[ Times@@ (First[#] ^(Last[#]-1)& /@  f), Times@@((#-1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)
  • PARI
    A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A322321(n) = lcm(A003557(n), A173557(n));

Formula

a(n) = lcm(A003557(n), A173557(n)) = lcm(A322351(n), A322352(n)).
a(n) = A000010(n) / A322320(n).

A322319 a(n) = lcm(A003557(n), A048250(n)).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 12, 14, 24, 24, 24, 18, 12, 20, 18, 32, 36, 24, 12, 30, 42, 36, 24, 30, 72, 32, 48, 48, 54, 48, 12, 38, 60, 56, 36, 42, 96, 44, 36, 24, 72, 48, 24, 56, 90, 72, 42, 54, 36, 72, 24, 80, 90, 60, 72, 62, 96, 96, 96, 84, 144, 68, 54, 96, 144, 72, 12, 74, 114, 120, 60, 96, 168, 80, 72
Offset: 1

Views

Author

Antti Karttunen, Dec 05 2018

Keywords

Links

Crossrefs

Cf. A001615, A003557, A048250, A322318.
Cf. also A322321, A322359.

Programs

  • Mathematica
    a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, LCM[ Times@@ (First[#] ^(Last[#]-1)& /@  f), Times@@((#+1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)
  • PARI
    A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A048250(n) = factorback(apply(p -> p+1, factor(n)[, 1]));
A322319(n) = lcm(A048250(n), A003557(n));

Formula

a(n) = lcm(A003557(n), A048250(n)).
a(n) = A001615(n) / A322318(n).
Showing 1-4 of 4 results.

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