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.

A303277 If n = Product (p_j^k_j) then a(n) = (Sum (k_j))^(Sum (p_j)).

Original entry on oeis.org

1, 1, 1, 4, 1, 32, 1, 9, 8, 128, 1, 243, 1, 512, 256, 16, 1, 243, 1, 2187, 1024, 8192, 1, 1024, 32, 32768, 27, 19683, 1, 59049, 1, 25, 16384, 524288, 4096, 1024, 1, 2097152, 65536, 16384, 1, 531441, 1, 1594323, 6561, 33554432, 1, 3125, 128, 2187, 1048576, 14348907, 1, 1024, 65536
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 20 2018

Keywords

Examples

			a(48) = a(2^4 * 3^1) = (4 + 1)^(2 + 3) = 5^5 = 3125.

Links

Crossrefs

Cf. A000142, A000312, A001222, A002110, A007504, A008472, A008474, A008477, A022559, A034387, A039697, A088865, A263653, A285769.

Programs

  • Mathematica
    Join[{1}, Table[PrimeOmega[n]^DivisorSum[n, # &, PrimeQ[#] &], {n, 2, 55}]]
  • PARI
    a(n) = my(f=factor(n)); vecsum(f[,2])^vecsum(f[,1]); \\ Michel Marcus, Apr 21 2018

Formula

a(n) = bigomega(n)^sopf(n) = A001222(n)^A008472(n).
a(p^k) = k^p where p is a prime.
a(A000312(k)) = a(k)*k^A008472(k).
a(A000142(k)) = A022559(k)^A034387(k).
a(A002110(k)) = k^A007504(k).

A039774 Numbers k such that phi(k) is equal to the product of (the sum of prime factors and the sum of exponents) of k-1.

Original entry on oeis.org

3, 5, 9, 25, 31, 57, 116, 144, 154, 288, 372, 414, 624, 792, 10032
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Next term if it exists is greater than 100000.
a(16) > 10^10, if it exists. - Amiram Eldar, Jun 10 2025

Examples

			25 is a term since phi(25) = 20, 24 = 2^3*3^1, (2+3)*(3+1) = 20.

Crossrefs

Cf. A000010, A039697, A039788, A039789.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;;, 1]]; e = f[[;;, 2]]; Total[p] * Total[e]]; Select[Range[3, 12000], EulerPhi[#] == s[#-1] &] (* Amiram Eldar, Jun 10 2025 *)
  • PARI
    isok(k) = if(k < 3, 0, my(f = factor(k-1)); eulerphi(k) == vecsum(f[,1]) * vecsum(f[,2])); \\ Amiram Eldar, Jun 10 2025

A039788 Numbers k such that phi(k) is equal to the product of (the sum of prime factors and the sum of exponents) of k.

Original entry on oeis.org

9, 16, 35, 45, 150, 154, 234, 264
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Next term if it exists is greater than 10^7. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 29 2004
a(9) > 10^10, if it exists. - Amiram Eldar, Jun 10 2025

Examples

			45 is a term since phi(45) = 24, 45 = 3^2*5^1, (3+5)*(2+1) = 24.

Crossrefs

Cf. A000010, A039697, A039774, A039789.

Programs

  • Mathematica
    q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;;, 1]]; e = f[[;;, 2]]; Times @@ ((p-1) * p^(e-1)) == Total[p] * Total[e]]; Select[Range[2, 300], q] (* Amiram Eldar, Jun 10 2025 *)
  • PARI
    for(n=1,10000000,f=factor(n);l=#f[,1];if(eulerphi(n)==sum(i=1,l,f[i,1])*sum(i=1,l,f[i,2]),print1(n,","))) (Ronaldo)
  • PARI
    isok(k) = if(k == 1, 0, my(f = factor(k)); eulerphi(f) == vecsum(f[,1]) * vecsum(f[,2])); \\ Amiram Eldar, Jun 10 2025

A039789 Integers k such that phi(k) is equal to the product of (the sum of prime factors and the sum of exponents) of k+1.

Original entry on oeis.org

7, 15, 62, 65, 76, 98, 260, 980
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Next term if it exists is greater than 1500000. - Reiner Martin, May 20 2001
No further terms up to 20000000. - Harvey P. Dale, Apr 19 2013
a(9) > 10^10, if it exists. - Amiram Eldar, Jun 10 2025

Examples

			62 is a term since phi(62) = 30, 63 = 3^2*7^1, (3+7)*(2+1) = 30.

Crossrefs

Cf. A000010, A039697, A039774, A039788.

Programs

  • Mathematica
    epQ[n_]:=Module[{fi=Transpose[FactorInteger[n+1]]},EulerPhi[n]== Total[ First[fi]]* Total[Last[fi]]]; Select[Range[1000],epQ] (* Harvey P. Dale, Apr 19 2013 *)
  • PARI
    isok(k) = my(f=factor(k+1)); eulerphi(k) == vecsum(f[,1]) * vecsum(f[,2]); \\ Michel Marcus, Oct 30 2022
Showing 1-4 of 4 results.

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

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