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.

A125070 a(n) = number of nonzero exponents in the prime factorization of n which are not primes.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 0, 2, 0, 1, 1, 3, 1, 0, 2, 2, 2, 0, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 2, 0, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 1, 0, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 0, 1, 3, 1, 1, 3
Offset: 1

Views

Author

Leroy Quet, Nov 18 2006

Keywords

Examples

			720 has the prime-factorization of 2^4 *3^2 *5^1. Two of these exponents, 4 and 1, are not primes. So a(720) = 2.
		

Crossrefs

Programs

  • Mathematica
    f[n_] := Length @ Select[Last /@ FactorInteger[n], ! PrimeQ[ # ] &];Table[f[n], {n, 110}] (* Ray Chandler, Nov 19 2006 *)
  • PARI
    A125070(n) = vecsum(apply(e -> if(isprime(e),0,1), factorint(n)[, 2])); \\ Antti Karttunen, Jul 07 2017

Formula

From Amiram Eldar, Sep 30 2023: (Start)
Additive with a(p^e) = A005171(e).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B - C), where B is Mertens's constant (A077761) and C = Sum_{p prime} (P(p) - P(p+1)) = 0.39847584805803104040..., where P(s) is the prime zeta function. (End)

Extensions

Extended by Ray Chandler, Nov 19 2006

A125030 a(n) = sum of exponents in the prime factorization of n that are noncomposite.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 0, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 1, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 0, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 1, 0, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 6, 1, 3, 3, 4, 1, 3, 1, 4, 3
Offset: 1

Views

Author

Leroy Quet, Nov 16 2006

Keywords

Examples

			a(720) = 3, since the prime factorization of 720 is 2^4 * 3^2 * 5^1 and two of the exponents in this factorization are noncomposites (the exponents 2 and 1, whose sum is 3).
		

Crossrefs

Programs

  • Mathematica
    f[n_] := Plus @@ Select[Last /@ FactorInteger[n], # == 1 || PrimeQ[ # ] &];Table[f[n], {n, 110}] (* Ray Chandler, Nov 19 2006 *)
  • PARI
    A125030(n) = vecsum(apply(e -> if((1==e)||isprime(e),e,0), factorint(n)[, 2])); \\ Antti Karttunen, Jul 07 2017

Formula

From Amiram Eldar, Sep 30 2023: (Start)
Additive with a(p^e) = e if e is composite, and 0 otherwise.
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = - P(2) + Sum_{p prime} p * (P(p) - P(p+1)) = 0.52262278983683613884..., where P(s) is the prime zeta function. (End)

Extensions

Extended by Ray Chandler, Nov 19 2006

A125073 a(n) = sum of the exponents in the prime factorization of n which are triangular numbers.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 3, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 4, 0, 2, 3, 1, 1, 3, 1, 0, 2, 2, 2, 0, 1, 2, 2, 4, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 4, 2, 4, 2, 2, 1, 2, 1, 2, 1, 6, 2, 3, 1, 1, 2, 3, 1, 3, 1, 2, 1, 1, 2, 3, 1, 1, 0, 2, 1, 2, 2, 2, 2, 4, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 0, 1, 3, 1, 4, 3
Offset: 1

Views

Author

Leroy Quet, Nov 18 2006

Keywords

Examples

			The prime factorization of 360 is 2^3 *3^2 *5^1. There are two exponents in this factorization which are triangular numbers, 1 and 3. So a(360) = 1 + 3 = 4.
		

Crossrefs

Programs

  • Mathematica
    f[n_] := Plus @@ Select[Last /@ FactorInteger[n], IntegerQ[Sqrt[8# + 1]] &];Table[f[n], {n, 110}] (* Ray Chandler, Nov 19 2006 *)
  • PARI
    A010054(n) = issquare(8*n + 1); \\ This function from Michael Somos, Apr 27 2000.
    A125073(n) = vecsum(apply(e -> (A010054(e)*e), factorint(n)[, 2])); \\ Antti Karttunen, Jul 08 2017

Formula

Additive with a(p^e) = A010054(e)*e. - Antti Karttunen, Jul 08 2017
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = -P(2) + Sum_{k>=2} (k*(k+1)/2) * (P(k*(k+1)/2) - P(k*(k+1)/2 + 1)) = -0.10099019472003733178..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 28 2023

Extensions

Extended by Ray Chandler, Nov 19 2006

A123391 a(n) = sum of exponents that are primes in the prime factorization of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 0, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 5, 0, 2, 2, 4, 0, 0, 0, 3, 0
Offset: 1

Views

Author

Leroy Quet, Nov 10 2006

Keywords

Examples

			36 = 2^2*3^2. Both exponents in this prime factorization are primes. So a(36) = 2+2 = 4.
		

Crossrefs

Programs

  • Mathematica
    f[n_] := Plus @@ Select[Last /@ FactorInteger[n], PrimeQ];Table[f[n], {n, 120}] (* Ray Chandler, Nov 11 2006*)
  • PARI
    {m=105; for(n=1, m, v=factor(n)[,2]; s=0; for(j=1, #v, if(isprime(v[j]), s=s+v[j])); print1(s,","))} \\ Klaus Brockhaus, Nov 14 2006
    
  • PARI
    A123391(n) = vecsum(apply(e -> isprime(e)*e, factorint(n)[, 2])); \\ Antti Karttunen, Jul 19 2017

Formula

Additive with a(p^e) = A010051(e)*e. - Antti Karttunen, Jul 19 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} p*(P(p)-P(p+1)) = 0.97487020987790163735..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 29 2023

Extensions

Extended by Ray Chandler and Klaus Brockhaus, Nov 11 2006
Showing 1-4 of 4 results.