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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A105555 Let d = number of divisors of n; a(n) = d-th prime.

Original entry on oeis.org

2, 3, 3, 5, 3, 7, 3, 7, 5, 7, 3, 13, 3, 7, 7, 11, 3, 13, 3, 13, 7, 7, 3, 19, 5, 7, 7, 13, 3, 19, 3, 13, 7, 7, 7, 23, 3, 7, 7, 19, 3, 19, 3, 13, 13, 7, 3, 29, 5, 13, 7, 13, 3, 19, 7, 19, 7, 7, 3, 37, 3, 7, 13, 17, 7, 19, 3, 13, 7, 19, 3, 37, 3, 7, 13, 13, 7, 19, 3, 29, 11, 7, 3, 37, 7, 7, 7, 19, 3
Offset: 1

Views

Author

Cino Hilliard, May 03 2005

Keywords

Examples

			n = 6 has 4 divisors, prime(4) = 7, so a(6) = 7.

Links

Crossrefs

Cf. A000005, A000040, A105560, A105561.

Programs

  • Mathematica
    Prime[DivisorSigma[0,Range[90]]] (* Harvey P. Dale, Jul 27 2011 *)
  • PARI
    d(n) = for(x=1,n,print1(prime(numdiv(x))","))
  • Python
    from sympy import prime, divisor_count
def a(n): return prime(divisor_count(n)) # Indranil Ghosh, May 25 2017

Formula

a(n) = A000040(A000005(n)). - Antti Karttunen, May 25 2017

A339876 a(n) = A336466(A122111(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 9, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1
Offset: 1

Views

Author

Antti Karttunen, Dec 25 2020

Keywords

Links

Crossrefs

Cf. A000265, A064989, A105560, A122111, A336466.
Cf. also A334107, A339877.

Programs

  • PARI
    A000265(n) = (n>>valuation(n,2));
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A336466(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]-1))^f[k,2])); };
A339876(n) = A336466(A122111(n));
  • PARI
    A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A105560(n) = if(1==n,n,prime(bigomega(n)));
A339876(n) = if(1==n,n,A000265(A105560(n)-1) * A339876(A064989(n)));

Formula

a(1) = 1, for n > 1, a(n) = A000265(A105560(n)-1) * a(A064989(n)).
a(n) = A336466(A122111(n)).

A105561 a(n) is the m-th prime, where m is the number of distinct prime factors of n (A001221), a(1) = 1.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 2, 3, 2, 5, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 2, 5, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 5, 2, 3, 3, 2, 3, 5, 2, 3, 3, 5, 2, 3, 2, 3, 3, 3, 3, 5, 2, 3, 2, 3, 2, 5, 3, 3, 3, 3, 2, 5, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 2, 5, 2, 3, 5, 3, 2, 3, 2, 5, 3, 3, 2, 5, 3, 3, 3, 3, 3, 5
Offset: 1

Views

Author

Cino Hilliard, May 03 2005

Keywords

Comments

Term a(1) = 1 prepended to match with the definition of A105560. - Antti Karttunen, May 25 2017

Examples

			Let n = 6; 6 has 2 different prime factors, therefore a(6) = 3, the second prime.

Links

Crossrefs

Cf. A000040, A001221, A105555, A105560.

Programs

  • Mathematica
    Table[Prime[Length[FactorInteger[n]]], {n, 2, 84}]
Prime[PrimeNu[Range[2,90]]] (* Harvey P. Dale, Oct 02 2013 *)
  • PARI
    A105561(n) = if(1==n,n,prime(omega(n))); \\ [After the original Pari-program given here.] - Antti Karttunen, May 25 2017
  • Python
    from sympy import prime, primefactors
def a(n): return 1 if n==1 else prime(len(primefactors(n))) # Indranil Ghosh, May 25 2017

Extensions

Edited by Stefan Steinerberger, Jun 15 2007
Term a(1) = 1 prepended (correcting also the indexing of the rest of terms), and data section extended to 120 terms by Antti Karttunen, May 25 2017

A339877 a(n) = A336467(A122111(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 9, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 9, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 9, 7, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 9, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 7, 1, 3, 9, 1, 1, 3, 1, 1, 9
Offset: 1

Views

Author

Antti Karttunen, Dec 25 2020

Keywords

Links

Crossrefs

Cf. A000265, A064989, A105560, A122111, A336467.
Cf. also A339876, A334108.

Programs

  • PARI
    A000265(n) = (n>>valuation(n,2));
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A336467(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]+1))^f[k,2])); };
A339877(n) = A336467(A122111(n));
  • PARI
    A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A105560(n) = if(1==n,n,prime(bigomega(n)));
A339877(n) = if(1==n||isprime(n),1,A000265(A105560(n)+1) * A339877(A064989(n)));

Formula

For noncomposite n, a(n) = 1, for composite n, a(n) = A000265(A105560(n)+1) * a(A064989(n)).
a(n) = A336467(A122111(n)).

A329033 a(n) = A003415(A122111(n)).

Original entry on oeis.org

0, 1, 4, 1, 12, 5, 32, 1, 6, 16, 80, 7, 192, 44, 21, 1, 448, 8, 1024, 24, 60, 112, 2304, 9, 27, 272, 10, 68, 5120, 31, 11264, 1, 156, 640, 81, 10, 24576, 1472, 384, 32, 53248, 92, 114688, 176, 45, 3328, 245760, 13, 108, 39, 912, 432, 524288, 12, 216, 92, 2112, 7424, 1114112, 41, 2359296, 16384, 140, 1, 540, 244, 4980736, 1024, 4800, 123
Offset: 1

Views

Author

Antti Karttunen, Nov 08 2019

Keywords

Links

Crossrefs

Cf. A003415, A064989, A105560, A122111.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A329033(n) = A003415(A122111(n));

Formula

a(n) = A003415(A122111(n)).
a(1) = 0; for n > 1, a(n) = A122111(A064989(n)) + (A105560(n) * a(A064989(n))).
Previous Showing 11-15 of 15 results.

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