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.

A081391 Numbers k such that the central binomial coefficient C(2*k,k) has only one prime divisor whose exponent equals one.

Original entry on oeis.org

3, 6, 7, 8, 9, 10, 11, 12, 16, 21, 22, 28, 29, 30, 31, 36, 37, 54, 55, 57, 58, 110, 171, 784, 786
Offset: 1

Views

Author

Labos Elemer, Mar 27 2003

Keywords

Comments

Numbers k such that C(2*k,k) has one non-unitary prime divisor.
Numbers k for which A081387(k) = 1.
No more terms through 10^6; conjecture: no terms after 786. - Jon E. Schoenfield, Jul 29 2017

Examples

			For k = 786, C(1572,786) = 2*2*2*2*m, where m is a squarefree product of 169 primes.

Crossrefs

Cf. A000108, A000984, A001405, A046098, A081386, A081387, A081388, A081389, A081390, A080664.

Programs

A081390 Number k such that the k-th Catalan number has only one non-unitary prime divisor; all the other prime divisors have exponent one.

Original entry on oeis.org

6, 10, 12, 15, 16, 20, 21, 22, 27, 28, 29, 30, 32, 33, 34, 36, 37, 39, 53, 54, 55, 56, 57, 58, 65, 67, 79, 80, 109, 110, 129, 135, 159, 161, 170, 171, 255, 783, 784, 785, 786, 902
Offset: 1

Views

Author

Labos Elemer, Mar 27 2003

Keywords

Comments

a(43) > 25000, if it exists. - Amiram Eldar, Jul 22 2024

Examples

			902 is a term because binomial(1804,902)/903 has 189 prime factor, 188 stand with exponent one, but 2 with 5: 2^5.

Crossrefs

Cf. A000108, A081386, A081387, A081388, A081389, A080405, A080664.

Programs

A081392 Numbers k such that the central binomial coefficient C(k, floor(k/2)) has only one prime divisor whose exponent is greater than one.

Original entry on oeis.org

6, 9, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 31, 32, 33, 35, 39, 41, 42, 43, 44, 55, 56, 57, 58, 59, 60, 61, 62, 65, 67, 72, 73, 74, 79, 107, 108, 109, 110, 113, 114, 115, 116, 131, 159, 219, 220, 271, 319, 341, 342, 1567, 1568, 1571, 1572
Offset: 1

Views

Author

Labos Elemer, Mar 27 2003

Keywords

Comments

As expected, the (single) non-unitary prime divisors for C(2k, k) and C(k, floor(k/2)) or for Catalan numbers equally come from the smallest prime(s).
Numbers k such that A001405(k) is in A190641. - Michel Marcus, Jul 30 2017
a(56) > 5*10^6 if it exists. - David A. Corneth, Apr 03 2021

Examples

			For k=341, binomial(341,170) = 2*2*2*2*M, where M is a squarefree product of 48 further prime factors.

Crossrefs

Cf. A000108, A000984, A001405, A046098, A080664, A081386-A081391, A190641.

Programs

  • Mathematica
    pde1Q[n_]:=Length[Select[FactorInteger[Binomial[n,Floor[n/2]]],#[[2]]> 1&]] == 1; Select[Range[1600],pde1Q] (* Harvey P. Dale, Jan 21 2019 *)
  • PARI
    isok(n) = my(f=factor(binomial(n, n\2))); #select(x->(x>1), f[,2]) == 1; \\ Michel Marcus, Jul 30 2017
  • PARI
    is(n) = { my(nf2 = n\2, nmnf2 = n-nf2, t); forprime(p = 2, n, if(val(n, p) - val(nf2, p) - val(nmnf2, p) > 1, t++; if(t > 1, return(0) ) ) ); t==1 }
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Apr 03 2021

Extensions

a(52)-a(55) from Michel Marcus, Jul 30 2017

A107042 First differences of indices of squarefree Catalan numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 6, 2, 12, 4
Offset: 0

Views

Author

Paul Barry, May 09 2005

Keywords

Comments

First differences of A080664. Next term is > 9965.

Crossrefs

Cf. A000108.

Programs

Showing 1-4 of 4 results.

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

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