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 1-6 of 6 results.

A265394 Positions of records in A265388.

Original entry on oeis.org

1, 2, 3, 6, 7, 15, 19, 31, 37, 79, 97, 139, 157, 199, 211, 229, 271, 307, 331, 337, 367, 379, 439, 499, 547, 577, 601, 607, 619, 661, 691, 727, 811, 829, 877, 937, 967, 997, 1009, 1069, 1171, 1237, 1279, 1297, 1399, 1429, 1459, 1531, 1609, 1627, 1657, 1759, 1867, 2011, 2029, 2089, 2131, 2137, 2179, 2221, 2281, 2311
Offset: 1

Views

Author

Antti Karttunen, Dec 08 2015

Keywords

Comments

Question: Apart from a(1)=1, a(4)=6 and a(6)=15, are all other terms primes? See also comments in A265395 & A265396.

Links

Crossrefs

Cf. A265388.
Cf. A265395 (gives the record values), A265396.

Programs

  • Mathematica
    DeleteDuplicates[Table[{n,GCD@@Array[Binomial[2 n,2 #]&,{n-1}]},{n,2350}],GreaterEqual[ #1[[2]],#2[[2]]]&][[All,1]] (* Harvey P. Dale, Jul 19 2022 *)
  • PARI
    lista(nn) = {r = 0; for (n=1, nn, nr = gcd(vector(n-1, k, binomial(2*n, 2*k))); if (nr > r, print1(n, ", "); r = nr););} \\ Michel Marcus, Dec 08 2015

A265395 Record values in A265388.

Original entry on oeis.org

0, 6, 15, 33, 91, 435, 703, 1891, 2701, 12403, 18721, 38503, 49141, 79003, 88831, 104653, 146611, 188191, 218791, 226801, 269011, 286903, 385003, 497503, 597871, 665281, 721801, 736291, 765703, 873181, 954271, 1056331, 1314631, 1373653, 1537381, 1755001, 1869211, 1987021, 2035153, 2284453, 2741311, 3059101, 3270403, 3363121, 3913003
Offset: 1

Views

Author

Antti Karttunen, Dec 08 2015

Keywords

Comments

Question: Are all terms apart from 0 and 435 (= 3*5*29) semiprimes? See also comments in A265394 & A265396.

Links

Crossrefs

Cf. A265388, A265394, A265396.

Programs

  • PARI
    lista(nn) = {r = 0; for (n=1, nn, nr = gcd(vector(n-1, k, binomial(2*n, 2*k))); if (nr > r, print1(nr, ", "); r = nr););} \\ Michel Marcus, Dec 08 2015. Note: this gives correct value for a(1) only from PARI version 2.8 onward, see A265388 for why. - Antti Karttunen, Dec 11 2015

Formula

a(n) = A265388(A265394(n)).

A265396 Numerator of A265395(n)/A265394(n), record values / record positions in A265388.

Original entry on oeis.org

0, 3, 5, 11, 13, 29, 37, 61, 73, 157, 193, 277, 313, 397, 421, 457, 541, 613, 661, 673, 733, 757, 877, 997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1933, 1993, 2017, 2137, 2341, 2473, 2557, 2593, 2797, 2857, 2917, 3061, 3217, 3253, 3313, 3517, 3733, 4021, 4057, 4177, 4261, 4273, 4357, 4441
Offset: 1

Views

Author

Antti Karttunen, Dec 11 2015

Keywords

Comments

It seems that n=4 is the only case where A265395(n)/A265394(n) is not integral (as 33/6 = 11/2), thus for other n, a(n) actually seems to give the value of A265395(n)/A265394(n).

Links

Crossrefs

Cf. A265388, A265394, A265395.
For many terms coincides with A005383, A079147, A113733 and A256072.

Programs

A265402 Fixed points of A265388: numbers n for which gcd{k=1..n-1} binomial(2*n, 2*k) = n.

Original entry on oeis.org

11, 17, 23, 29, 43, 47, 53, 59, 67, 71, 73, 83, 89, 101, 103, 107, 109, 113, 127, 131, 137, 149, 151, 163, 167, 173, 179, 191, 193, 197, 223, 227, 233, 239, 241, 251, 257, 263, 269, 277, 281, 283, 293, 311, 317, 347, 349, 353, 359, 373, 383, 389, 397, 401, 409, 419, 431, 433, 443, 449, 457, 461, 463, 467, 479, 487, 491, 503
Offset: 1

Views

Author

Antti Karttunen, Dec 08 2015

Keywords

Links

Crossrefs

Fixed points of A265388. Cf. also A265403.

Programs

  • Mathematica
    Select[Range@ 504, GCD @@ Array[Function[k, Binomial[2 #, 2 k]], {# - 1}] == # &] (* Michael De Vlieger, Dec 11 2015 *)
  • PARI
    isok(n) = (n>1) && gcd(vector(n-1, k, binomial(2*n, 2*k))) == n; \\ Michel Marcus, Dec 08 2015, edited by Antti Karttunen, Dec 11 2015 (see A265388 for why).

A265403 Numbers n for which gcd{k=1..n-1} binomial(2*n, 2*k) = 2n-1.

Original entry on oeis.org

10, 12, 21, 22, 24, 30, 34, 36, 40, 51, 52, 55, 57, 69, 70, 76, 82, 84, 87, 90, 96, 99, 100, 106, 112, 114, 115, 117, 120, 129, 132, 136, 141, 142, 147, 154, 156, 159, 166, 174, 177, 180, 184, 187, 192, 195, 201, 205, 210, 216, 217, 220, 222, 225, 231, 232, 234, 240, 244, 246, 250, 252, 255, 261, 262, 274, 279, 282, 285, 286, 294, 297, 300
Offset: 1

Views

Author

Antti Karttunen, Dec 08 2015

Keywords

Links

Crossrefs

Cf. A265388.
Cf. also A265402.

Programs

  • Mathematica
    Select[Range@ 300, GCD @@ Array[Function[k, Binomial[2 #, 2 k]], {# - 1}] == 2 # - 1 &] (* Michael De Vlieger, Dec 11 2015 *)
  • PARI
    isok(n) = (n>1) && gcd(vector(n-1, k, binomial(2*n, 2*k))) == 2*n-1; \\ Michel Marcus, Dec 08 2015, edited by Antti Karttunen, Dec 11 2015 (see A265388 for why).

A265401 Numbers n for which gcd{k=1..n-1} binomial(2*n, 2*k) = 1.

Original entry on oeis.org

20, 26, 33, 35, 38, 39, 44, 46, 48, 50, 56, 58, 60, 62, 68, 72, 74, 77, 78, 80, 86, 88, 92, 93, 94, 95, 98, 102, 104, 105, 108, 110, 111, 116, 118, 119, 124, 130, 133, 134, 138, 140, 143, 144, 146, 148, 150, 152, 155, 158, 160, 161, 164, 165, 168, 170, 171, 176, 178, 182, 183, 185, 186, 188, 189, 194, 198, 200
Offset: 1

Views

Author

Antti Karttunen, Dec 08 2015

Keywords

Comments

Contains no primes or prime powers. - Robert Israel, Dec 10 2015

Links

Crossrefs

Cf. A265388.

Programs

  • Maple
    select(n -> igcd(seq(binomial(2*n, 2*k), k=1..n-1)) = 1, [$1..200]); # Robert Israel, Dec 10 2015
  • Mathematica
    Select[Range@ 200, GCD @@ Table[Binomial[2 #, 2 k], {k, # - 1}] == 1 &] (* Michael De Vlieger, Dec 09 2015, modified to match the new corrected definition by Antti Karttunen, Dec 11 2015 *)
  • PARI
    isok(n) = (n>1) && gcd(vector(n-1, k, binomial(2*n, 2*k))) == 1; \\ Michel Marcus, Dec 08 2015, edited by Antti Karttunen, Dec 11 2015 (see A265388 for why).
Showing 1-6 of 6 results.

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