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 1-5 of 5 results.

A136003 Primes that are not the sum, minus 1, of a Pythagorean triple.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 31, 37, 41, 43, 53, 61, 67, 73, 97, 101, 103, 109, 113, 127, 137, 151, 157, 163, 173, 193, 211, 229, 241, 257, 271, 277, 281, 283, 293, 313, 317, 331, 337, 353, 367, 397, 401, 409, 421, 433, 457, 463, 487, 499, 521, 523, 541, 547, 557
Offset: 1

Views

Author

Omar E. Pol, Dec 16 2007

Keywords

Comments

Primes in A136002.

Links

Crossrefs

Cf. A009096, A010814, A136002.

Programs

  • Mathematica
    q[n_] := PrimeQ[n] && (n == 2 || Module[{d = Divisors[(n+1)/2]}, AllTrue[Range[3, Length[d]], d[[#]] >= 2 * d[[#-1]] &]]); Select[Range[600], q] (* Amiram Eldar, Oct 19 2024 *)

Extensions

Extended by Ray Chandler, Dec 13 2008

A136005 Numbers of the form 2^p - 1, where p is a prime number that is not the sum, minus 1, of a Pythagorean triple.

Original entry on oeis.org

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 137438953471, 2199023255551, 8796093022207, 9007199254740991, 2305843009213693951, 147573952589676412927, 9444732965739290427391, 158456325028528675187087900671, 2535301200456458802993406410751
Offset: 1

Views

Author

Omar E. Pol, Dec 17 2007

Keywords

Comments

See A136003 for the values of p.

Examples

			a(3) = 31 because A136003(3) = 5 and 2^5 = 32 and 32-1 = 31.

Links

Crossrefs

Cf. A000079, A000668, A136002, A136003.

Programs

  • Mathematica
    q[n_] := PrimeQ[n] && (n == 2 || Module[{d = Divisors[(n+1)/2]}, AllTrue[Range[3, Length[d]], d[[#]] >= 2 * d[[#-1]] &]]); 2^Select[Range[100], q] - 1 (* Amiram Eldar, Oct 20 2024 *)

Formula

a(n) = 2^A136003(n) - 1.

Extensions

a(16)-a(17) from Amiram Eldar, Oct 20 2024

A136000 a(n) = A010814(n) - 1.

Original entry on oeis.org

11, 23, 29, 35, 39, 47, 55, 59, 69, 71, 79, 83, 89, 95, 107, 111, 119, 125, 131, 139, 143, 149, 153, 155, 159, 167, 175, 179, 181, 191, 197, 199, 203, 207, 209, 215, 219, 223, 227, 233, 239, 251, 259, 263, 269, 275, 279, 285, 287, 299, 305, 307, 311, 319, 323
Offset: 1

Views

Author

Omar E. Pol, Dec 10 2007

Keywords

Comments

Numbers of the form P-1 in increasing order, where P is the sum of a Pythagorean triple. Also P is the perimeter of a Pythagorean triangle. The open triangle represent a triangle instrument and, in general, any musical instrument. Positive integers are musician numbers or dancer number A136002.

Examples

			a(1) = 11 because {3,4,5} is a Pythagorean triple and 3+4+5 = 12 is the sum of a Pythagorean triple and 11+1 = 12, then we can write 3+4+5 = 11+1.

Links

Crossrefs

Cf. A010814, A136001, A136002, A009096 (perimeters of Pythagorean triangles).

Programs

  • Mathematica
    q[n_] := OddQ[n] && Module[{d = Divisors[(n+1)/2]}, AnyTrue[Range[3, Length[d]], d[[#]] < 2 * d[[#-1]] &]]; Select[Range[350], q] (* Amiram Eldar, Oct 19 2024 *)

Extensions

Definition corrected by R. J. Mathar, Dec 12 2007
Extended by Ray Chandler, Dec 13 2008

A136007 Primes of the form 2^p - 1, where p is a prime number that is not the sum, minus 1, of a Pythagorean triple.

Original entry on oeis.org

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 170141183460469231731687303715884105727, 6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151
Offset: 1

Views

Author

Omar E. Pol, Dec 17 2007

Keywords

Comments

Primes in A136005.

Links

Crossrefs

Cf. A000079, A000668, A136002, A136003, A136005, A152961.

Programs

  • Mathematica
    q[n_] := n == 2 || Module[{d = Divisors[(n+1)/2]}, AllTrue[Range[3, Length[d]], d[[#]] >= 2 * d[[#-1]] &]]; 2^Select[MersennePrimeExponent[Range[13]], q] - 1 (* Amiram Eldar, Oct 20 2024 *)

Formula

a(n) = 2^A152961(n) - 1. - Amiram Eldar, Oct 20 2024

Extensions

Extended by Ray Chandler, Dec 13 2008

A152961 Base-2 logarithm of A136007(n)+1.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 31, 61, 127, 521, 2281, 3217, 4253, 9941, 19937, 21701, 23209, 44497, 110503, 216091, 859433, 1257787, 3021377, 6972593, 13466917, 20996011, 30402457, 32582657, 42643801, 57885161
Offset: 1

Views

Author

Omar E. Pol, Dec 16 2008, Dec 20 2008

Keywords

Comments

The first 10 terms coincide with A109799.

Crossrefs

Intersection of A000043 and A136002.
Cf. A000523, A109799, A136003, A136005, A136007.

Programs

  • Mathematica
    q[n_] := n == 2 || Module[{d = Divisors[(n+1)/2]}, AllTrue[Range[3, Length[d]], d[[#]] >= 2 * d[[#-1]] &]]; Select[MersennePrimeExponent[Range[48]], q] (* Amiram Eldar, Oct 20 2024 *)

Formula

a(n) = A000523(A136007(n)+1). - Michel Marcus, Apr 14 2021

Extensions

a(12)-(a31) from Amiram Eldar, Oct 20 2024
Showing 1-5 of 5 results.

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

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