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 1-10 of 60 results. Next

A005110 Class 2- primes (for definition see A005109).

Original entry on oeis.org

11, 29, 31, 41, 43, 53, 61, 71, 79, 101, 103, 113, 127, 131, 137, 149, 151, 157, 181, 191, 197, 211, 223, 229, 239, 241, 251, 271, 281, 293, 307, 313, 337, 379, 389, 401, 409, 421, 439, 443, 449, 457, 491, 521, 541, 547, 571, 593, 601, 613, 631, 641, 647, 653, 673
Offset: 1

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

References

  • R. K. Guy, Unsolved Problems in Number Theory, A18.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A005113, A056637, A005109, A005111, A005112, A081424, A081425, A081426, A081427, A081428, A081429, A081430.

Programs

  • Mathematica
    PrimeFactors[n_Integer] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]];
f[n_Integer] := Block[{m = n}, If[m == 0, m = 1, While[ IntegerQ[m/2], m /= 2];
While[ IntegerQ[m/3], m /= 3]];
Apply[Times, PrimeFactors[m] - 1]];
ClassMinusNbr[n_] := Length[NestWhileList[f, n, UnsameQ, All]] - 3;
Prime[ Select[ Range[122], ClassMinusNbr[ Prime[ # ]] == 2 &] ] (* Robert G. Wilson v *)

Extensions

Edited and extended by Robert G. Wilson v, Mar 20 2003
Corrected by R. J. Mathar, Feb 01 2007

A005111 Class 3- primes (for definition see A005109).

Original entry on oeis.org

23, 59, 67, 83, 89, 107, 173, 199, 227, 233, 263, 311, 317, 331, 349, 353, 367, 373, 383, 397, 419, 431, 463, 479, 503, 509, 523, 563, 569, 587, 607, 617, 619, 661, 683, 727, 733, 739, 743, 787, 809, 821, 823, 853, 859, 881, 887, 907, 929, 947, 977, 983, 991, 1031, 1033
Offset: 1

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

References

  • R. K. Guy, Unsolved Problems in Number Theory, A18.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A005113, A056637, A005109, A005110, A005112, A081424, A081425, A081426, A081427, A081428, A081429, A081430.

Programs

  • Mathematica
    PrimeFactors[n_Integer] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; f[n_Integer] := Block[{m = n}, If[m == 0, m = 1, While[ IntegerQ[m/2], m /= 2]; While[ IntegerQ[m/3], m /= 3]]; Apply[Times, PrimeFactors[m] - 1]]; ClassMinusNbr[n_] := Length[NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[175], ClassMinusNbr[ Prime[ # ]] == 3 &]]

Extensions

Edited and extended by Robert G. Wilson v, Mar 20 2003
Corrected by R. J. Mathar, Feb 01 2007

A005112 Class 4- primes (for definition see A005109).

Original entry on oeis.org

47, 139, 167, 179, 269, 277, 347, 461, 467, 499, 599, 643, 691, 709, 797, 827, 829, 839, 857, 863, 967, 997, 1013, 1019, 1039, 1063, 1069, 1151, 1163, 1181, 1289, 1367, 1381, 1399, 1427, 1487, 1493, 1499, 1579, 1609, 1619, 1657, 1867, 1877, 1889, 1933, 1979
Offset: 1

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

References

  • R. K. Guy, Unsolved Problems in Number Theory, A18.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A005113, A056637, A005109, A005110, A005111, A081424, A081425, A081426, A081427, A081428, A081429, A081430

Programs

  • Mathematica
    PrimeFactors[n_Integer] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; f[n_Integer] := Block[{m = n}, If[m == 0, m = 1, While[ IntegerQ[m/2], m /= 2]; While[ IntegerQ[m/3], m /= 3]]; Apply[Times, PrimeFactors[m] - 1]]; ClassMinusNbr[n_] := Length[NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[300], ClassMinusNbr[ Prime[ # ]] == 4 &]]

Extensions

Edited and extended by Robert G. Wilson v, Mar 20 2003

A081424 Class 5- primes (for definition see A005109).

Original entry on oeis.org

283, 359, 557, 659, 941, 1109, 1129, 1223, 1433, 1663, 1669, 1693, 1787, 1997, 2027, 2039, 2069, 2083, 2153, 2339, 2351, 2503, 2539, 2579, 2633, 2767, 2777, 2803, 2837, 2999, 3229, 3581, 3761, 3767, 3779, 3989, 4127, 4157, 4231, 4253, 4283, 4297, 4513
Offset: 1

Views

Author

Robert G. Wilson v, Mar 20 2003

Keywords

References

  • R. K. Guy, Unsolved Problems in Number Theory, A18.

Links

Crossrefs

Cf. A005113, A056637, A005109, A005110, A005111, A005112, A081425, A081426, A081427, A081428, A081429, A081430.

Programs

  • Mathematica
    PrimeFactors[n_Integer] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; f[n_Integer] := Block[{m = n}, If[m == 0, m = 1, While[ IntegerQ[m/2], m /= 2]; While[ IntegerQ[m/3], m /= 3]]; Apply[Times, PrimeFactors[m] - 1]]; ClassMinusNbr[n_] := Length[NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[700], ClassMinusNbr[ Prime[ # ]] == 5 &]]

A081425 Class 6- primes (for definition see A005109).

Original entry on oeis.org

719, 1319, 1699, 2447, 3343, 4079, 4139, 4457, 4517, 4679, 4703, 5273, 5647, 6653, 6793, 7523, 7529, 7559, 8599, 9227, 9587, 9623, 9839, 10159, 10343, 10723, 10771, 11069, 11213, 11279, 11321, 11489, 11863, 11887, 12163, 12917, 12919, 13163
Offset: 1

Views

Author

Robert G. Wilson v, Mar 20 2003

Keywords

References

  • R. K. Guy, Unsolved Problems in Number Theory, A18.

Links

Crossrefs

Cf. A005113, A056637, A005109, A005110, A005111, A005112, A081424, A081426, A081427, A081428, A081429, A081430.

Programs

  • Mathematica
    PrimeFactors[n_Integer] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; f[n_Integer] := Block[{m = n}, If[m == 0, m = 1, While[ IntegerQ[m/2], m /= 2]; While[ IntegerQ[m/3], m /= 3]]; Apply[Times, PrimeFactors[m] - 1]]; ClassMinusNbr[n_] := Length[NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[1700], ClassMinusNbr[ Prime[ # ]] == 6 &]]

A122257 Characteristic function of Pierpont primes (A005109).

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 29 2006

Keywords

Links

Crossrefs

Cf. A005109, A010051, A065333, A122258 (partial sums).

Programs

  • Mathematica
    smooth3Q[n_] := n == 2^IntegerExponent[n, 2]*3^IntegerExponent[n, 3];
a[n_] := Boole[PrimeQ[n] && smooth3Q[n - 1]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 16 2021 *)
  • PARI
    is3smooth(n) = my(m = n >> valuation(n, 2)); m == 3^valuation(m, 3);
a(n) = isprime(n) && is3smooth(n-1); \\ Amiram Eldar, May 14 2025
  • Scheme
    (define (A122257 n) (if (= 1 n) 0 (if (= 1 (A065333 (- n 1))) (A010051 n) 0)))
(define (A065333 n) (if (= 1 (A038502 (A000265 n))) 1 0))
;; Antti Karttunen, Dec 07 2017

Formula

a(n) = A010051(n) * A065333(n-1).
a(n) = if (n is prime) and (n-1 is 3-smooth) then 1 else 0.
a(n) = if n=1 then 0 else A122258(n) - A122258(n-1);
a(A122259(n)) = 0, a(A005109(n)) = 1.

A113433 Semi-Pierpont semiprimes: products of exactly two Pierpont primes A005109.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 25, 26, 34, 35, 38, 39, 49, 51, 57, 65, 74, 85, 91, 95, 111, 119, 133, 146, 169, 185, 194, 218, 219, 221, 247, 259, 289, 291, 323, 326, 327, 361, 365, 386, 481, 485, 489, 511, 514, 545, 579, 629, 679, 703, 763, 771, 815, 866, 949, 965
Offset: 1

Views

Author

Jonathan Vos Post, Nov 01 2005

Keywords

Comments

Semiprime both of whose prime factors are Pierpont primes (A005109), which are primes of the form (2^K)*(3^L)+1. Not to be confused with A113432: Pierpont semiprimes [Semiprimes of the form (2^K)*(3^L)+1]. This terminology itself is by analogy to what Tomaszewski used for the Sophie Germain counterparts A111153 and A111206.

Examples

			a(1) = 4 = 2^2 = [(2^0)*(3^0)+1]*[(2^1)*(3^0)+1] = A005109(1)*A005109(1).
a(2) = 6 = 2*3 = [(2^0)*(3^0)+1]*[(2^1)*(3^0)+1] = A005109(1)*A005109(2).
a(3) = 9 = 3^2 = [(2^1)*(3^0)+1]*[(2^1)*(3^0)+1] = A005109(2)*A005109(2).
a(4) = 10 = 2*5 = [(2^0)*(3^0)+1]*[(2^2)*(3^0)+1] = A005109(1)*A005109(3).
a(5) = 14 = 2*7 = [(2^0)*(3^0)+1]*[(2^1)*(3^1)+1] = A005109(1)*A005109(4).
a(6) = 15 = 3*5 = [(2^1)*(3^0)+1]*[(2^2)*(3^0)+1] = A005109(2)*A005109(3).

Links

Crossrefs

Cf. A001358, A003586, A005109, A055600, A111153, A111206, A113432, A113434.

Programs

  • Mathematica
    Select[Range[10^3], Plus @@ Last /@ FactorInteger[ # ] == 2 && And @@ (Max @@ First /@ FactorInteger[ # - 1] < 5 &) /@ First /@ FactorInteger[ # ] &] (* Ray Chandler, Jan 24 2006 *)

Formula

{a(n)} = Semiprimes A001358 both of whose factors are of the form (2^K)*(3^L)+1. {a(n)} = {A005109(i)*A005109(j) for integers i and j not necessarily distinct}.

A217035 Generalized cuban primes (A007645) which are also Class 1- (or Pierpont) primes (A005109).

Original entry on oeis.org

3, 7, 13, 19, 37, 73, 97, 109, 163, 193, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 139969, 147457, 209953, 331777, 472393, 629857, 746497, 786433, 839809, 995329, 1179649, 1492993, 1769473
Offset: 1

Views

Author

Jonathan Vos Post, Sep 24 2012

Keywords

Comments

Is this the union of A058383 and {3}? - R. J. Mathar, Sep 28 2012
Yes, it is, because the only Fermat prime == 0 or 1 mod 3 is 3. - Robert Israel, Mar 02 2018
Generalized cuban primes are primes of the form x^2 + xy + y^2; or: primes of form x^2 + 3*y^2; or: primes == 0 or 1 mod 3. Class 1- (or Pierpont) primes: primes of the form 2^t*3^u + 1.

Links

Crossrefs

Cf. A007645, A005109, A058383.

Programs

  • Mathematica
    nn = 100000; t1 = Join[{3}, Select[Prime[Range[nn]], MemberQ[{1}, Mod[#, 3]] &]]; t2 = Select[Prime[Range[nn]], Max @@ First /@ FactorInteger[# - 1] < 5 &]; Intersection[t1, t2] (* T. D. Noe, Sep 26 2012 *)

Formula

A007645 INTERSECTION A005109.

A172167 Partial sums of A005109.

Original entry on oeis.org

2, 5, 10, 17, 30, 47, 66, 103, 176, 273, 382, 545, 738, 995, 1428, 1915, 2492, 3261, 4414, 5711, 7170, 9763, 12680, 16137, 20026, 30395, 42684, 60181, 78614, 117981, 170470, 236007, 375976, 523433, 733386, 1065163, 1537556, 2167413, 2913910
Offset: 1

Views

Author

Jonathan Vos Post, Jan 28 2010

Keywords

Examples

			a(20) = 2 + 3 + 5 + 7 + 13 + 17 + 19 + 37 + 73 + 97 + 109 + 163 + 193 + 257 + 433 + 487 + 577 + 769 + 1153 + 1297 = 5711.

Crossrefs

Cf. A005109.

Formula

a(n) = Sum_{i=1..n} A005109(i).

Extensions

More terms from R. J. Mathar, Feb 07 2010

A338931 Least b such that b^(2^n) + 1 is an odd Pierpont prime (A005109).

Original entry on oeis.org

2, 2, 2, 2, 2, 54, 162, 8310407949893763072, 46438023168, 65229815808, 396718580736, 629856, 152461794335880672662217818112
Offset: 0

Views

Author

Jeppe Stig Nielsen, Nov 16 2020

Keywords

Comments

Every term is even (A005843) and 3-smooth (A003586).
For n = 0, 1, 2, 3, 4, 7, 8, 9, 12, ..., the corresponding number b^(2^n) + 1 is also a Proth prime (A080076), while for n = 5, 6, 10, 11, ..., it is a non-Proth.
The form b^(2^n) + 1 is called a generalized Fermat number.

Examples

			a(7) corresponds to prime 8310407949893763072^128 + 1 = (2^47*3^10)^128 + 1.

Crossrefs

Cf. A005109, A056993, A334053.
Showing 1-10 of 60 results. Next

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

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