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 11 results. Next

A056637 a(n) is the least prime of class n-, according to the Erdős-Selfridge classification of primes.

Original entry on oeis.org

2, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, 14920303, 36449279, 377982107, 1432349099, 22111003847, 110874748763
Offset: 1

Views

Author

Robert G. Wilson v, Jan 31 2001

Keywords

Comments

A prime p is in class 1- if p-1 has no prime factor larger than 3. If p-1 has other prime factors, p is in class (c+1)-, where c- is the largest class of its prime factors. See also A005109.
1432349099 < a(16) <= 25782283783.
a(18) <= 619108107719, a(19) <= 19811459447009, a(20) <= 152772264735359. These upper limits can be found by generating class (n+1)- primes from a list of n- class primes; if the latter is sufficiently complete, one can deduce that there is no smaller (n+1)- prime. - M. F. Hasler, Apr 05 2007

Crossrefs

Cf. A005113, A005109, A005110, A005111, A005112, A081424, A081425, A081426, A081427, A081428, A081429, A081430.
Cf. A082449, A129246, A081640, A129248.

Programs

  • Mathematica
    PrimeFactors[n_Integer] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; NextPrime[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; 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; a = Table[0, {15}]; a[[1]] = 2; k = 5; Do[c = ClassMinusNbr[ k]; If[ a[[c]] == 0, a[[c]] = k]; k = NextPrime[k], {n, 3, 7223000}]; a

Formula

a(n+1) >= 2*a(n)+1, since a(n+1)-1 is even and must have a factor of class n- which is odd (n>1) and >= a(n). a(n+1) <= min { p = 2*k*a(n)+1 | k=1,2,3... such that p is prime }, since a(n) is a prime of class n-. - M. F. Hasler, Apr 05 2007

Extensions

Extended by Robert G. Wilson v, Mar 20 2003
More terms from Don Reble, Apr 11 2003
a(16) and a(17) from M. F. Hasler, Apr 21 2007

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

A081426 Class 7- primes.

Original entry on oeis.org

1439, 8629, 10067, 14683, 17257, 19577, 20389, 22643, 23743, 27103, 28219, 29399, 31657, 32633, 33107, 33113, 33863, 34259, 34513, 35951, 36137, 36887, 37379, 40127, 40637, 40759, 42179, 42209, 42767, 44519, 44579, 45139, 49019, 49669
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, A081425, 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[5200], ClassMinusNbr[ Prime[ # ]] == 7 &]]

A081427 Class 8- primes.

Original entry on oeis.org

2879, 20147, 25903, 34537, 46049, 58733, 63317, 65267, 69029, 69073, 74759, 80537, 86291, 86341, 103549, 106487, 108413, 112877, 120877, 131687, 135859, 138053, 140939, 141023, 147647, 155413, 157427, 165527, 172681, 187163, 189949, 207079
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, A081425, A081426, 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[20000], ClassMinusNbr[ Prime[ # ]] == 8 &]]

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 &]]

A081430 Class 11- primes.

Original entry on oeis.org

1266767, 1520159, 2486717, 3316619, 4144541, 4512947, 4836779, 5389519, 5638379, 6218827, 6448979, 6633457, 6771419, 6907247, 7460149, 7462639, 7600597, 7739033, 7874627, 8153567, 8291573, 9110639, 9112319, 9121003
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, A081425, A081426, A081427, A081429.

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[300000, 1000000], ClassMinusNbr[ Prime[ # ]] == 1 &]]

A081428 Class 9- primes.

Original entry on oeis.org

34549, 86371, 103613, 120919, 138059, 149519, 172583, 172741, 224563, 276293, 282059, 282143, 293659, 299417, 316691, 352399, 368513, 379903, 397303, 403061, 414577, 451499, 483179, 486527, 489431, 500947, 506537, 517747, 518047, 541799
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, A081425, A081426, A081427, 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[50000], ClassMinusNbr[ Prime[ # ]] == 9 &]]
Showing 1-10 of 11 results. Next

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

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