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

A081633 Class 5+ primes (for definition see A005105).

Original entry on oeis.org

1021, 1321, 1381, 1459, 1877, 2467, 2503, 2657, 2707, 3253, 3313, 3547, 3701, 3733, 3907, 4561, 4817, 4937, 5441, 5443, 5527, 5693, 5839, 5861, 6037, 6131, 6211, 6217, 6277, 6361, 6373, 6569, 6653, 7057, 7243, 7591, 7753, 7817, 7883, 8101, 8123, 8221
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, A005105, A005106, A005107, A005108, A081634, A081635, A081636, A081637, A081638, A081639.

Programs

  • Maple
    For Maple program see Mathar link in A005105.
  • 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]]; ClassPlusNbr[n_] := Length[ NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[1040], ClassPlusNbr[ Prime[ # ]] == 5 &]]

A081638 Class 10+ primes.

Original entry on oeis.org

1065601, 2424973, 5114881, 7222561, 8124481, 8524091, 8647411, 8650321, 9190681, 9287521, 9590417, 10617601, 10929817, 11996161, 12349093, 12508081, 12786181, 12971117, 13570681, 14113027, 14308123, 14312743, 14476807
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, A005105, A005106, A005107, A005108, A081633, A081634, A081635, A081636, A081637, A081639.

Programs

  • Maple
    For Maple program see Mathar link in A005105.
  • 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]]; ClassPlusNbr[n_] := Length[ NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[150000], ClassPlusNbr[ Prime[ # ]] == 10 &]]

A081634 Class 6+ primes.

Original entry on oeis.org

2917, 4933, 5413, 7507, 8167, 8753, 10567, 10627, 11047, 11261, 11677, 12073, 12251, 12421, 12433, 12553, 12721, 14293, 15017, 17041, 18181, 18493, 19267, 19333, 20023, 21193, 21313, 21661, 22397, 24481, 25933, 26261, 26437, 27361
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, A005105, A005106, A005107, A005108, A081633, A081635, A081636, A081637, A081638, A081639.

Programs

  • Maple
    For Maple program see Mathar link in A005105.
  • 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]]; ClassPlusNbr[n_] := Length[ NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[3000], ClassPlusNbr[ Prime[ # ]] == 6 &]]

A081636 Class 8+ primes.

Original entry on oeis.org

49681, 109441, 120103, 151561, 198733, 210193, 246241, 255043, 266401, 280243, 295873, 326659, 326701, 347773, 355171, 360421, 368881, 397633, 397673, 423001, 441877, 447137, 471241, 480541, 489989, 499397, 508037, 511507, 532757, 539401
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, A005105, A005106, A005107, A005108, A081633, A081634, A081635, A081637, A081638, A081639.

Programs

  • Maple
    For Maple program see Mathar link in A005105.
  • 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]]; ClassPlusNbr[n_] := Length[ NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[44535], ClassPlusNbr[ Prime[ # ]] == 8 &]]

A081637 Class 9+ primes.

Original entry on oeis.org

532801, 710341, 720617, 1212487, 1261157, 1372081, 1457293, 1490429, 1532173, 1657801, 1788547, 1789093, 1809601, 1829293, 1887877, 1944181, 1960141, 1997587, 2121853, 2161853, 2474413, 2484049, 2557441, 2578801, 2613607
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, A005105, A005106, A005107, A005108, A081633, A081634, A081635, A081636, A081638, A081639.

Programs

  • Maple
    For Maple program see Mathar link in A005105.
  • 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]]; ClassPlusNbr[n_] := Length[ NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[196000], ClassPlusNbr[ Prime[ # ]] == 9 &]]

A084071 Class 12+ primes.

Original entry on oeis.org

68198461, 115084901, 138358573, 156811273, 213397621, 220576331, 234432217, 260050573, 282261961, 290996753, 330864497, 353653063, 371500819, 383616341, 406915273, 426240379, 445800983, 446707201, 449558323, 460339577, 472782553
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, A005105, A005106, A005107, A005108, A081633, A081634, A081635, A081636, A081637, A081638, A081639.

Programs

  • Maple
    For Maple program see Mathar link in A005105.
  • 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]]; ClassPlusNbr[n_] := Length[ NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[25000000], ClassPlusNbr[ Prime[ # ]] == 12 &]]

A129470 Primes p such that the largest prime factor of p+1 has Erdős-Selfridge class+ < N-1 if p is of class N+.

Original entry on oeis.org

883, 1747, 2417, 2621, 3181, 3301, 3533, 3571, 3691, 3853, 4027, 4133, 4513, 4783, 4861, 4957, 5303, 5381, 5393, 5563, 5641, 5821, 6067, 6577, 6991, 7177, 7253, 7331, 8059, 8093, 8377, 8731, 8839, 8929, 8969, 9221, 9281, 9397, 9613, 9931
Offset: 1

Views

Author

M. F. Hasler, Apr 16 2007

Keywords

Comments

In practice the class+ of a prime p is most often given by 1 + the class of the largest prime factor of p+1; terms of this sequence are counterexamples to this "rule". Terms of this sequence are at least of class 3+, since primes of class 1+ and 2+ have all prime factors of p+1 of class 1+. Terms a(k) of this sequence are >= -1 + 2*A005113(N-1) * nextprime(A005113(N-1)), where N is the class of a(k).

Examples

			a(3) = 883 = -1 + 2*13*17 is a prime of class 3+ since 13 is of class 2+, but the largest divisor of 883+1 is 17 which is only of class 1+.

Crossrefs

Cf. A005105, A005106, A005107, A005108, A005113, A081633, A081634, A081635, A081636, A081637, A081638, A081639, A084071, A090468, A129469, A129471, A129472, A129473, A129474, A129475, A129477, A129478.

Programs

  • PARI
    class(n,s=1)={n=factor(n+s)[,1];if(n[ #n]<=3,1,for(i=2,#n,n[1]=max(class(n[i],s)+1,n[1]));n[1])}; A129470(n=100,p=1,a=[])={ local(f); while( #a 3, f=factor(1+p=nextprime(p+1))[,1]); forstep( i=#f,2,-1, f[i]=class( f[i] ); if( f[i] > f[ #f], a=concat(a,p); /*print(#a," ",p);*/ break))); a}

A129478 Primes p of Erdos-Selfridge class 7+ with largest prime factor of p+1 not of class 6+.

Original entry on oeis.org

17227801, 18207913, 18592957, 19433053, 19608073, 19678081, 20028121, 20518177, 20658193, 20833213, 21043237, 21218257, 21533293, 21743317, 22128361, 22303381, 23668537, 25068697, 25418737, 25453741
Offset: 1

Views

Author

M. F. Hasler, Apr 17 2007

Keywords

Comments

See A129470

Examples

			a(1) = 17227801 = -1+2*2917*2953 = A129469[7] is a prime of class 7+ since 17227801+1 has prime factor 2917 = A081634[1] = A005113[6] of class 6+, but the largest prime factor of 17227801+1 is 2953 = A005107[175] of class 3+.

Crossrefs

Subsequence of A129470; see also A129471-A129473, A129477, A129469, A005113, A005105-A005108, A081633-A081635.

Programs

  • PARI
    class(n,s=1)={n=factor(n+s)[,1];if(n[ #n]<=3,1,for(i=2,#n,n[1]=max(class(n[i],s)+1,n[1]));n[1])}; a129478(n=100,p=1,a=[])={local(f,a6=A005113[6]);p=max(p,a6*nextprime(a6+1)*2-2); while( #a2 & f[ #f-1] >= a6 & 6 > class(f[ #f]), f=factor(1+p=nextprime(p+1))[,1]); forstep( i=#f-1,2,-1, if( 7 < f[1] = max( f[1],1+class( f[i] )), next(2))); if( f[1] == 7, a=concat(a,p); print(#a," ",p))); a}

A081640 a(n) = n-th prime of class 12- according to the Erdős-Selfridge classification.

Original entry on oeis.org

14920303, 18224639, 24867247, 26532953, 34548443, 38003011, 39800743, 41319599, 41443483, 45604771, 46432667, 47247763, 49734341, 49734493, 49749439, 51591833, 53014667, 55257977, 59681383, 59700749, 60804817
Offset: 1

Views

Author

Robert G. Wilson v, Mar 23 2003

Keywords

Comments

The first 184 resp. 300 terms of A081430 allow us to deduce 44 resp. 84 consecutive terms of this sequence. - M. F. Hasler, Apr 05 2007

Examples

			a(1) = 14920303 = 1+2*A081430(3)*3 is the smallest 12- prime

References

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

Links

Crossrefs

Cf. A005113, A005105, A005106, A005107, A005108, A081633, A081633, A081635, A081636, A081637, A081638.
Cf. A056637, A081430, A081641.

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[3610000], ClassMinusNbr[ Prime[ # ]] == 12 &]]
  • PARI
    nextclassminus( a, p=1, n=[] )={ while( p, n=concat(n,p); p=0; for( i=1,#a, if( p & 2*a[i] >= p-1, break); for( k=ceil(n[ #n]/2/a[i]),a[ #a]/a[i], if( p & 2*k*a[i] >= p-1, break); if( isprime(2*k*a[i]+1), p=2*k*a[i]+1; break(1+(k==1)); ))));vecextract(n,"^1")}; A081640 = nextclassminus(A081430) \\ M. F. Hasler, Apr 05 2007

Formula

{ a(n) } = { p = 2*m*A081430(k)+1 | k=1,2,...,oo and m=1,2,... such that p is prime and m has no factor of class > 11- } - M. F. Hasler, Apr 05 2007

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 21 2007

A081641 a(n) = n-th prime of class 13- according to the Erdős-Selfridge classification.

Original entry on oeis.org

36449279, 53065907, 59681213, 69096887, 132756479, 135388367, 164255999, 179043637, 188991053, 207290663, 241560239, 279709259, 309550999, 364492781, 372993983, 377982103, 398007431, 406165099, 425633717, 445901987, 447609067, 516737983
Offset: 1

Views

Author

Robert G. Wilson v, Mar 23 2003

Keywords

References

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

Links

Crossrefs

Cf. A005113, A005105, A005106, A005107, A005108, A081633, A081633, A081635, A081636, A081637, A081638.
Cf. A056637, A081640, A129248.

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[23733333], ClassMinusNbr[ Prime[ # ]] == 12 &]]
  • PARI
    A081641 = nextclassminus(A081640) /* cf. A081640 - M. F. Hasler, Apr 05 2007 */

Extensions

Edited by N. J. A. Sloane, May 14 2008 at the suggestion of R. J. Mathar.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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