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.

A103667 Primes p such that the largest prime divisor of p-1 is greater than the largest prime divisor of p+1.

Original entry on oeis.org

7, 11, 23, 29, 31, 47, 53, 59, 71, 79, 83, 89, 103, 107, 127, 131, 139, 149, 167, 173, 179, 191, 199, 223, 227, 233, 239, 263, 269, 293, 307, 311, 317, 347, 349, 359, 367, 373, 383, 389, 419, 431, 439, 449, 461, 467, 479, 499, 503, 509, 557, 563, 569, 571, 587
Offset: 1

Views

Author

Hugo Pfoertner, Feb 19 2005

Keywords

Comments

Primes of the form 2*A070087(n)+1 for some n. - Charles R Greathouse IV, Dec 22 2022
Conjecture: this sequence is of positive relative density in the primes, perhaps even 1/2. - Charles R Greathouse IV, Dec 22 2022

Examples

			a(1)=7 because the largest prime divisor of 6 is greater than the largest prime divisor of 8.

Links

Crossrefs

Cf. A023503 (greatest prime divisor of n-th prime - 1), A023509 (greatest prime divisor of n-th prime + 1), A103666, A070087.

Programs

  • Maple
    filter:= p -> isprime(p) and max(numtheory:-factorset(p-1)) > max(numtheory:-factorset(p+1)):
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Jan 15 2024
  • Mathematica
    Select[Prime@Range[2, 107], If[FactorInteger[#-1][[-1, 1]]>FactorInteger[#+1][[-1, 1]], True]&] (* James C. McMahon, Jan 15 2024 *)

A103664 Primes p such that the number of divisors of p-1 is less than the number of divisors of p+1.

Original entry on oeis.org

2, 3, 5, 11, 17, 23, 29, 47, 53, 59, 71, 79, 83, 89, 107, 131, 139, 149, 167, 173, 179, 191, 197, 223, 227, 233, 239, 251, 263, 269, 293, 311, 317, 347, 359, 367, 383, 389, 419, 431, 439, 443, 449, 461, 467, 479, 499, 503, 509, 557, 563, 569, 587, 593, 599, 607
Offset: 1

Views

Author

Hugo Pfoertner, Feb 19 2005

Keywords

Comments

Mathematica coding by Wouter Meeussen and Robert G. Wilson v.

Examples

			a(1)=2 because d(1)=1 < d(3)=2; a(2)=3 because d(2)=2 < d(4)=3.

Links

Crossrefs

Cf. A008328 number of divisors of p-1, A008329 number of divisors of p+1, A103665, A103666, A103667.

Programs

  • Maple
    with(numtheory): p:=proc(n) if isprime(n) and tau(n-1)Emeric Deutsch, Feb 22 2005
  • Mathematica
    Select[Prime[Range[1, 140]], Length[Divisors[ # - 1]] < Length[Divisors[ # + 1]] &]
Select[Prime[Range[200]],DivisorSigma[0,#-1]Harvey P. Dale, May 31 2019 *)

A103665 Primes p such that the number of divisors of p-1 is greater than the number of divisors of p+1.

Original entry on oeis.org

13, 31, 37, 43, 61, 67, 73, 97, 101, 109, 113, 127, 151, 157, 163, 181, 193, 211, 229, 241, 257, 271, 277, 281, 283, 313, 331, 337, 353, 373, 379, 397, 401, 409, 421, 433, 457, 463, 487, 521, 523, 541, 547, 571, 577, 601, 613, 617, 631, 641, 661, 673, 677
Offset: 1

Views

Author

Hugo Pfoertner, Feb 19 2005

Keywords

Comments

Mathematica coding by Wouter Meeussen and Robert G. Wilson v.

Examples

			a(1)=13 because d(12)=6 > d(14)=4.

Links

Crossrefs

Cf. A008328 number of divisors of p-1, A008329 number of divisors of p+1, A103664, A103666, A103667.

Programs

  • Mathematica
    Select[Prime[Range[2, 140]], Length[Divisors[ # - 1]] > Length[Divisors[ # + 1]] &]
Select[Prime[Range[200]],DivisorSigma[0,#-1]>DivisorSigma[0,#+1]&] (* Harvey P. Dale, Aug 21 2022 *)
  • PARI
    forprime (k=2,700,if(numdiv(k-1)>numdiv(k+1),print1(k,", ")))
\\ Hugo Pfoertner, Nov 30 2017

A180641 Primes P such that P > (largest prime factor of (P-1)) * (largest prime factor of (P+1)).

Original entry on oeis.org

7, 17, 19, 31, 41, 53, 71, 79, 89, 97, 101, 109, 127, 151, 163, 181, 191, 197, 199, 239, 241, 251, 257, 271, 307, 337, 349, 379, 401, 419, 431, 433, 449, 461, 463, 487, 491, 499, 521, 571, 577, 593, 599, 601, 631, 641, 647, 659, 683, 701, 727, 751, 769, 809
Offset: 1

Views

Author

Karl Hovekamp, Sep 14 2010

Keywords

Examples

			Example: For n = 3, a(3)=19.
The prime P = 19
P-1 = 18 (largest prime factor of 18 is 3)
P+1 = 20 (largest prime factor of 20 is 5)
19 > 3*5.

Links

Crossrefs

Cf. A180640. See also A103666, A103667.

Programs

  • Mathematica
    lpfQ[n_]:=Module[{a=FactorInteger[n-1][[-1,1]],b=FactorInteger[n+1][[-1,1]]},n>a*b]; Select[Prime[Range[200]],lpfQ] (* Harvey P. Dale, Aug 16 2013 *)
  • PARI
    lpf(n) = {f = factor(n); return (f[#f~, 1]);}
lista(nn) = {forprime(p=3, nn,if ((p > lpf(p-1)*lpf(p+1)), print1(p, ", ");););} \\ Michel Marcus, Jul 25 2013

A180640 Primes P such that P < (largest prime factor of (P-1)) * (largest prime factor of (P+1)).

Original entry on oeis.org

2, 3, 5, 11, 13, 23, 29, 37, 43, 47, 59, 61, 67, 73, 83, 103, 107, 113, 131, 137, 139, 149, 157, 167, 173, 179, 193, 211, 223, 227, 229, 233, 263, 269, 277, 281, 283, 293, 311, 313, 317, 331, 347, 353, 359, 367, 373, 383, 389, 397, 409, 421, 439, 443, 457, 467
Offset: 1

Views

Author

Karl Hovekamp, Sep 14 2010

Keywords

Examples

			For n = 3, a(3)=11.
The prime P = 11
P-1 = 10 (largest prime factor of 10 is 5)
P+1 = 12 (largest prime factor of 12 is 3)
11 < 5*3.

Crossrefs

Cf. A180641. See also A103666, A103667.

Programs

  • Mathematica
    Select[Prime[Range[100]],#<(FactorInteger[#-1][[-1,1]] FactorInteger[#+1][[-1,1]])&]  (* Harvey P. Dale, Feb 22 2011 *)
  • PARI
    isok(p) = (p==2) || (isprime(p) && (p < vecmax(factor(p-1)[, 1]) * vecmax(factor(p+1)[, 1]))); \\ Michel Marcus, Oct 29 2022

Extensions

Initial term, i.e., 2, added by Harvey P. Dale, Feb 22 2011
Showing 1-5 of 5 results.

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