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 1-3 of 3 results.

A079151 Primes p such that p-1 has at most 3 prime factors, counted with multiplicity; i.e., primes p such that bigomega(p-1) = A001222(p-1) <= 3.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 43, 47, 53, 59, 67, 71, 79, 83, 103, 107, 131, 139, 149, 167, 173, 179, 191, 223, 227, 239, 263, 269, 283, 293, 311, 317, 347, 359, 367, 383, 389, 419, 431, 439, 443, 467, 479, 499, 503, 509, 557, 563, 587, 599, 607, 619, 643
Offset: 1

Views

Author

Cino Hilliard, Dec 27 2002

Keywords

Comments

Can it be proved that this is a subsequence of A301590, except for a(5) = 13? (Checked up to A301591(10^4) = 427421.) - M. F. Hasler, Aug 14 2021

Examples

			149 is in the sequence because 149 - 1 = 2*2*37 has 3 prime factors.

Links

Crossrefs

Cf. A079148, A079150, A079152, A079153.

Programs

  • Mathematica
    s[n_] := Reap[sr = 0; ct = 0; For[x = 2, x <= n, x = NextPrime[x], If[PrimeOmega[x - 1] < 4, Sow[x]; sr += 1.0/x; ct += 1]]][[2, 1]]; s[700] (* Jean-François Alcover, Jun 08 2013, translated and adapted from Pari *)
Select[Prime[Range[120]],PrimeOmega[#-1]<4&] (* Harvey P. Dale, Oct 02 2017 *)
  • PARI
    s(n)=sr=0; ct=0; forprime(x=2,n, if(bigomega(x-1) < 4, print1(x" "); sr+=1.0/x; ct+=1; ); ); print(); print(ct" "sr); \\ Lists primes p<=n such that p-1 has at most 3 prime factors.
  • PARI
    list(lim)=my(v=List([2,3]),t); forprime(p=2,(lim-1)\2, if(isprime(t=2*p+1), listput(v,t))); forprime(p=2,(lim-1)\4, forprime(q=2,min(p,(lim-1)\2\p), if(isprime(t=2*p*q+1), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Mar 06 2017

Extensions

Typos in definition corrected by Harvey P. Dale, Oct 02 2017

A079150 Primes p such that p+1 has at most 3 prime factors, counted with multiplicity; i.e., primes p such that bigomega(p+1) = A001222(p+1) <= 3.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 37, 41, 43, 61, 67, 73, 97, 101, 109, 113, 137, 157, 163, 173, 181, 193, 211, 229, 241, 257, 277, 281, 283, 313, 317, 331, 337, 353, 373, 397, 401, 409, 421, 433, 457, 523, 541, 547, 577, 601, 613, 617, 641, 653, 661, 673, 677
Offset: 1

Views

Author

Cino Hilliard, Dec 27 2002

Keywords

Examples

			173 is in the sequence because 173+1 = 2*3*29 has 3 prime factors.

Links

Crossrefs

Cf. A001222, A079147, A079151, A079152, A079153.

Programs

  • Mathematica
    Select[Prime[Range[200]],PrimeOmega[#+1]<4&] (* Harvey P. Dale, Feb 02 2012 *)
  • PARI
    s(n) = {sr=0; ct=0; forprime(x=2,n, if(bigomega(x+1) < 4, print1(x" "); sr+=1.0/x; ct+=1; ); ); print(); print(ct" "sr); } \\ Lists primes p<=n such that p+1 has at most 3 prime factors.

A079152 Primes p such that either p-1 or p+1 has at most 3 prime factors, counted with multiplicity; i.e., primes p such that either bigomega(p-1) <= 3 or bigomega(p+1) <= 3, where bigomega(n) = A001222(n).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 157, 163, 167, 173, 179, 181, 191, 193, 211, 223, 227, 229, 239, 241, 257, 263, 269, 277, 281, 283, 293, 311, 313, 317, 331
Offset: 1

Views

Author

Cino Hilliard, Dec 27 2002

Keywords

Comments

Up to 83, this is the sequence of prime numbers A000040. 89 is not in the sequence because both 89-1 = 88 = 2*2*2*11 and 89+1 = 90 = 2*3*3*5 have 4 prime factors.

Examples

			97 is in the sequence because 97+1 = 98 = 2*7*7 has 3 prime factors.

Links

Crossrefs

Union of A079150 and A079151. Cf. A079149, A079153.

Programs

  • Magma
    bg:=func; [2] cat [p: p in PrimesInInterval(3,340)| bg(p-1) le 3 or bg(p+1) le 3]; // Marius A. Burtea, Jan 16 2020
  • Mathematica
    Select[Prime /@ Range[70], PrimeOmega[# - 1] <= 3 || PrimeOmega[# + 1] <= 3 & ] (* Jean-François Alcover, Jul 02 2013 *)
  • PARI
    s(n) = {sr=0; ct=0; forprime(x=2,n, if(bigomega(x-1) < 4 || bigomega(x+1) < 4, print1(x, ", "); sr+=1.0/x; ct+=1; ); ); print(); print(ct" "sr); } \\ Lists primes p<=n such that either p-1 or p+1 has at most 3 prime factors.
Showing 1-3 of 3 results.

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