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 1-6 of 6 results.

A180065 Smallest strong pseudoprime to base 2 with n prime factors.

Original entry on oeis.org

2047, 15841, 800605, 293609485, 10761055201, 5478598723585, 713808066913201, 90614118359482705, 5993318051893040401, 24325630440506854886701, 27146803388402594456683201, 4365221464536367089854499301, 2162223198751674481689868383601, 548097717006566233800428685318301
Offset: 2

Views

Author

Kevin Batista (kevin762401(AT)yahoo.com), Aug 09 2010

Keywords

Examples

			800605 is the third term because 800605 = 5 * 13 * 109 * 113, more prime factors than smaller 2-strong pseudoprimes.

Links

Crossrefs

Cf. A007011, A001262.

Programs

  • PARI
    strong_check(p, base, e, r) = my(tv=valuation(p-1, 2)); tv > e && Mod(base, p)^((p-1)>>(tv-e)) == r;
strong_fermat_psp(A, B, k, base) = A=max(A, vecprod(primes(k))); (f(m, l, lo, k, e, r) = my(list=List()); my(hi=sqrtnint(B\m, k)); if(lo > hi, return(list)); if(k==1, forstep(p=lift(1/Mod(m, l)), hi, l, if(isprimepower(p) && gcd(m*base, p) == 1 && strong_check(p, base, e, r), my(n=m*p); if(n >= A && (n-1) % znorder(Mod(base, p)) == 0, listput(list, n)))), forprime(p=lo, hi, base%p == 0 && next; strong_check(p, base, e, r) || next; my(z=znorder(Mod(base, p))); gcd(m,z) == 1 || next; my(q=p, v=m*p); while(v <= B, list=concat(list, f(v, lcm(l, z), p+1, k-1, e, r)); q *= p; Mod(base, q)^z == 1 || break; v *= p))); list); my(res=f(1, 1, 2, k, 0, 1)); for(v=0, logint(B, 2), res=concat(res, f(1, 1, 2, k, v, -1))); vecsort(Set(res));
a(n) = if(n < 2, return()); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=strong_fermat_psp(x, y, n, 2)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Mar 04 2023

Extensions

a(6)-a(10) and editing from Charles R Greathouse IV, Aug 29 2010
a(11)-a(15) from Daniel Suteu, Sep 24 2022

A271873 Square array A(n, k) read by antidiagonals downwards: smallest base-n Fermat pseudoprime with k distinct prime factors for k, n >= 2.

Original entry on oeis.org

341, 561, 91, 11305, 286, 15, 825265, 41041, 435, 124, 45593065, 825265, 11305, 561, 35, 370851481, 130027051, 418285, 41041, 1105, 6, 38504389105, 2531091745, 30534805, 2203201, 25585, 561, 21, 7550611589521, 38504389105, 370851481, 68800501, 682465, 62745, 105, 28
Offset: 2

Views

Author

Felix Fröhlich, Apr 16 2016

Keywords

Examples

			The array A(n, k) starts as follows:
   k  =  2    3     4        5         6
n = 2: 341  561 11305   825265  45593065
n = 3:  91  286 41041   825265 130027051
n = 4:  15  435 11305   418285  30534805
n = 5: 124  561 41041  2203201  68800501
n = 6:  35 1105 25585   682465  12306385

Links

Crossrefs

Cf. A007011 (row n=2), A271874.

Programs

  • PARI
    minpsp(n, k) = forcomposite(c=1, , if(Mod(n, c)^(c-1)==1, if(omega(c)==k, return(c))))
a(n, k) = for(x=2, n, for(y=2, k, print1(minpsp(x, y), ", ")); print(""))
a(6, 6) \\ print array up to n = 6, k = 6
  • PARI
    fermat_psp(A, B, k, base) = A=max(A, vecprod(primes(k))); (f(m, l, lo, k) = my(list=List()); my(hi=sqrtnint(B\m, k)); if(lo > hi, return(list)); if(k==1, forstep(p=lift(1/Mod(m, l)), hi, l, if(isprimepower(p) && gcd(m*base, p) == 1, my(n=m*p); if(n >= A && (n-1) % znorder(Mod(base, p)) == 0, listput(list, n)))), forprime(p=lo, hi, base%p == 0 && next; my(z=znorder(Mod(base, p))); gcd(m,z) == 1 || next; my(q=p, v=m*p); while(v <= B, list=concat(list, f(v, lcm(l, z), p+1, k-1)); q *= p; Mod(base, q)^z == 1 || break; v *= p))); list); vecsort(Set(f(1, 1, 2, k)));
T(n,k) = if(n < 2, return()); my(x=vecprod(primes(k)), y=2*x); while(1, my(v=fermat_psp(x, y, k, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x);
print_table(n, k) = for(x=2, n, for(y=2, k, print1(T(x, y), ", ")); print(""));
for(k=2, 9, for(n=2, k, print1(T(n, k-n+2)", "))); \\ Daniel Suteu, Dec 01 2023

Extensions

a(16)-a(37) from Daniel Suteu, Sep 02 2022

A080748 Number of distinct prime factors of n-th Fermat pseudoprime to base 2.

Original entry on oeis.org

2, 3, 3, 3, 2, 3, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 3, 4, 3, 3, 2, 3, 2, 2, 3, 3, 4, 2, 2, 4, 2, 3, 3, 3, 3, 2, 2, 2, 3, 3, 2, 4, 4, 3, 2, 3, 2, 2, 3, 4, 3, 2, 2, 4, 4, 2, 2, 3, 4, 3, 4, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3
Offset: 1

Views

Author

Benoit Cloitre, Mar 08 2003

Keywords

Comments

What is the average order of this sequence? In particular, is it bounded? For extremal behavior, see A007011. - Charles R Greathouse IV, Sep 12 2012

Links

Crossrefs

Cf. A001567.

A242276 Irregular array of factors of n-th Poulet number read by rows, where row n corresponds to A001567(n).

Original entry on oeis.org

11, 31, 3, 11, 17, 3, 5, 43, 5, 13, 17, 19, 73, 7, 13, 19, 3, 5, 127, 23, 89, 5, 17, 29, 37, 73, 7, 13, 31, 29, 113, 37, 109, 17, 257, 3, 31, 47, 31, 151, 43, 127, 7, 23, 41, 73, 109, 53, 157, 3, 11, 257, 7, 19, 67, 31, 331, 5, 29, 73, 5, 7, 17, 19, 3, 17, 251, 7, 13, 151, 59, 233, 11, 31, 41, 43, 337, 23, 683, 7, 31, 73, 5, 13
Offset: 1

Views

Author

Felix Fröhlich, Aug 16 2014

Keywords

Examples

			The first three Poulet numbers (2-pseudoprimes) are 341 = 11*31, 561 = 3*11*17, and 645 = 3*5*43, so the sequence begins:
11, 31;
3, 11, 17;
3, 5, 43;
etc.

Links

Crossrefs

Cf. A001567, A007011, A084653, A084654, A084655.

Programs

  • PARI
    forcomposite(n=1, 1e4, if(Mod(2, n)^(n-1)==1, f=factor(n)[, 1]; for(i=1, #f, print1(f[i], ", "))))

A328665 Least super-Poulet number (A050217) with n distinct prime factors.

Original entry on oeis.org

341, 294409, 9972894583, 1264022137981459, 14054662152215842621
Offset: 2

Views

Author

Amiram Eldar, Oct 24 2019

Keywords

Comments

a(7) <= 1842158622953082708177091, and a(8) <= 317565023788749598474704753433331761 (from Michon's site).
From Daniel Suteu, Oct 28 2019: (Start)
a(8) <= 192463418472849397730107809253922101,
a(9) <= 1347320741392600160214289343906212762456021,
a(10) <= 70865138168006643427403953978871929070133095474701,
a(11) <= 3363391752747838578311772729701478698952546288306688208857,
a(12) <= 132153369641266990823936945628293401491197666138621036175881960329,
a(13) <= 9105096650335639994239038954861714246150666715328403635257215036295306537. (End)

Links

Crossrefs

Cf. A001567, A007011, A050217, A178997, A300327.

Programs

  • Mathematica
    a[n_] := Module[{k=1}, While[PrimeNu[k] < n || PowerMod[2, k - 1, k] != 1 || Union @ PowerMod[2, Rest[Divisors[k]], k] != {2}, k++]; k]; Array[a, 3, 2]
  • PARI
    isok(k, n) = if (omega(k) == n, fordiv(k, d, if(Mod(2, d)^d!=2, return(0))); return(1));
a(n) = my(k=4); while (!isok(k, n), k++); k; \\ Michel Marcus, Oct 28 2019
  • PARI
    isupperbound(n,k) = my(f=factor(k)); omega(f) == n && Mod(2, k)^gcd(vector(#f~, i, f[i,1]-1)) == 1; \\ Daniel Suteu, Oct 28 2019

A353409 Smallest overpseudoprime to base 2 (A141232) with n distinct prime factors.

Original entry on oeis.org

2047, 13421773, 14073748835533
Offset: 2

Views

Author

Daniel Suteu, May 07 2022

Keywords

Comments

a(5) > 2^64.
a(5) <= 1376414970248942474729,
a(6) <= 48663264978548104646392577273,
a(7) <= 294413417279041274238472403168164964689,
a(8) <= 98117433931341406381352476618801951316878459720486433149,
a(9) <= 1252977736815195675988249271013258909221812482895905512953752551821.

Crossrefs

Cf. A001567, A007011, A141232, A180065, A328665.
Showing 1-6 of 6 results.

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