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

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

Author

Felix Fröhlich, Apr 16 2016

Keywords

• nonn
• tabl

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

• Daniel Suteu, Table of n, a(n) for n = 2..407

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