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
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
Programs
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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
Comments