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 1-4 of 4 results.

A360184 Square array A(n, k) read by antidiagonals downwards: smallest base-n strong Fermat pseudoprime with k distinct prime factors for k, n >= 2.

Original entry on oeis.org

2047, 15841, 703, 800605, 8911, 341, 293609485, 152551, 4371, 781, 10761055201, 41341321, 129921, 24211, 217, 5478598723585, 12283706701, 9224391, 4382191, 29341, 325, 713808066913201, 1064404682551, 2592053871, 381347461, 3405961, 58825, 65, 90614118359482705
Offset: 2

Views

Author

Daniel Suteu, Mar 04 2023

Keywords

Comments

The array A(n, k) starts as follows:
k = 2 3 4 5 6
n = 2: 2047 15841 800605 293609485 10761055201
n = 3: 703 8911 152551 41341321 12283706701
n = 4: 341 4371 129921 9224391 2592053871
n = 5: 781 24211 4382191 381347461 9075517561
n = 6: 217 29341 3405961 557795161 333515107081

Links

Crossrefs

Cf. A001262, A180065 (row n=2), A271873.

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));
T(n, k) = if(n < 2, return()); my(x=vecprod(primes(k)), y=2*x); while(1, my(v=strong_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)", ")));

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.

A356866 Smallest Carmichael number (A002997) with n prime factors that is also a strong pseudoprime to base 2 (A001262).

Original entry on oeis.org

15841, 5310721, 440707345, 10761055201, 5478598723585, 713808066913201, 1022751992545146865, 5993318051893040401, 120459489697022624089201, 27146803388402594456683201, 14889929431153115006659489681
Offset: 3

Views

Author

Daniel Suteu, Oct 01 2022

Keywords

Crossrefs

Cf. A001262, A002997, A006931, A063847, A180065.

Programs

  • PARI
    carmichael_strong_psp(A, B, k, base) = A=max(A, vecprod(primes(k+1))\2); (f(m, l, p, k, k_exp, congr, u=0, v=0) = my(list=List()); if(k==1, forprime(q=u, v, my(t=m*q); if((t-1)%l == 0 && (t-1)%(q-1) == 0, my(tv=valuation(q-1, 2)); if(tv > k_exp && Mod(base, q)^(((q-1)>>tv)< k_exp && Mod(base, q)^(((q-1)>>tv)<u, u=r); list=concat(list, f(t, L, r, k-1, k_exp, congr, u, v)))))))); list); my(res=f(1, 1, 3, k, 0, 1)); for(v=0, logint(B, 2), res=concat(res, f(1, 1, 3, k, v, -1))); vecsort(Vec(res));
a(n,base=2) = if(n < 3, return()); my(x=vecprod(primes(n+1))\2,y=2*x); while(1, my(v=carmichael_strong_psp(x,y,n,base)); if(#v >= 1, return(v[1])); x=y+1; y=2*x);

Formula

a(n) >= max(A180065(n), A006931(n)).

A361256 Smallest base-n strong Fermat pseudoprime with n distinct prime factors.

Original entry on oeis.org

2047, 8911, 129921, 381347461, 333515107081, 37388680793101, 713808066913201, 665242007427361, 179042026797485691841, 8915864307267517099501, 331537694571170093744101, 2359851544225139066759651401, 17890806687914532842449765082011
Offset: 2

Views

Author

Daniel Suteu, Mar 06 2023

Keywords

Comments

Main diagonal of A360184.

Crossrefs

Cf. A001262, A180065, A271874, A360184.

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, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x);
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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