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.
smallf(q,nmax) = {my(qq=q,j=0); forprime (p=2, nmax, my(k=0); while (qq%p==0, k++; qq/=p); if (k>0, j++;)); [j,qq]};
a362957(upto) = {my(nfmax=0); for (n=1, upto, forprime (p=2, oo, my(f=p^n+1, s=smallf(f,p)); if (s[1]nfmax, print1(p,", "); nfmax=nf; break)))};
a362957(12)
a(3) = 9 because 9^3 + 1 = 2 * 5 * 73 and 9 is the least number with this property.
a(n) = my(k=1); while (!issquarefree(k^n+1) || omega(k^n+1) != n, k++); k;
a(1) = 3; 3^1 + 2 = 5. a(2) = 2; 2^2 + 2 = 2 * 3. a(3) = 7; 7^3 + 2 = 3 * 5 * 23. a(4) = 71; 71^4 + 2 = 3 * 11 * 19 * 40529.
Table[b=2;y[a_]:=FactorInteger[Prime[a]^n+b];k=1;Monitor[Parallelize[While[True,If[And[Length[y[k]]==n,Count[Flatten[y[k]],1]==n],Break[]];k++];k],k]//Prime,{n,1,10}]
a(n) = forprime(p=2, , my(f=factor(p^n + 2)); if (issquarefree(f) && (omega(f) == n), return(p))); \\ Michel Marcus, Nov 29 2022
a(3) = 47 since 47^(2*3-1) - 1 = 229345006 = 2*11*23*31*14621 is the product of 5 distinct primes and 47 is the smallest prime number with this property.
isok(p, n) = my(f=factor(p^(2*n-1)-1)); issquarefree(f) && (omega(f) == 2*n-1); a(n) = my(p=2); while (!isok(p, n), p=nextprime(p+1)); p; \\ Michel Marcus, Dec 15 2022
a(1) = 3; 3^1 + 4 = 7. a(2) = 19; 19^2 + 4 = 5 * 73. a(3) = 11; 11^3 + 4 = 3 * 5 * 89. a(4) = 29; 29^4 + 4 = 5 * 17 * 53 * 157.
Table[b=4;y[a_]:=FactorInteger[Prime[a]^n+b];k=1;Monitor[Parallelize[While[True,If[And[Length[y[k]]==n,Count[Flatten[y[k]],1]==n],Break[]];k++];k],k]//Prime,{n,1,10}]
a(n) = forprime(p=2, , my(f=factor(p^n + 4)); if (issquarefree(f) && (omega(f) == n), return(p)));
a(1) = 5; 5^1 + 6 = 11. a(2) = 2; 2^2 + 6 = 2 * 5. a(3) = 23; 23^3 + 6 = 7 * 37 * 47. a(4) = 127; 127^4 + 6 = 7 * 131 * 367 * 773.
Table[b=6;y[a_]:=FactorInteger[Prime[a]^n+b];k=1;Monitor[Parallelize[While[True,If[And[Length[y[k]]==n,Count[Flatten[y[k]],1]==n],Break[]];k++];k],k]//Prime,{n,1,10}]
a(n) = forprime(p=2, , my(f=factor(p^n + 6)); if (issquarefree(f) && (omega(f) == n), return(p)));
a(5) = 7 is the smallest number of the set {k(i)} = {7, 14, 24, 26, 46, 51, ...} where k(i)^5 + 1 has exactly 5 prime factors counted with multiplicity.
a(n) = my(k=1); while (bigomega(k^n+1) != n, k++); k;
a(3) = 5 is the smallest prime of the set {p(i)} = {5, 11, 13, 19, 23, ...} where omega(p(i)^3 + 1) = 3.
a[n_] := Module[{p = 2}, While[PrimeNu[p^n + 1] != n, p = NextPrime[p]]; p]; Print[Array[a, 11]]
a(n) = forprime(p=2, oo, if(omega(p^n+1) == n, return(p)));
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