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.
nn = 500; c[_] = False; u = v = 2; a[1] = 1;
Monitor[Reap[
Do[k = u;
While[Or[c[k],
! CoprimeQ[k, Product[a[h], {h, n - Min[k, n - 1], n - 1}] ] ],
If[k > n - 1, k = v, k++]];
Set[{a[n], c[k]}, {k, True}];
If[CompositeQ[k], Sow[k]];
If[k == u, While[c[u], u++]];
If[k == v, While[Or[c[v], CompositeQ[v]], v++]], {n, 2, nn}] ][[-1, 1]], n] (* Michael De Vlieger, Feb 14 2025 *)
from math import gcd
from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
alst, aset, an, m = [1], {1}, 1, 2
for n in count(2):
if an > 3 and not isprime(an):
yield an
an = next(k for k in count(m) if k not in aset and all(gcd(alst[-j], k) == 1 for j in range(1, min(k, n-1)+1)))
alst.append(an)
aset.add(an)
while m in aset: m += 1
print(list(islice(agen(), 64))) # Michael S. Branicky, Feb 14 2025
nn = 500; c[_] = False; i = 0; u = v = 2; a[1] = 1;
Monitor[Reap[
Do[k = u;
While[Or[c[k],
! CoprimeQ[k, Product[a[h], {h, n - Min[k, n - 1], n - 1}] ] ],
If[k > n - 1, k = v, k++]];
Set[{a[n], c[k]}, {k, True}];
If[CompositeQ[k], Sow[i]; i = 0, i++];
If[k == u, While[c[u], u++]];
If[k == v, While[Or[c[v], CompositeQ[v]], v++]], {n, 2, nn}] ][[-1, 1]], n] (* Michael De Vlieger, Feb 14 2025 *)
Table listing n and S(n), where i = pi(sqrt(S(n))) and S = A381019. Asterisks denote confirmed S(n) = prime(i)^2 coprime to P(r)/prime(i), where P = A002110 and r, the index of the largest prime in S(1..n-1).
n i S(n)
--------------------------
7 1 2^2 = 4 *
13 2 3^2 = 9 *
30 3 5^2 = 25 *
55 4 7^2 = 49 *
178 6 13^2 = 169 *
468 5 11^2 = 121
541 9 23^2 = 529 *
854 10 29^2 = 841 *
1454 7 17^2 = 289
2099 8 19^2 = 361
3744 18 61^2 = 3721 *
7330 11 31^2 = 961
9091 12 37^2 = 1369
10138 13 41^2 = 1681
11917 29 109^2 = 11881
14154 14 43^2 = 1849
14350 15 47^2 = 2209
19363 34 139^2 = 19321
21555 16 53^2 = 2809
23553 17 59^2 = 3481
26615 38 163^2 = 26569
36109 21 73^2 = 5329
36533 43 191^2 = 36481
37302 44 193^2 = 37249
51588 49 227^2 = 51529
52576 20 71^2 = 5041
57183 52 239^2 = 57121
58064 19 67^2 = 4489
58144 53 241^2 = 58081
63067 54 251^2 = 63001
s = {1}; nn = 4000; r = 1; u = v = 2; c[_] = False;
MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, True}] &, s];
While[c[u], u++]; While[Or[c[v], CompositeQ[v]], v++];
Monitor[Reap[
Do[k = u; q = Product[a[h], {h, n - Min[k, n - 1], n - 1}];
While[Or[c[k], ! CoprimeQ[k, q]],
If[k > n - 1, k = v; q = Product[a[i], {i, r}],
k++; q *= a[n - k] ] ];
Set[{a[n], c[k]}, {k, True}];
If[And[PrimeQ[k], # > r], r = #] &[PrimePi[k]];
If[PrimeQ@ Sqrt[k], Sow[n]];
If[k == u, While[c[u], u++]];
If[k == v, While[Or[c[v], CompositeQ[v]], v++]],
{n, Length[s] + 1, nn}] ][[-1, 1]], n]
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