A382712 Regarding A381019 as a permutation of the natural numbers, this is the cycle with smallest term 8, read in the reverse direction.
8, 16, 64, 975, 126300
Offset: 1
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A381019(47) = 10 = 2*prime(3), so a(3) = 47.
7643 is a term, since 75, 7643 and 58 are three consecutive terms of A381019, where A381019(1033) = 7643 is prime, while A381019(1032) = 75 and A381019(1034) = 58 are both composite numbers.
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]
For n = 2, we observe that 9, 31, 37, and 8 are four consecutive terms of A381019, where 31 and 37 are exactly two consecutive primes and represent the first occurrence of two consecutive terms that are prime. So, a(2) = 31.
The number a(25) = 6 shares a factor with a(16) = 8, and therefore must be at "distance" > 8 (i.e., separated by 8 relatively prime terms) from a(16). This is the first example where the smaller of two terms sharing a common factor occurs after the larger one.
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