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A381095 Indices of prime squares in A381019.

%I A381095 #15 Apr 01 2025 03:28:20
%S A381095 7,13,30,55,178,468,541,854,1454,2099,3744,7330,9091,10138,11917,
%T A381095 14154,14350,19363,21555,23553,26615,36109,36533,37302,51588,52576,
%U A381095 57183,58064,58144,63067,69927,70135,80174,81920,85923,89936,93749,99240,121884,124693,151411
%N A381095 Indices of prime squares in A381019.
%C A381095 Let S = A381019.
%C A381095 Observation: S(n) < n for S(n) = prime(i)^2 for some i.
%e A381095 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).
%e A381095       n    i             S(n)
%e A381095   --------------------------
%e A381095       7    1     2^2 =     4 *
%e A381095      13    2     3^2 =     9 *
%e A381095      30    3     5^2 =    25 *
%e A381095      55    4     7^2 =    49 *
%e A381095     178    6    13^2 =   169 *
%e A381095     468    5    11^2 =   121
%e A381095     541    9    23^2 =   529 *
%e A381095     854   10    29^2 =   841 *
%e A381095    1454    7    17^2 =   289
%e A381095    2099    8    19^2 =   361
%e A381095    3744   18    61^2 =  3721 *
%e A381095    7330   11    31^2 =   961
%e A381095    9091   12    37^2 =  1369
%e A381095   10138   13    41^2 =  1681
%e A381095   11917   29   109^2 = 11881
%e A381095   14154   14    43^2 =  1849
%e A381095   14350   15    47^2 =  2209
%e A381095   19363   34   139^2 = 19321
%e A381095   21555   16    53^2 =  2809
%e A381095   23553   17    59^2 =  3481
%e A381095   26615   38   163^2 = 26569
%e A381095   36109   21    73^2 =  5329
%e A381095   36533   43   191^2 = 36481
%e A381095   37302   44   193^2 = 37249
%e A381095   51588   49   227^2 = 51529
%e A381095   52576   20    71^2 =  5041
%e A381095   57183   52   239^2 = 57121
%e A381095   58064   19    67^2 =  4489
%e A381095   58144   53   241^2 = 58081
%e A381095   63067   54   251^2 = 63001
%t A381095 s = {1}; nn = 4000; r = 1; u = v = 2; c[_] = False;
%t A381095 MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, True}] &, s];
%t A381095 While[c[u], u++]; While[Or[c[v], CompositeQ[v]], v++];
%t A381095 Monitor[Reap[
%t A381095   Do[k = u; q = Product[a[h], {h, n - Min[k, n - 1], n - 1}];
%t A381095     While[Or[c[k], ! CoprimeQ[k, q]],
%t A381095       If[k > n - 1, k = v; q = Product[a[i], {i, r}],
%t A381095         k++; q *= a[n - k] ] ];
%t A381095     Set[{a[n], c[k]}, {k, True}];
%t A381095     If[And[PrimeQ[k], # > r], r = #] &[PrimePi[k]];
%t A381095     If[PrimeQ@ Sqrt[k], Sow[n]];
%t A381095     If[k == u, While[c[u], u++]];
%t A381095     If[k == v, While[Or[c[v], CompositeQ[v]], v++]],
%t A381095   {n, Length[s] + 1, nn}] ][[-1, 1]], n]
%Y A381095 Cf. A001248, A381019, A381116, A381119.
%K A381095 nonn
%O A381095 1,1
%A A381095 _Michael De Vlieger_, Feb 16 2025
%E A381095 More terms from _Jinyuan Wang_, Feb 25 2025