A381095 Indices of prime squares in A381019.
7, 13, 30, 55, 178, 468, 541, 854, 1454, 2099, 3744, 7330, 9091, 10138, 11917, 14154, 14350, 19363, 21555, 23553, 26615, 36109, 36533, 37302, 51588, 52576, 57183, 58064, 58144, 63067, 69927, 70135, 80174, 81920, 85923, 89936, 93749, 99240, 121884, 124693, 151411
Offset: 1
Keywords
Examples
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
Programs
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Mathematica
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]
Extensions
More terms from Jinyuan Wang, Feb 25 2025
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