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%I #18 Jun 16 2020 03:05:55
%S 4,1,13,229,3259,15739,449569,6958841,130259903,2404517671,
%T 56014949761,538155413969
%N a(n) is the least k such that A335294(k) = n, or -1 if no such k exists.
%C Conjecture: a(n) > 0 for all n.
%H Altug Alkan, Andrew R. Booker, and Florian Luca, <a href="https://arxiv.org/abs/2006.08013">On a recursively defined sequence involving the prime counting function</a>, arXiv:2006.08013 [math.NT], 2020.
%t upto[nn_] := Block[{A = <|1->1|>, v, i=0, is=0, sp=2, p=2, s=1}, Print@{1, 1}; Do[ If[n == p, i++; p = NextPrime@ p]; v = i-is; If[! KeyExistsQ[A, v], A[v] = n; Print[{n, v}]]; s += v; While[s >= sp, is++; sp = NextPrime@ sp], {n, 2, nn}]; Last /@ Sort@ Normal@ A]; upto[450000] (* _Giovanni Resta_, Jun 02 2020 *)
%o (PARI) a(n) = { my (pn=0, ps=0, s=0, v); for (k=1, oo, pn+=isprime(k); v=if (k==1, 1, pn - ps); if (v==n, return (k)); ps+=sum(k=s+1, s+=v, isprime(k))) } \\ _Rémy Sigrist_, Jun 02 2020
%Y Cf. A335294.
%K nonn,more
%O 0,1
%A _Altug Alkan_, Jun 02 2020
%E a(7)-a(11) from _Giovanni Resta_, Jun 02 2020
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