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A308617 Integers i such that the equation A088387(i) = p has N > 1 solutions in the interval prevprime(i)..nextprime(i).

%I A308617 #81 Jun 20 2024 18:50:13
%S A308617 140,147,621,630,2184,2197,2511,2520,3230,3249,3740,3757,4180,4199,
%T A308617 5750,5775,9975,10000,19635,19652,26600,26625,30600,30625,40040,40053,
%U A308617 43355,43384,45900,45927,50232,50255,50600,50625,64515,64538,67320,67337,68400,68425
%N A308617 Integers i such that the equation A088387(i) = p has N > 1 solutions in the interval prevprime(i)..nextprime(i).
%C A308617 Conjecture: N = 2. Checked up to 10^8.
%H A308617 Robert Israel, <a href="/A308617/b308617.txt">Table of n, a(n) for n = 1..1148</a>
%e A308617 Between primes 139 and 149: A088387(140) = A088387(147) = 7.
%e A308617 Between primes 619 and 631: A088387(621) = A088387(630) = 3.
%e A308617 Between primes 8752871 and 8752987: A088387(8752880) = A088387(8752951) = 71 and A088387(8752926) = A088387(8752967) = 41.
%e A308617 Between primes 33622489 and 33622607: A088387(33622507) = A088387(33622600) = 31.
%p A308617 A88387:= proc(n) local F,j;
%p A308617   F:= ifactors(n)[2];
%p A308617   F[max[index](map(t -> t[1]^t[2],F)),1]
%p A308617 end proc:
%p A308617 R:= NULL: count:= 0:
%p A308617 q:= 2:
%p A308617 while count < 100 do
%p A308617   p:= nextprime(q);
%p A308617   L:= [$(q+1)..(p-1)];
%p A308617   V:= map(A88387,L);
%p A308617   S:= select(t -> numboccur(t,V) > 1, convert(V,set));
%p A308617   J:= select(i -> member(V[i],S),[$1..p-q-1]);
%p A308617   count:= count+nops(J);
%p A308617   R:= R, op(L[J]);
%p A308617   q:= p;
%p A308617 od:
%p A308617 R; # _Robert Israel_, Jun 20 2024
%t A308617 A088387[n_] := MaximalBy[FactorInteger[n], Power @@ # &][[1, 1]]; A034699[n_] := If[n == 1, 1, Max[#[[1]]^#[[2]] & /@ FactorInteger@n]]; t = Table[Table[A088387[n],{n, Prime[k], Prime[k + 1]-1}], {k, 2,12000}  ]; duplicates = Select[t, Not@DuplicateFreeQ[#] &]; a = {}; pickFrom[list_] := Do[If[Count[list, list[[k]]] > 1 , a = Append[a, k - 1 + First[list]]], {k, 2, Length[list]}]; pickFrom /@ duplicates; a (* _Jianglin Luo_, Dec 01 2023 *)
%o A308617 (MATLAB)
%o A308617 n = 0; ip = 0;
%o A308617 for m = 1:oo
%o A308617 if isprime(m) ip = ip + 1; end
%o A308617 if A088387(m) == m & m > 1
%o A308617   for i = A007917(ip):A007918(ip)
%o A308617    for j = A007917(ip):A007918(ip)
%o A308617     if A088387(i) == A088387(j) & i ~= j
%o A308617      n = n + 1; a(n) = i;
%o A308617     end
%o A308617    end
%o A308617   end
%o A308617 end
%o A308617 end
%o A308617 (PARI) plppf(n) = if(1==n, 1, my(f=factor(n), p=0); isprimepower(vecmax(vector(#f[, 1], i, f[i, 1]^f[i, 2])), &p); (p)); \\ A088387
%o A308617 lista(nn) = {for (n=1, nn, my(p = prime(n), q = nextprime(p+1)); my(v = vector(q-p-1, k, plppf(k+p)), vs = vecsort(v,,8)); if (#v != #vs, for (i=1, #vs, my(vx = select(x->(x==vs[i]), v, 1)); if (#vx > 1, for (j=1, #vx, print1(p+vx[j], ", "));););););} \\ _Michel Marcus_, Jun 27 2019
%Y A308617 Cf. A007917 (prevprime), A007918 (nextprime), A088387, A034699, A308752 (analog), A038610.
%K A308617 nonn
%O A308617 1,1
%A A308617 _I. V. Serov_ & _Michel Marcus_, Jun 25 2019