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Numbers k such that there is no prime between the k-th perfect power and the (k+1)-st perfect power.
9

%I #28 Nov 06 2024 04:34:46

%S 3,6,8,14,60,72,216,328,361,23268

%N Numbers k such that there is no prime between the k-th perfect power and the (k+1)-st perfect power.

%C The perfect powers are given in A001597.

%C Numbers k such that A080769(k) = 0.

%C Index of A116086(n) in A001597.

%C Numbers k such that A000720(A001597(k)) = A000720(A001597(k+1)).

%C It has been conjectured that 23268 is the last term of the sequence.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Redmond-Sun_conjecture">Redmond-Sun conjecture</a>.

%F a(n) = A069623(A116086(n)). - _Pontus von Brömssen_, Nov 05 2024

%e 25 is the 6th perfect power, i.e., 25 = A001597(6), and there is no prime between 25 and the next larger perfect power A001597(7) = 27, so 25 is a term of A116086, and thus 6 is a term of this sequence.

%t Position[Count[#, _?PrimeQ] & /@ Range @@@ # &@ Partition[#, 2, 1] &@ Select[Range[10^5], # == 1 || GCD @@ FactorInteger[#][[All, 2]] > 1 &], 0] // Flatten (* _Michael De Vlieger_, Jun 30 2016 *)

%o (PARI) a001597(n) = my(i=0, k=0); while(1, if(ispower(k) || k==1, i++); if(i==n, return(k)); k++)

%o a080769(n) = primepi(a001597(n+1))-primepi(a001597(n))

%o is(n) = a080769(n)==0

%Y Cf. A000720, A001597, A069623, A080769, A116086.

%K nonn,more

%O 1,1

%A _Felix Fröhlich_, Jun 30 2016