A274605 Numbers k such that there is no prime between the k-th perfect power and the (k+1)-st perfect power.

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

Offset: 1

Author

Felix Fröhlich, Jun 30 2016

Keywords

• nonn
• more

Comments

The perfect powers are given in A001597.

Numbers k such that A080769(k) = 0.

Index of A116086(n) in A001597.

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

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

Examples

			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.
		

Links

• Wikipedia, Redmond-Sun conjecture.

Crossrefs

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

Programs

Mathematica

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 *)

PARI

a001597(n) = my(i=0, k=0); while(1, if(ispower(k) || k==1, i++); if(i==n, return(k)); k++)
a080769(n) = primepi(a001597(n+1))-primepi(a001597(n))
is(n) = a080769(n)==0

Formula

a(n) = A069623(A116086(n)). - Pontus von Brömssen, Nov 05 2024