A097056 - OEIS

%I #23 Jan 22 2021 17:18:26

%S 5,11,46,2536,558640,572783,3362407,7928108,8928803,67460050,

%T 106938971,1763350849,2501641555,2756149047,4584349318,5713606932,

%U 17941228664,375376083513,411124334926,452894760105,1167680330892,1933159894790,1946131548918,2506032014606,2507269866902,8217688694093

%N Numbers n such that the interval n^2 < x < (n+1)^2 contains two or more distinct nonsquare perfect powers A097054.

%C Empirically, there seem to be no intervals between consecutive squares containing more than two nonsquare perfect powers.

%C It is easy to see that two distinct powers between n^2 and (n+1)^2 are necessarily of the form x^p and y^q where p, q are distinct odd primes. Among the first 180 terms, only 4 are of type (p,q) = (3,7) and all others are of type (3,5). The first term with q = 11, if it exists, is > (1e6)^(11/2) = 1e33. - _M. F. Hasler_, Jan 18 2021

%H T. D. Noe, <a href="/A097056/b097056.txt">Table of n, a(n) for n = 1..180</a> (using the b-file from A117934)

%e a(1) = 5: 5^2 < 3^3 < 2^5 < 6^2,

%e a(2) = 11: 11^2 < 5^3 < 2^7 < 12^2,

%e a(3) = 46: 46^2 = 2116 < 3^7 = 2187 < 13^3 = 2197 < 47^2 = 2209.

%e a(4) = 2536: 2536^2 = 6431296 < 186^3 = 6434856 < 23^5 = 6436343 < 2537^2 = 6436369.

%e 22 is not in the sequence because 2^9 and 8^3 (22^2 < 512 < 23^2) are not distinct.

%e Also, 181 is not listed since between 181^2 and 182^2 there is only 32^3 = 8^5.

%o (PARI) is(n)=my(s,t); forprime(p=3,2*log(n+1.5)\log(2), t=floor((n+1)^(2/p)); if(t^p>n^2 && !ispower(t) && s++ > 1, return(1))); 0 \\ _Charles R Greathouse IV_, Dec 11 2012

%o (PARI) haspow(lower,upper,eMin,eMax)=if(sqrtnint(upper,3)^3>lower, return(1)); forprime(e=eMin,eMax, if(sqrtnint(upper,e)^e>lower, return(1))); 0

%o list(lim)=lim\=1; my(v=List(),M=(lim+1)^2,L=logint(M,2),s); forprime(e=5,L, forprime(p=2,sqrtnint(M,e), s=sqrtint(p^e); if(haspow(s^2,(s+1)^2-1,e+1,L) && s<=lim, listput(v,s)))); Set(v) \\ _Charles R Greathouse IV_, Nov 05 2015

%Y Cf. A000290, A097054, A097055.

%Y Cf. A173341 (q=5), A173342 (q=7): y with a(n)^2 < y^q < (a(n)+1)^2.

%K nonn

%O 1,1

%A _Hugo Pfoertner_, Jul 21 2004

%E a(5)-a(20) from _Don Reble_

%E a(21)-a(26) from _David Wasserman_, Dec 17 2007