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A097056
Numbers n such that the interval n^2 < x < (n+1)^2 contains two or more distinct nonsquare perfect powers A097054.
9
5, 11, 46, 2536, 558640, 572783, 3362407, 7928108, 8928803, 67460050, 106938971, 1763350849, 2501641555, 2756149047, 4584349318, 5713606932, 17941228664, 375376083513, 411124334926, 452894760105, 1167680330892, 1933159894790, 1946131548918, 2506032014606, 2507269866902, 8217688694093
OFFSET
1,1
COMMENTS
Empirically, there seem to be no intervals between consecutive squares containing more than two nonsquare perfect powers.
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
LINKS
T. D. Noe, Table of n, a(n) for n = 1..180 (using the b-file from A117934)
EXAMPLE
a(1) = 5: 5^2 < 3^3 < 2^5 < 6^2,
a(2) = 11: 11^2 < 5^3 < 2^7 < 12^2,
a(3) = 46: 46^2 = 2116 < 3^7 = 2187 < 13^3 = 2197 < 47^2 = 2209.
a(4) = 2536: 2536^2 = 6431296 < 186^3 = 6434856 < 23^5 = 6436343 < 2537^2 = 6436369.
22 is not in the sequence because 2^9 and 8^3 (22^2 < 512 < 23^2) are not distinct.
Also, 181 is not listed since between 181^2 and 182^2 there is only 32^3 = 8^5.
PROG
(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
(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
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
CROSSREFS
Cf. A173341 (q=5), A173342 (q=7): y with a(n)^2 < y^q < (a(n)+1)^2.
Sequence in context: A333114 A276300 A222476 * A092358 A079029 A106953
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Jul 21 2004
EXTENSIONS
a(5)-a(20) from Don Reble
a(21)-a(26) from David Wasserman, Dec 17 2007
STATUS
approved