|
| |
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
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
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
