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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. 7
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.

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(4)=2536: 2536^2<x<2537^2 (6431296,6436369) contains 23^5=6436343 and 186^3=6434856.

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

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. A000290, A097054, A097055.

Sequence in context: A222368 A276300 A222476 * A092358 A079029 A106953

Adjacent sequences:  A097053 A097054 A097055 * A097057 A097058 A097059

KEYWORD

nonn

AUTHOR

Hugo Pfoertner, Jul 21 2004

EXTENSIONS

a(5)-a(20) from Don Reble (djr(AT)nk.ca)

a(21)-a(26) from David Wasserman, Dec 17 2007

STATUS

approved

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Last modified August 19 21:13 EDT 2017. Contains 290821 sequences.