cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A097056 Numbers n such that the interval n^2 < x < (n+1)^2 contains two or more distinct nonsquare perfect powers A097054.

Original entry on oeis.org

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

Views

Author

Hugo Pfoertner, Jul 21 2004

Keywords

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

Examples

			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.

Links

Crossrefs

Cf. A000290, A097054, A097055.
Cf. A173341 (q=5), A173342 (q=7): y with a(n)^2 < y^q < (a(n)+1)^2.

Programs

  • 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

Extensions

a(5)-a(20) from Don Reble
a(21)-a(26) from David Wasserman, Dec 17 2007

A173341 Numbers n such that n^5 and a cube are between consecutive squares.

Original entry on oeis.org

2, 23, 199, 201, 408, 575, 603, 1354, 1628, 4995, 5745, 7320, 7994, 12634, 42637, 44217, 45962, 67132, 82131, 82351, 91116, 91134, 146521, 177682, 229863, 359373, 394826, 458908, 462763, 512012, 665719, 728982, 1009965, 1156978, 1450803
Offset: 1

Views

Author

T. D. Noe, Feb 16 2010

Keywords

Comments

This sequence appears to be infinite. Sequence A117594 is a subsequence. The corresponding sequence for n^7 is A173342. Are there ever more than two perfect powers between consecutive squares?

Examples

			2 is here because 2^5=32 and 3^3=27 are between 5^2=25 and 6^2=36.
23 is here because 23^5 and 186^3 are between 2536^2 and 2537^2.
199 is here because 199^5 and 6783^3 are between 558640^2 and 558641^2.

Crossrefs

A097056, A117896

Programs

  • Mathematica
    t={}; Do[n2=Floor[n^(5/2)]; n3=Round[n^(5/3)]; If[n2^2

A173353 Numbers x such that 0 < |x^7 - y^3| < x^(11/3) for some number y.

Original entry on oeis.org

2, 3, 32, 33, 34, 88, 442, 498, 942, 2266144, 12527271, 20160899, 272585923
Offset: 1

Views

Author

T. D. Noe, Feb 22 2010

Keywords

Comments

No additional terms < 10^7. See A173348 for more information.
a(14) > 10^9. - Robert Price, Apr 20 2021

Crossrefs

Cf. A173342 (n^7 and a cube are between consecutive squares)
Cf. A078933, A116884, A173348-A173379.

Programs

  • Mathematica
    Solutions[7,3,10^7] (* see A173348 *)

Extensions

a(11)-a(13) from Robert Price, Apr 20 2021
Showing 1-3 of 3 results.

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