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Search: id:A117934
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| A117934 |
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Perfect powers that are close, that is, between consecutive squares. |
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+0
2
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| 27, 32, 125, 128, 2187, 2197, 6434856, 6436343, 312079600999, 312079650687, 328080401001, 328080696273, 11305786504384, 11305787424768, 62854898176000, 62854912109375, 79723529268319, 79723537443243, 4550858390629024
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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It appears that all pairs of close powers involve a cube. For three pairs, the other power is a 7th power. For all remaining pairs, the other power is a 5th power. If this is true, then three powers are never close.
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EXAMPLE
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27 and 32 are close because they are between 25 and 36.
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MATHEMATICA
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nMax=10^14; lst={}; log2Max=Ceiling[Log[2, nMax]]; bases=Table[2, {log2Max}]; powers=bases^Range[log2Max]; powers[[1]]=Infinity; currPP=1; cnt=0; While[nextPP=Min[powers]; nextPP <= nMax, pos=Flatten[Position[powers, nextPP]]; If[MemberQ[pos, 2], cnt=0, cnt++ ]; If[cnt>1, AppendTo[lst, {currPP, nextPP}]]; Do[k=pos[[i]]; bases[[k]]++; powers[[k]]=bases[[k]]^k, {i, Length[pos]}]; currPP=nextPP]; Flatten[lst]
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CROSSREFS
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Cf. A117896 (number of perfect powers between consecutive squares n^2 and (n+1)^2).
Sequence in context: A095387 A031408 A144862 this_sequence A030134 A024796 A025322
Adjacent sequences: A117931 A117932 A117933 this_sequence A117935 A117936 A117937
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KEYWORD
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nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Apr 03 2006
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