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A217297
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Triprimes (numbers that are a product of exactly three primes: A014612) that become cubes when their central digit or central pair of digits is deleted.
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2
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207, 604, 654, 2007, 2037, 2057, 2067, 2097, 2107, 2197, 2247, 2337, 2367, 2387, 2397, 2527, 2547, 2597, 2607, 2637, 2667, 2697, 2717, 2737, 2817, 2847, 2877, 2937, 2967, 6014, 6034, 6044, 6054, 6094, 6114, 6124, 6154, 6194, 6214, 6234, 6254, 6284, 6294, 6394
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OFFSET
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1,1
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COMMENTS
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If a number (a product of exactly three primes) has an odd number of digits, only its central digit is deleted to test for status as a cube; if such a number has an even number of digits, its two central digits are deleted to test whether that's a cube. - Harvey P. Dale, Dec 19 2020
In theory, a cube with an even number of digits could be represented in the sequence by up to 110 numbers by inserting {0,1,...,9} and {00,01,...,99}. In the first 10000 terms, 1079^3 has a record 46 representatives, though it is unlikely that this is a global record.
The cubes of 10, 20 and 48 are the first three cubes not represented in the sequence.
It would be nice to have a proof that this sequence is infinite. - N. J. A. Sloane, Dec 19 2020
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LINKS
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EXAMPLE
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207 = 3*3*23 is a term: it becomes the cube 27 when the central digit is deleted.
2007 = 3*3*223 is a term: it becomes the cube 27 when the two central digits are deleted.
Here is a larger example taken at random from the b-file:
4178131923 = (3) (7) (198958663)
Delete the central pair of digits and we get a cube: 41781923 = 347^3. - N. J. A. Sloane, Dec 19 2020
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MATHEMATICA
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cdn[n_]:=Module[{idn=IntegerDigits[n], len}, len=Length[idn]; If[OddQ[ len], FromDigits[ Drop[idn, {(len+1)/2}]], FromDigits[Drop[idn, {len/2, len/2+1}]]]]; Select[Range[100, 100000], PrimeOmega[#]==3 && IntegerQ[ Surd[ cdn[#], 3]]&] (* Harvey P. Dale, Dec 19 2020 *)
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PROG
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(R)library(gmp);
removecentraldigit<-function(x) { s=as.character(x); n=nchar(s);
as.bigz(paste(substr(s, 1, ifelse(n%%2==0, n/2-1, (n-1)/2)), substr(s, ifelse(n%%2==0, n/2+2, (n+3)/2), n), sep=""))};
istriprime=function(x) ifelse(as.bigz(x)<8, F, length(factorize(x))==3);
iscube<-function(x) ifelse(as.bigz(x)<2, T, all(table(as.numeric(factorize(x)))%%3==0));
which(sapply(201:6400, function(x) istriprime(x) & iscube(removecentraldigit(x))))+200
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CROSSREFS
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KEYWORD
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nonn,base,less
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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