A217297 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.

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

Offset

1,1

Comments

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

Links

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Example

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

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A014612 ("triprimes"), A225082, A080603, A000578, A339578.

Sequence in context: A179171 A204876 A204869 * A205198 A205194 A204868

Adjacent sequences: A217294 A217295 A217296 * A217298 A217299 A217300

Keyword

nonn,base,less

Author

Kevin L. Schwartz and Christian N. K. Anderson, May 03 2013

Extensions

Edited by N. J. A. Sloane, Dec 19 2020

Status

approved