A217297 - OEIS

%I #48 Dec 20 2020 06:09:49

%S 207,604,654,2007,2037,2057,2067,2097,2107,2197,2247,2337,2367,2387,

%T 2397,2527,2547,2597,2607,2637,2667,2697,2717,2737,2817,2847,2877,

%U 2937,2967,6014,6034,6044,6054,6094,6114,6124,6154,6194,6214,6234,6254,6284,6294,6394

%N 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.

%C 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

%C 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.

%C The cubes of 10, 20 and 48 are the first three cubes not represented in the sequence.

%C It would be nice to have a proof that this sequence is infinite. - _N. J. A. Sloane_, Dec 19 2020

%H Christian N. K. Anderson, <a href="/A217297/b217297.txt">Table of n, a(n) for n = 1..10000</a>

%e 207 = 3*3*23 is a term: it becomes the cube 27 when the central digit is deleted.

%e 2007 = 3*3*223 is a term: it becomes the cube 27 when the two central digits are deleted.

%e Here is a larger example taken at random from the b-file:

%e              4178131923 = (3)  (7)  (198958663)

%e Delete the central pair of digits and we get a cube: 41781923 = 347^3. - _N. J. A. Sloane_, Dec 19 2020

%t 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 *)

%o (R)library(gmp);

%o removecentraldigit<-function(x) { s=as.character(x); n=nchar(s);

%o     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=""))};

%o istriprime=function(x) ifelse(as.bigz(x)<8,F,length(factorize(x))==3);

%o iscube<-function(x) ifelse(as.bigz(x)<2,T,all(table(as.numeric(factorize(x)))%%3==0));

%o which(sapply(201:6400, function(x) istriprime(x) & iscube(removecentraldigit(x))))+200

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

%K nonn,base,less

%O 1,1

%A _Kevin L. Schwartz_ and _Christian N. K. Anderson_, May 03 2013

%E Edited by _N. J. A. Sloane_, Dec 19 2020