OFFSET
1,1
COMMENTS
This is a kind of inverse to A217297.
From David A. Corneth, Dec 20 2020: (Start)
If a(n) > 100 then gcd(a(n), 10) = 1. Proof: Suppose gcd(a(n), 10)) > 1 then a(n) is divisible by 2 or 5. As the last three digits of m (as defined in name) will have the last three digits such that m is divisible by 2^3 = 8 or 5^3 = 125 we have a number that is a product of strictly more than three primes. Contradiction.
Furthermore, a(n)^3 has an even number of digits as after deletion of the middle digits there always is an even number of digits left. (End)
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000
EXAMPLE
3 is a term because 3^3 = 27 can be obtained by deleting the central digit of 3*3*23 = 207.
4 is a term because 4^3 = 64 can be obtained by deleting the central pair of digits of 2*31*97 = 6014.
20 is not a term because 20^3 = 8000, and any candidate for m will end in 00, and therefore will have at least four prime factors.
MATHEMATICA
p3oQ[x_]:=AnyTrue[Table[With[{td=TakeDrop[IntegerDigits[x], IntegerLength[ x]/2]}, FromDigits[ Flatten[ Join[ {td[[1]], {v}, td[[2]]}]]]], {v, 0, 9}], PrimeOmega[#] == 3&]; p3eQ[x_]:= AnyTrue[ Table[With[ {td=TakeDrop[ IntegerDigits[x], IntegerLength[ x]/2]}, FromDigits[Flatten[ Join[ {td[[1]], PadLeft[{w}, 2], td[[2]]}]]]], {w, 0, 99}], PrimeOmega[#]==3&]; p3Q[m_]:=p3oQ[m] || p3eQ[m]; Module[{rng={Ceiling[Reduce[k^3>#, k][[-1]]], Floor[Reduce[k^3<10#, k][[-1]]] }&/@ Table[10^(2n-1), {n, 4}]}, Surd[#, 3]&/@Select[Flatten[Table[w^3, {w, #[[1]], #[[2]]}]&/@ rng], p3Q]] // Quiet (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 20 2020 *)
PROG
(PARI) is(n) = { my(d = digits(n^3), half); if(#d % 2 == 1, return(0)); half = #d \ 2; left = vector(half, i, d[i]); right = vector(half, i, d[i + half]); for(j = 1, 2, c = fromdigits(left) * 10^(half + j) + fromdigits(right); for(i = 0, 10^j - 1, if(bigomega(c + i*10^half) == 3, print(c + i*10^half); return(1) ) ) ); 0 } \\ David A. Corneth, Dec 20 2020
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Harvey P. Dale and N. J. A. Sloane, Dec 20 2020
STATUS
approved