A328198 Numbers of the form N = a+b+c such that N^3 = concat(a,b,c); a, b, c > 0.

8, 45, 1611, 4445, 4544, 4949, 5049, 5455, 5554, 7172, 19908, 55556, 60434, 77778, 422577, 427868, 461539, 478115, 488214, 494208, 543752, 559846, 598807, 664741, 757835, 791505, 807598, 4927940, 5555555, 6183170, 25252524, 27272728, 27282727, 28201724, 30731977

Offset: 1

Author

M. F. Hasler, Oct 07 2019

Keywords

• nonn
• base

Comments

A variant of Kaprekar and pseudo-Kaprekar triples, cf. A006887 and A060768.

Leading zeros as in A006887(4), 26198073 = (26+198+073)^3, are not allowed here.

Is it a coincidence that a(2)^3 = 91125 also verifies sqrt(91125) = 9*sqrt(1125)?

See A328199 for the triples (a,b,c) and A328200 for the cubes / concatenations.

Examples

			5 + 1 + 2 = 512^(1/3) = 8,
9 + 11 + 25 = 91125^(1/3) = 45,
418 + 1062 + 131 = (4181062131)^(1/3) = 1611, ...
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..239 (terms < 10^12)
• Números y algo mas, 9 + 11 + 25 = 91125^(1/3) etc, post on facebook.com, Sep 30 2019.

Crossrefs

Cf. A328199 (corresponding a,b,c), A328200 (cubes / concatenations), A006887 & A291461 (Kaprekar numbers), A060768 (pseudo Kaprekar numbers); A000578 (the cubes), A055642 (number of digits of n).

Programs

PARI

is(n,Ln=A055642(n),n3=n^3,Ln3=A055642(n3))={my(ab,c); for(Lc=Ln3-2*Ln,Ln, [ab,c]=divrem(n3, 10^Lc); n-c<10^(Ln-1) || c < 10^(Lc-1) || for( Lb=Ln3-Ln-Lc,Ln, vecsum(divrem(ab,10^Lb)) == n-c && ab%10^Lb>=10^(Lb-1)&& return(1)))} \\ A055642(n)=logint(n,10)+1 = #digits(n)
for( Ln=1,oo, for( n=10^(Ln-1),10^Ln-1, is(n,Ln)&& print1(n", ")))

Extensions

a(31)-a(35) from Giovanni Resta, Oct 09 2019