A366512 Numbers k such that the square of the sum of the digits times the sum of the cubes of the digits equals k.

1, 32144, 37000, 111616, 382360

Offset: 1

Author

René-Louis Clerc, Oct 11 2023

Keywords

• nonn
• base
• fini
• full

Comments

There are exactly 5 such numbers (Property 14 of Clerc).

Examples

			32144 = ((3+2+1+4+4)^2)*(3^3 + 2^3 + 1^3 + 4^3 + 4^3) = 196*164.
		

Links

• René-Louis Clerc, Quelques nombres de Niven-Harshad particuliers, pp. 1-17, hal-04235744, 2023.
• René-Louis Clerc, Nombres S+P, maxSP, minSP et |P-S|, hal-04507547 [math.nt], 2024. (In French)

Crossrefs

Cf. A115518, A257766, A061209, A061210, A254000, A130680.

Programs

Mathematica

Select[Range[10^6], #1 == Total[#2]^2*Total[#2^3] & @@ {#, IntegerDigits[#]} &] (* Michael De Vlieger, Mar 25 2024 *)

PARI

niven23()={for(a=0,9,for(b=0,9,for(c=0,9,for(d=0,9,for(e=0,9,for(f=0,9,for(g=0,9,for(h=0,9,if((a+b+c+d+e+f+g+h)^2*(a^3+b^3+c^3+d^3+e^3+f^3+g^3+h^3)==10000000*a+1000000*b+100000*c+10000*d+1000*e+100*f+10*g+h,print1(10000000*a+1000000*b+100000*c+10000*d+1000*e+100*f+10*g+h,", "))))))))))}

PARI

isok(k) = my(d=digits(k)); vecsum(d)^2*sum(i=1, #d, d[i]^3) == k; \\ Michel Marcus, Oct 12 2023