%I #33 Mar 26 2026 09:10:58

%S 1,32144,37000,111616,382360

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

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

%H René-Louis Clerc, <a href="https://zenodo.org/records/19018760">Quelques nombres de Niven-Harshad particuliers</a>, pp. 1-17, zenodo.19018760, 2026.

%H René-Louis Clerc, <a href="https://zenodo.org/records/19020223">Nombres S+P, maxSP, minSP et |P-S|</a>, zenodo.19020223, 2026. (In French)

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

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

%o (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,", "))))))))))}

%o (PARI) isok(k) = my(d=digits(k)); vecsum(d)^2*sum(i=1, #d, d[i]^3) == k; \\ _Michel Marcus_, Oct 12 2023

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

%K nonn,base,fini,full

%O 1,2

%A _René-Louis Clerc_, Oct 11 2023