A364601 Numbers m such that, if k is the number of digits of m, then for some r > 1, the sum of the k-th powers of the digits of m^r is equal to m.

1, 7, 8, 9, 180, 205, 38998, 45994, 89080, 726191, 5540343, 7491889, 8690141, 167535050, 749387107, 9945245922

Offset: 1

Author

René-Louis Clerc, Jul 29 2023

Keywords

• nonn
• base
• fini
• more

Comments

Corresponding r's: any, 4, 3, 2, 6, 2, 2, 2, 2, 2, 3, 2, 3, 3, 4, 3.

Examples

			180 with r=6 satisfies: 180^6 = 34012224000000, 3^3 + 4^3 + 1 + 2^3 + 2^3 + 2^3 + 4^3 = 180.
		

Links

• René-Louis Clerc, Pleasant transformations and a new class of narcissistic perfect numbers, hal-03619147, 2022-2023.
• René-Louis Clerc, The Perfect R-Narcissistic Numbers, 2023.
• René-Louis Clerc, Perfect r-narcissistic numbers in any base, hal-04376934, 2024.
• René-Louis Clerc, Nombres S+P, maxSP, minSP et |P-S|, hal-04507547 [math.nt], 2024. (In French)

Crossrefs

Cf. A005188 (Armstrong's numbers, case r=1 in our terminology).

Cf. A066003, A066004, A065999 (for terms 7, 8 and 9).

Programs

PARI

SomP(n,p)={resu=0;for(i=1,#digits(n),resu+=(digits(n)[i])^p);resu}
Ppdi(k,r)={for(n=10^(k-1),10^k,if(SomP(n^r,k)==n,print1(n,";")))}