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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.

%I A364601 #58 Apr 24 2024 14:54:17
%S A364601 1,7,8,9,180,205,38998,45994,89080,726191,5540343,7491889,8690141,
%T A364601 167535050,749387107,9945245922
%N 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.
%C A364601 Corresponding r's: any, 4, 3, 2, 6, 2, 2, 2, 2, 2, 3, 2, 3, 3, 4, 3.
%H A364601 René-Louis Clerc, <a href="https://hal.science/hal-03619147">Pleasant transformations and a new class of narcissistic perfect numbers</a>, hal-03619147, 2022-2023.
%H A364601 René-Louis Clerc, <a href="https://hal.science/hal-04229895">The Perfect R-Narcissistic Numbers</a>, 2023.
%H A364601 René-Louis Clerc, <a href="https://hal.science/hal-04376934">Perfect r-narcissistic numbers in any base</a>, hal-04376934, 2024.
%H A364601 René-Louis Clerc, <a href="https://ut3-toulouseinp.hal.science/hal-04507547">Nombres S+P, maxSP, minSP et |P-S|</a>, hal-04507547 [math.nt], 2024. (In French)
%e A364601 180 with r=6 satisfies: 180^6 = 34012224000000, 3^3 + 4^3 + 1 + 2^3 + 2^3 + 2^3 + 4^3 = 180.
%o A364601 (PARI) SomP(n,p)={resu=0;for(i=1,#digits(n),resu+=(digits(n)[i])^p);resu}
%o A364601 Ppdi(k,r)={for(n=10^(k-1),10^k,if(SomP(n^r,k)==n,print1(n,";")))}
%Y A364601 Cf. A005188 (Armstrong's numbers, case r=1 in our terminology).
%Y A364601 Cf. A066003, A066004, A065999 (for terms 7, 8 and 9).
%K A364601 nonn,base,fini,more
%O A364601 1,2
%A A364601 _René-Louis Clerc_, Jul 29 2023