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 A007532 Handsome numbers: sum of positive powers of its digits; a(n) = Sum_{i=1..k} d[i]^e[i] where d[1..k] are the decimal digits of a(n), e[i] > 0. (Formerly M0487) 16
 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, 132, 135, 153, 175, 209, 224, 226, 262, 264, 267, 283, 332, 333, 334, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 518, 598, 629, 739, 794, 849, 935, 994, 1034 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The previous name was "Powerful numbers, Definition (2). Cf. A001694, A023052. - N. J. A. Sloane, Jan 16 2022 J. Randle has suggested the name "powerful numbers" for the perfect digital invariants A023052, equal to the sum of a fixed power of the digits. However, "powerful" usually refers to a prime factorization related property, cf. A001694 (and references there as well as on the MathWorld page). C. Rivera has suggested the name "handsome" for these numbers (in view of narcissistic numbers A005188) in his prime puzzle #15: see also contributed comments concerning terminology on that page. - M. F. Hasler, Nov 21 2019 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS David W. Wilson, Table of n, a(n) for n = 1..10000 Giovanni Resta, d-powerful numbers, the 30067 terms and sums up to 10^6. Carlos Rivera, Puzzle 15.- Narcissistic and Handsome Primes, The Prime Puzzles and Problems Connection. Eric Weisstein's World of Mathematics, Powerful Number. FORMULA If n = d_1 d_2 ... d_k in decimal, then there are integers m_1, m_2, ..., m_k > 0 such that n = d_1^m_1 + ... + d_k^m_k. EXAMPLE 43 = 4^2 + 3^3 is OK; 254 = 2^7 + 5^3 + 4^0 is not OK since one of the powers is 0. MAPLE N:= 10000; # to get all entries <= N Sums:= proc(L, N)   option remember;   local x1, L1;   x1:= L;   if x1 = 1 then L1:= {1}   else L1:= {seq(x1^j, j=1..floor(log[x1](N)))};   fi;   if nops(L) = 1 then L1   else select(`<=`, {seq(seq(a+b, a=L1), b=Sums(L[2..-1], N))}, N)   fi end proc; filter:= proc(x, N)    local L;    L:= sort(subs(0=NULL, convert(x, base, 10))) ;    member(x, Sums(L, N)); end proc; A007532:= select(filter, [\$1..N], N); # Robert Israel, Apr 13 2014 MATHEMATICA Select[Range@1000, (s=#; MemberQ[Total/@(a^#&/@Tuples[Range@If[#==1||#==0, 1, Floor[Log[#, s]]]&/@(a=IntegerDigits[s])]), s])&] (* Giorgos Kalogeropoulos, Aug 18 2021 *) PROG (Haskell) a007532 n = a007532_list !! (n-1) a007532_list = filter f [1..] where    f x = g x 0 where      g 0 v = v == x      g u v = if d <= 1 then g u' (v + d) else v <= x && h d              where h p = p <= x && (g u' (v + p) || h (p * d))                    (u', d) = divMod u 10 -- Reinhard Zumkeller, Jun 02 2013 (Python) from itertools import count, takewhile def cands(n, d):     return takewhile(lambda x: x<=n, (d**i for i in count(1))) def handsome(s, t):     if s == "":         return t == 0     if s in "01":         return handsome(s[1:], t - int(s))     return any(handsome(s[1:], t - p) for p in cands(t, int(s))) def ok(n):     return n and handsome(str(n), n) print(list(filter(ok, range(1035)))) # Michael S. Branicky, Aug 18 2021 CROSSREFS Cf. A001694, A005934, A005188, A003321, A014576, A023052, A046074, A050240 (>= 2 reps.), A050241. Different from A061862. Sequence in context: A228187 A134703 A061862 * A349279 A347189 A068189 Adjacent sequences:  A007529 A007530 A007531 * A007533 A007534 A007535 KEYWORD base,nonn,nice AUTHOR STATUS approved

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Last modified October 6 21:32 EDT 2022. Contains 357270 sequences. (Running on oeis4.)