

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
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OFFSET

1,2


COMMENTS

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, dpowerful numbers the 30067 terms and sums up to 10^6.
Carlos Rivera, Puzzle 15. Narcissistic and Handsome Primes, PrimePuzzles.net, 1998.
Eric Weisstein's World of Mathematics, Powerful Number.
Index entries for sequences related to powerful numbers


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[1];
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


PROG

(Haskell)
a007532 n = a007532_list !! (n1)
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


CROSSREFS

Cf. A001694, A005934, A005188, A003321, A014576, A023052, A046074.
Different from A061862.
Sequence in context: A228187 A134703 A061862 * A068189 A069716 A095289
Adjacent sequences: A007529 A007530 A007531 * A007533 A007534 A007535


KEYWORD

base,nonn,nice


AUTHOR

N. J. A. Sloane, Robert G. Wilson v


STATUS

approved



