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A006753
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Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted with multiplicity).
(Formerly M3582)
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82
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4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985, 1086, 1111, 1165, 1219
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OFFSET
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1,1
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COMMENTS
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Of course primes also have this property, trivially.
a(133809) = 4937775 is the first Smith number historically: 4937775 = 3*5*5*65837 and 4+9+3+7+7+7+5 = 3+5+5+(6+5+8+3+7) = 42, Albert Wilansky coined the term Smith number when he noticed the defining property in the phone number of his brother-in-law Harold Smith: 493-7775.
There are 248483 7-digit Smith numbers, corresponding to US phone numbers without area codes (like 4937775). - Charles R Greathouse IV, May 19 2013
3^3, 3^6, 3^9, 3^27 are in the sequence. - Sergey Pavlov, Apr 01 2017
As mentioned by Giovanni Resta, there are no other terms of the form 3^t for 0 < t < 300000 and, probably, no other terms of such form for t >= 300000. It seems that, if there exists any other term of form 3^t with integer t, then t == 0 (mod 3) or, perhaps, t = {3^k; 2*3^k} where k is integer, k > 10. - Sergey Pavlov, Apr 03 2017
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REFERENCES
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M. Gardner, Penrose Tiles to Trapdoor Ciphers. Freeman, NY, 1989, p. 300.
R. K. Guy, Unsolved Problems in the Theory of Numbers, Section B49.
C. A. Pickover, "A Brief History of Smith Numbers" in "Wonders of Numbers: Adventures in Mathematics, Mind and Meaning", pp. 247-248, Oxford University Press, 2000.
J. E. Roberts, Lure of the Integers, pp. 269-270 MAA 1992.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. D. Spencer, Key Dates in Number Theory History, Camelot Pub. Co. FL, 1995, pp. 94.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 180.
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LINKS
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C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
A. Wilansky, Smith numbers, Two-Year Coll. Math. J., 13 (1982), p. 21.
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EXAMPLE
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58 = 2*29; sum of digits of 58 is 13, sum of digits of 2 + sum of digits of 29 = 2+11 is also 13.
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MAPLE
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q:= n-> not isprime(n) and (s-> s(n)=add(s(i[1])*i[2], i=
ifactors(n)[2]))(h-> add(i, i=convert(h, base, 10))):
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MATHEMATICA
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fQ[n_] := !PrimeQ@ n && n>1 && Plus @@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]] }] & /@ FactorInteger@ n] == Plus @@ IntegerDigits@ n; Select[ Range@ 1200, fQ]
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PROG
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(Sage) is_A006753 = lambda n: n > 1 and not is_prime(n) and sum(n.digits()) == sum(sum(p.digits())*m for p, m in factor(n)) # D. S. McNeil, Dec 28 2010
(Haskell)
a006753 n = a006753_list !! (n-1)
a006753_list = [x | x <- a002808_list,
a007953 x == sum (map a007953 (a027746_row x))]
(PARI) isA006753(n) = if(isprime(n), 0, my(f=factor(n)); sum(i=1, #f[, 1], sumdigits(f[i, 1])*f[i, 2]) == sumdigits(n)); \\ Charles R Greathouse IV, Jan 03 2012; updated by Max Alekseyev, Oct 21 2016
(Python)
from sympy import factorint
def sd(n): return sum(map(int, str(n)))
def ok(n):
f = factorint(n)
return sum(f[p] for p in f) > 1 and sd(n) == sum(sd(p)*f[p] for p in f)
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CROSSREFS
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Cf. A002808, A007953, A019506, A050218, A050224, A050255, A098834-A098840, A103123-A103126, A104166-A104171, A104390, A104391, A202387, A202388.
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KEYWORD
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nonn,base,nice,easy
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AUTHOR
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STATUS
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approved
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