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%I M3582 #98 Apr 03 2023 10:36:09
%S 4,22,27,58,85,94,121,166,202,265,274,319,346,355,378,382,391,438,454,
%T 483,517,526,535,562,576,588,627,634,636,645,648,654,663,666,690,706,
%U 728,729,762,778,825,852,861,895,913,915,922,958,985,1086,1111,1165,1219
%N 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).
%C Of course primes also have this property, trivially.
%C 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.
%C There are 248483 7-digit Smith numbers, corresponding to US phone numbers without area codes (like 4937775). - _Charles R Greathouse IV_, May 19 2013
%C A007953(a(n)) = Sum_{k=1..A001222(a(n))} A007953(A027746(a(n),k)), and A066247(a(n))=1. - _Reinhard Zumkeller_, Dec 19 2011
%C 3^3, 3^6, 3^9, 3^27 are in the sequence. - _Sergey Pavlov_, Apr 01 2017
%C 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
%D M. Gardner, Penrose Tiles to Trapdoor Ciphers. Freeman, NY, 1989, p. 300.
%D R. K. Guy, Unsolved Problems in the Theory of Numbers, Section B49.
%D 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.
%D J. E. Roberts, Lure of the Integers, pp. 269-270 MAA 1992.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D D. D. Spencer, Key Dates in Number Theory History, Camelot Pub. Co. FL, 1995, pp. 94.
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 180.
%H T. D. Noe, <a href="/A006753/b006753.txt">Table of n, a(n) for n = 1..10000</a>
%H K. S. Brown's Mathpages, <a href="http://www.mathpages.com/home/kmath007/kmath007.htm">Smith Numbers and Rhonda Numbers</a>
%H C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=SmithNumber">Smith number</a>
%H P. J. Costello, <a href="https://web.archive.org/web/20020527191732/http://www.math.eku.edu/PJCostello/smith.htm">Smith Numbers</a>
%H M. Gardner, <a href="/A005130/a005130_1.pdf">Letter to N. J. A. Sloane</a>, Jun 20 1991.
%H Ely Golden, <a href="/A006753/a006753_1.sagews.txt">General program for generating Smith number sequences</a>
%H S. S. Gupta, <a href="http://www.shyamsundergupta.com/smith.htm">Smith Numbers</a>
%H T. Jason, <a href="http://everything2.net/index.pl?node_id=1104442&displaytype=printable&lastnode_id=1104442">Smith number</a>
%H Madras Math's Amazing Number Facts, <a href="http://www.madrasmaths.com/activities/number_facts/fact_42.html">Smith Numbers</a>
%H Sham Oltikar, and Keith Wayland, <a href="http://www.jstor.org/stable/2690265">Construction of Smith Numbers</a>, Mathematics Magazine, vol. 56(1), 1983, pp. 36-37.
%H C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0983.00008&format=complete">Zentralblatt review</a>
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_107.htm">Problem 107: Consecutive Smith numbers</a>, The Prime Puzzles and Problems Connection.
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_108.htm">Problem 108: Methods for generating Smith numbers</a>, The Prime Puzzles and Problems Connection.
%H W. Schneider, <a href="http://web.archive.org/web/2004/www.wschnei.de/digit-related-numbers/smith-numbers.html">Smith Numbers</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmithNumber.html">Smith Number</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Smith_number">Smith number</a>
%H A. Wilansky, <a href="http://www.jstor.org/stable/3026531">Smith numbers</a>, Two-Year Coll. Math. J., 13 (1982), p. 21.
%H A. Witno, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Witno/witno6.html">A Family of Sequences Generating Smith Numbers</a>, J. Int. Seq. 16 (2013) #13.4.6
%e 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.
%p q:= n-> not isprime(n) and (s-> s(n)=add(s(i[1])*i[2], i=
%p ifactors(n)[2]))(h-> add(i, i=convert(h, base, 10))):
%p select(q, [$1..2000])[]; # _Alois P. Heinz_, Apr 22 2021
%t fQ[n_] := !PrimeQ@ n && n>1 && Plus @@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]] }] & /@ FactorInteger@ n] == Plus @@ IntegerDigits@ n; Select[ Range@ 1200, fQ]
%o (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
%o (Haskell)
%o a006753 n = a006753_list !! (n-1)
%o a006753_list = [x | x <- a002808_list,
%o a007953 x == sum (map a007953 (a027746_row x))]
%o -- _Reinhard Zumkeller_, Dec 19 2011
%o (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
%o (Python)
%o from sympy import factorint
%o def sd(n): return sum(map(int, str(n)))
%o def ok(n):
%o f = factorint(n)
%o return sum(f[p] for p in f) > 1 and sd(n) == sum(sd(p)*f[p] for p in f)
%o print(list(filter(ok, range(1220)))) # _Michael S. Branicky_, Apr 22 2021
%Y Cf. A002808, A007953, A019506, A050218, A050224, A050255, A098834-A098840, A103123-A103126, A104166-A104171, A104390, A104391, A202387, A202388.
%K nonn,base,nice,easy
%O 1,1
%A _N. J. A. Sloane_
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