

A006753


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)


70



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
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OFFSET

1,1


COMMENTS

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 brotherinlaw Harold Smith: 4937775.
There are 248483 7digit Smith numbers, corresponding to US phone numbers without area codes (like 4937775).  Charles R Greathouse IV, May 19 2013
A007953(a(n)) = Sum_{k=1..A001222(a(n))} A007953(A027746(a(n),k)), and A066247(a(n))=1.  Reinhard Zumkeller, Dec 19 2011
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


REFERENCES

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. 247248, Oxford University Press, 2000.
J. E. Roberts, Lure of the Integers, pp. 269270 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.


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
K. S. Brown's Mathpages, Smith Numbers and Rhonda Numbers
C. K. Caldwell, The Prime Glossary, Smith number
P. J. Costello, Smith Numbers
Ely Golden, General program for generating Smith number sequences
S. S. Gupta, Smith Numbers
T. Jason, Smith number
Madras Math's Amazing Number Facts, Smith Numbers
Sham Oltikar, and Keith Wayland, Construction of Smith Numbers, Mathematics Magazine, vol. 56(1), 1983, pp. 3637.
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
C. Rivera, PrimePuzzles.Net, Problem 107:Consecutive Smith numbers
C. Rivera, PrimePuzzles.Net, Problem 108:Methods for generating Smith numbers
W. Schneider, Smith Numbers
Eric Weisstein's World of Mathematics, Smith Number
Wikipedia, Smith number
A. Wilansky, Smith numbers, TwoYear Coll. Math. J., 13 (1982), p. 21.


EXAMPLE

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.


MATHEMATICA

fQ[n_] := !PrimeQ@ n && n>1 && Plus @@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]] }] & /@ FactorInteger@ n] == Plus @@ IntegerDigits@ n; Select[ Range@ 1200, fQ]


PROG

(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 !! (n1)
a006753_list = [x  x < a002808_list,
a007953 x == sum (map a007953 (a027746_row x))]
 Reinhard Zumkeller, Dec 19 2011
(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


CROSSREFS

Cf. A002808, A019506, A050218, A050224, A050255, A098834A098840, A103123A103126, A104166A104171, A104390, A104391, A202387, A202388.
Sequence in context: A244411 A213240 A279314 * A098836 A204341 A036920
Adjacent sequences: A006750 A006751 A006752 * A006754 A006755 A006756


KEYWORD

nonn,base,nice,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



