|
| |
|
|
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)
|
|
54
| |
|
|
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
(list; graph; refs; listen; history; internal format)
|
|
|
|
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 brother-in-law Harold Smith: 493-7775.
|
|
|
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.
Oltikar, Sham and Keith Wayland. "Construction of Smith Numbers," Mathematics Magazine, vol. 56(1), 1983, pp. 36-37.
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.
A. Wilansky, Smith numbers, Two-Year Coll. Math. J., 13 (1982), 21.
|
|
|
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
S. S. Gupta, Smith Numbers
Jason T., Smith number
Madras Math's Amazing Number Facts, Smith Numbers
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
C. Riveras, PrimePuzzles.Net, Problem 107:Consecutive Smith numbers
C. Riveras, PrimePuzzles.Net, Problem 108:Methods for generating Smith numbers
W. Schneider, Smith Numbers
Eric Weisstein's World of Mathematics, Smith Number
Wikipedia, Smith number
|
|
|
FORMULA
| A007953(a(n)) = sum(A007953(A027746(a(n),k)): k=1..A001222(a(n))) and A066247(a(n))=1. [Reinhard Zumkeller, Dec 19 2011]
|
|
|
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 !! (n-1)
a006753_list = [x | x <- a002808_list,
a007953 x == sum (map a007953 (a027746_row x))]
-- Reinhard Zumkeller, Dec 19 2011
(PARI) dsum(n)=my(s); while(n, s+=n%10; n\=10); s
is(n)=if(isprime(n), 0, my(f=factor(n)); sum(i=1, #f[, 1], dsum(f[i, 1])*f[i, 2])==dsum(n))
\\ Charles R Greathouse IV, Jan 03 2012
|
|
|
CROSSREFS
| Cf. A019506, A050224, A050255, A098834-A098840, A103123-A103126, A104166-A104171, A104390, A104391, A002808, A050218, A202387, A202388.
Sequence in context: A009925 A059653 A022385 * A098836 A204341 A036920
Adjacent sequences: A006750 A006751 A006752 * A006754 A006755 A006756
|
|
|
KEYWORD
| nonn,base,nice,easy
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
