A006753 Smith (or joke) numbers: composite numbers k such that sum of digits of k = sum of digits of prime factors of k (counted with multiplicity).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base
• nice
• easy

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.

There are 248483 7-digit 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 an integer, k > 10. - Sergey Pavlov, Apr 03 2017

Examples

			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.
		

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. 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.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 3.1.14 and 3.1.16 on pages 84-85.
• 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
• M. Gardner, Letter to N. J. A. Sloane, Jun 20 1991.
• 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.
• Wayne L. McDaniel, The Existence of Infinitely Many k-Smith Numbers, The Fibonacci Quarterly, 25(1), 76-80, (1987).
• Sham Oltikar, and Keith Wayland, Construction of Smith Numbers, Mathematics Magazine, vol. 56(1), 1983, pp. 36-37.
• C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
• Carlos Rivera, Problem 107: Consecutive Smith numbers, The Prime Puzzles and Problems Connection.
• Carlos Rivera, Problem 108: Methods for generating Smith numbers, The Prime Puzzles and Problems Connection.
• W. Schneider, Smith Numbers
• Eric Weisstein's World of Mathematics, Smith Number
• Wikipedia, Smith number
• A. Wilansky, Smith numbers, Two-Year Coll. Math. J., 13 (1982), p. 21.
• A. Witno, A Family of Sequences Generating Smith Numbers, J. Int. Seq. 16 (2013) #13.4.6

Crossrefs

Cf. A002808, A007953, A019506, A050218, A050224, A050255, A098834-A098840, A103123-A103126, A104166-A104171, A104390, A104391, A202387, A202388.

Programs

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

Maple

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))):
select(q, [$1..2000])[];  # Alois P. Heinz, Apr 22 2021

Mathematica

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

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)
print(list(filter(ok, range(1220)))) # Michael S. Branicky, Apr 22 2021

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