A104171 -id:A104171 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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.

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.

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

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

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

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

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

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

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

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

A098834 Palindromic Smith numbers.

Original entry on oeis.org

4, 22, 121, 202, 454, 535, 636, 666, 1111, 1881, 3663, 7227, 7447, 9229, 10201, 17271, 22522, 24142, 28182, 33633, 38283, 45054, 45454, 46664, 47074, 50305, 51115, 51315, 54645, 55055, 55955, 72627, 81418, 82628, 83038, 83938, 90409, 95359, 96169, 164461

Offset: 1

Shyam Sunder Gupta, Oct 10 2004

			a(3) = 121 because 121 is a Smith number as well as a palindromic number.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.

Cf. A006753.

Subsequence of A104171. Supersequence of A104166.

d[n_] := IntegerDigits[n]; tr[n_] := Transpose[FactorInteger[n]]; Select[Range[2, 1.7*10^5], !PrimeQ[#] && Reverse[x=d[#]] == x && Total[x] == Total[d@tr[#][[1]]*tr[#][[2]], 2]&] (* Jayanta Basu, Jun 04 2013 *)

from sympy import factorint
from itertools import product
def sd(n): return sum(map(int, str(n)))
def smith(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)
def palsto(limit):
  yield from range(min(limit, 9)+1)
  midrange = [[""], [str(i) for i in range(10)]]
  for digs in range(2, 10**len(str(limit))):
    for p in product("0123456789", repeat=digs//2):
      left = "".join(p)
      if left[0] == '0': continue
      for middle in midrange[digs%2]:
        out = int(left + middle + left[::-1])
        if out > limit: return
        yield out
print(list(filter(smith, palsto(164461)))) # Michael S. Branicky, Apr 22 2021

A104166 Repdigit Smith numbers.

Original entry on oeis.org

4, 22, 666, 1111, 6666666, 4444444444, 44444444444444444444, 555555555555555555555555555, 55555555555555555555555555555555, 4444444444444444444444444444444444444444444444444444444

Offset: 1

Shyam Sunder Gupta, Mar 10 2005

Shyam Sunder Gupta, Smith Numbers.
Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.

Cf. A006753.

Subsequence of both A098834 and A104171.

d[n_]:=IntegerDigits[n]; tr[n_]:=Transpose[FactorInteger[n]]; a[n_]:=NestList[FromDigits[Flatten[d[{#,n}]]]&,n,55]; t={}; Do[If[!PrimeQ[n]&&Total[d[n]]==Total[d@tr[n][[1]]*tr[n][[2]],2],AppendTo[t,n]],{n,Drop[Union[Flatten[Table[a[k],{k,9}]]],1]}]; t (* Jayanta Basu, Jun 04 2013 *)

from sympy import factorint
from itertools import product
def sd(n): return sum(map(int, str(n)))
def smith(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)
def repsto(limit):
  yield from range(min(limit, 9)+1)
  for rep in range(2, 10**len(str(limit))):
    for digit in "123456789":
      out = int(digit*rep)
      if out > limit: return
      yield out
print(list(filter(smith, repsto(10**32)))) # Michael S. Branicky, Apr 22 2021

A337295 Reversible binary Smith numbers: binary Smith numbers (A278909) whose binary reversal (A030101) is also a binary Smith number.

Original entry on oeis.org

15, 51, 85, 159, 190, 249, 303, 471, 489, 639, 679, 763, 765, 771, 799, 843, 893, 917, 951, 995, 1010, 1017, 1023, 1167, 1203, 1285, 1467, 1501, 1615, 1630, 1641, 1707, 1742, 1773, 1788, 1929, 1939, 1970, 2015, 2167, 2319, 2367, 2493, 2787, 2931, 2975, 3033, 3055

Offset: 1

Amiram Eldar, Aug 21 2020

			159 is a binary Smith number: 159 = 3 * 53 is in binary representation 10011111 = 11 * 110101, and (1 + 0 + 0 + 1 + 1 + 1 + 1 + 1) = (1 + 1) + (1 + 1 + 0 + 1 + 0 + 1) = 6. The binary reversal of 159 = 10011111_2 is 249 = 11111001_2 which is also a binary Smith number: 249 = 3 * 83 is in binary representation 11111001 = 11 * 1010011, and (1 + 1 + 1 + 1 + 1 + 0 + 0 + 1) = (1 + 1) + (1 + 0 + 1 + 0 + 0 + 1 + 1) = 6. Therefore, 159 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

The binary version of A104171.
Subsequence of A278909.
A334530 is a subsequence.
Cf. A030101.

binSmithQ[n_] := CompositeQ[n] && Plus @@ (Last @#* DigitCount[First@#, 2, 1] & /@ FactorInteger[n]) == DigitCount[n, 2, 1]; rev[n_] := FromDigits[Reverse @ IntegerDigits[n, 2], 2]; Select[Range[3000], binSmithQ[#] && binSmithQ[rev[#]] &]

cp's OEIS Frontend

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).

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A098834 Palindromic Smith numbers.

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A104166 Repdigit Smith numbers.

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A337295 Reversible binary Smith numbers: binary Smith numbers (A278909) whose binary reversal (A030101) is also a binary Smith number.

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Mathematica