A104391 -id:A104391 - 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

A104390 2-Smith numbers.

Original entry on oeis.org

32, 42, 60, 70, 104, 152, 231, 315, 316, 322, 330, 342, 361, 406, 430, 450, 540, 602, 610, 612, 632, 703, 722, 812, 1016, 1027, 1029, 1108, 1162, 1190, 1246, 1261, 1304, 1314, 1316, 1351, 1406, 1470, 1510, 1603, 2013, 2054, 2065, 2070, 2071, 2106, 2114

Offset: 1

Eric W. Weisstein, Mar 04 2005 and Shyam Sunder Gupta, Mar 11 2005

			32 is a 2-Smith number because the sum of the digits of its prime factors, i.e., Sp (32) = Sp(2*2*2*2*2)= 2 + 2 + 2 + 2 + 2 = 10, which is equal to twice the digit sum of 32, i.e., 2*S(32) = 2*(3 + 2) = 10.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)
Shyam Sunder Gupta, Smith Numbers.
Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.
Wayne L. McDaniel, The Existence of infinitely Many k-Smith numbers, Fibonacci Quarterly, Vol. 25, No. 1 (1987), pp. 76-80.

Cf. A006753, A104391.

d[n_]:=IntegerDigits[n]; tr[n_]:=Transpose[FactorInteger[n]]; Select[Range[2120],2Total[d[#]]==Total[d@tr[#][[1]]*tr[#][[2]],2]&] (* Jayanta Basu, Jun 04 2013 *)

A195191 Smallest n-Smith number.

Original entry on oeis.org

32, 402, 2401, 2030, 10112, 10, 200, 10200, 10010, 100200, 1000110, 1000200, 100, 20000, 10200000, 1001000, 100200000, 1000110000, 1000200000, 1000, 2000000, 10200000000, 100100000, 100200000000, 1000110000000, 1000000000100, 10000, 200000000

Offset: 2

Kausthub Gudipati, Sep 11 2011

The smallest number for which the sum of the digits of its prime factors equals n multiplied by the sum of its digits.

			The first term of A104390, the first term of A104391, the first term of A103125 etc.

Shyam Sunder Gupta, Smith numbers

A007953 := proc(n) add(d,d=convert(n,base,10)) ; end proc:
A118503 := proc(n) a := 0 ; for p in ifactors(n)[2] do a := a+ op(2,p)*A007953(op(1,p)) ; end do: a ; end proc:
A195191 := proc(n) for k from 1 do if A118503(k) = n*A007953(k) then return k; end if; end do: end proc: # R. J. Mathar, Sep 14 2011

a(12)-a(29) from Donovan Johnson, Sep 15 2011

A385932 Composite numbers m such that the sum of digits of m divides the sum of digits of prime factors of m (counted with multiplicity).

Original entry on oeis.org

4, 10, 22, 27, 32, 42, 58, 60, 70, 85, 94, 100, 104, 121, 152, 166, 200, 202, 231, 265, 274, 315, 316, 319, 322, 330, 342, 346, 355, 361, 378, 382, 391, 402, 406, 430, 438, 450, 454, 483, 510, 517, 526, 535, 540, 562, 576, 588, 602, 610, 612, 627, 632, 634, 636, 645, 648

Offset: 1

Stefano Spezia, Jul 12 2025

Equivalently, numbers m such that A007953(m) | A118503(m).

Union of the k-Smith numbers for all the positive integers k.

			10 = 2*5 is a term since it is a 7-Smith number: 1 + 0 = 1 | 7 = 2 + 5;
60 = 2^2*3*5 is term since it is a 2-Smith number: 6 + 0 = 6 | 12 = 2 + 2 + 3 + 5;
382 = 2*191 is a term since it is a Smith number (k=1): 3 + 8 + 2 = 13 | 13 = 2 + 1 + 9 + 1;
635 = 5*127 is not a term since 6 + 3 + 5 = 14 does not divide 15 = 5 + 1 + 2 + 7.

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.

Wayne L. McDaniel, The Existence of Infinitely Many k-Smith Numbers, The Fibonacci Quarterly, 25(1), 76-80, (1987).

Supersequence of A006753, A103125, A103126, A104390, A104391.

Cf. A002808, A007953, A118503.

fQ[n_]:=!PrimeQ[n] && n>1 && Divisible[Total[Flatten[IntegerDigits[Table[#[[1]], {#[[2]]}]] & /@ FactorInteger[n]]], Total[IntegerDigits[n]]]; Select[ Range@ 650, fQ]

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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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A104390 2-Smith numbers.

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A195191 Smallest n-Smith number.

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A385932 Composite numbers m such that the sum of digits of m divides the sum of digits of prime factors of m (counted with multiplicity).

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