A006753 -id:A006753 - cp's OEIS frontend

A278909 Binary Smith numbers: composite numbers n such that sum of bits of n = sum of bits of prime factors of n (counted with multiplicity).

15, 51, 55, 85, 125, 159, 185, 190, 205, 215, 222, 238, 246, 249, 253, 287, 303, 319, 374, 407, 438, 442, 469, 471, 475, 489, 494, 501, 507, 591, 623, 639, 670, 679, 687, 699, 730, 745, 755, 763, 765, 771, 799, 807, 822, 830, 843, 867, 890, 893, 917, 923, 925, 935, 939, 951, 970, 973, 979, 986, 989, 995, 1010, 1015, 1017, 1020, 1023, 1135, 1167, 1203, 1243

Offset: 1

Ely Golden, Nov 30 2016

Binary equivalent of A006753 as well as A176670. (Since bits can only be 0 or 1, having equal sums of bits is logically equivalent to having the same nonzero bits.)

There are 615 terms up to 10^4, 6412 up to 10^5, 66369 up to 10^6, 630106 up to 10^7, 6268949 up to 10^8, 62159262 up to 10^9, and 596587090 up to 10^10. - Charles R Greathouse IV, Dec 09 2016

			a(1) = 15, as 15 (1111) in binary has the same number of 1 bits as its prime factors (11 and 101).

Ely Golden, Table of n, a(n) for n = 1..10000

Cf. A006753, A176670.

Select[Range@ 1250, And[CompositeQ@ #, DigitCount[#, 2, 1] = Total@ Flatten@ Apply[DigitCount[#, 2, 1] & /@ ConstantArray[#1, #2] &, FactorInteger@ #, 1]] &] (* Michael De Vlieger, Dec 02 2016 *)

is(n) = my(f=factor(n)[, 1]~, expo=factor(n)[, 2]~, v=[], s=0); for(k=1, #f, while(expo[k] > 0, expo[k]--; v=concat(v, f[k]))); for(k=1, #v, v[k]=binary(v[k])); my(w=[]); for(y=1, #v, w=concat(w, v[y])); if(vecsum(w)==vecsum(binary(n)), return(1), return(0))
terms(n) = my(i=0); forcomposite(c=1, , if(is(c), print1(c, ", "); i++; if(i==n, break)))
/* Print initial 70 terms as follows: */
terms(70) \\ Felix Fröhlich, Dec 01 2016

is(n)=my(f=factor(n),t=#f~); (t>1 || (t==1 && f[1,2]>1)) && hammingweight(n)==sum(i=1,t, hammingweight(f[i,1])*f[i,2]) \\ Charles R Greathouse IV, Dec 02 2016

from sympy import factorint
def sbd(n): return bin(n).count('1')
def ok(n):
  f = factorint(n)
  return sum(f[p] for p in f) > 1 and sbd(n) == sum(sbd(p)*f[p] for p in f)
print(list(filter(ok, range(1244)))) # Michael S. Branicky, Apr 22 2021

def numfactorbits(x):
    if(x<2):
        return 0;
    s=0;
    f=list(factor(x));
    #ensures inequality of numfactorbits(x) and bin(x).count("1") if x is prime
    if((len(f)==1)&(f[0][1]==1)):
        return 0;
    for c in range(len(f)):
        s+=bin(f[c][0]).count("1")*f[c][1]
    return s;
counter=2
index=1
while(index<=10000):
    if(numfactorbits(counter)==bin(counter).count("1")):
        print(str(index)+" "+str(counter))
        index+=1;
    counter+=1;

A067077 Numbers whose product of distinct prime factors is equal to its sum of digits.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 24, 375, 392, 640, 2401, 4802, 4913, 6400, 7744, 17576, 42592, 64000, 106496, 234256, 295936, 468750, 546875, 628864, 640000, 877952, 1124864, 1966080, 2839714, 3687936, 4687500, 4816896, 4952198, 6400000, 6453888

Offset: 1

Joseph L. Pe, Feb 18 2002

The product of the distinct prime factors of n (the squarefree kernel of n) is also denoted by rad(n) = A007947(n). - Giovanni Resta, Apr 21 2017

			The prime factors of 375 are 3,5, which have product = 15, the sum of the digits of 375, so 375 is a term of the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 71 terms from Harry J. Smith)

Cf. A007947, A006753, A057531, A057532, A050689, A070274, A070275, A063737, A285494.

f[n_] := Times@@ (First/@ FactorInteger[n]); g[n_] := Plus @@ IntegerDigits[n]; Select[Range[10^5], f[#] == g[#] &] (* or *)
nd=12; up=10^nd; L={1}; Do[If[SquareFreeQ[su], ps = First /@ FactorInteger[su]; nps = Length@ ps; Clear[ric]; ric[n_, i_] := Block[{e = 0, m}, If[i > nps, If[Plus @@ IntegerDigits[su n] == su, Sow[su n]], While[ (m = n ps[[i]]^e ) su < up, ric[m, i+1]; e++]]]; z = Reap[ ric[1, 1]][[2]]; If[z != {}, L = Union[L, z[[1]]]]], {su, 2, 9 nd}]; L (* fast, terms < 10^12, Giovanni Resta, Apr 21 2017 *)
Select[Range[65*10^5],Times@@FactorInteger[#][[All,1]]==Total[ IntegerDigits[ #]]&] (* Harvey P. Dale, Dec 16 2018 *)

isok(k)={vecprod(factor(k)[,1]) == sumdigits(k)} \\ Harry J. Smith, May 06 2010

a(19)-a(35) from Donovan Johnson, Sep 29 2009

a(1)=1 prepended by Giovanni Resta, Apr 21 2017

A174460 Smith numbers of order 2.

Original entry on oeis.org

56, 58, 810, 822, 1075, 1519, 1752, 2145, 2227, 2260, 2483, 2618, 2620, 3078, 3576, 3653, 3962, 4336, 4823, 4974, 5216, 5242, 5386, 5636, 5719, 5762, 5935, 5998, 6220, 6424, 6622, 6845, 7015, 7251, 7339, 7705, 7756, 8460, 9254, 9303, 9355, 10481, 10626, 10659

Offset: 1

Paul Weisenhorn, Dec 20 2010

Composite numbers a(n) such that the sum of digits^2 equals the sum of digits^2 of its prime factors without the numbers of A176670 that have the same digits as its prime factors (without the zero digit).

It seems as though as the order n approaches infinity, the sequence of n-order Smith numbers approaches A176670. Is there a value of n where the only n-order Smith numbers are members of A176670? - Ely Golden, Dec 07 2016

			a(2) = 58 = 2*29 is a Smith number of order 2 because 5^2 + 8^2 = 2^2 + 2^2 + 9^2 = 89.

Ely Golden and Donovan Johnson, Table of n, a(n) for n = 1..10000 (terms 1 to 1000 by Donovan Johnson)
Patrick Costello, A new largest Smith number, Fibonacci Quarterly 40(4) (2002), 369-371.
Underwood Dudley, Smith numbers, Mathematics Magazine 67(1) (1994), 62-65.
Ely Golden, Smith number sequence generator optimized for order 2.
S. S. Gupta, Smith Numbers, Mathematical Spectrum 37(1) (2004/5), 27-29.
S. S. Gupta, Smith Numbers.
Eric Weisstein's World of Mathematics, Smith number.
Wikipedia, Smith number.
A. Wilansky, Smith Numbers, Two-Year College Math. J. 13(1) (1982), p. 21.
Amin Witno, Another simple construction of Smith numbers, Missouri J. Math. Sci. 22(2) (2010), 97-101.
Amin Witno, Smith multiples of a class of primes with small digital sum, Thai Journal of Mathematics 14(2) (2016), 491-495.

Cf. A006753 (Smith numbers), A176670, A178213, A178193, A178203, A178204.

for s from 2 to 10000 do g:=nops(ifactors(s)[2]): qsp:=0: for u from 1 to g do z:=ifactors(s)[2,u][1]: h:=0: while (z>0) do z:=iquo(z,10,'r'): h:=h+r^2: end do: h:=h*ifactors(s)[2,u][2]: qsp:=qsp+h: end do: z:=s: qs:=0: while (z>0) do z:=iquo(z,10,'r'): qs:=qs+r^2: end do: if (qsp=qs) then print(s): end if: end do:

With[{k = 2},Select[Range[12000], Function[n, And[Total@ Map[#^k &, IntegerDigits@ n] == Total@ Map[#^k &, Flatten@ IntegerDigits[#]], Not[Sort@ DeleteCases[#, 0] &@ IntegerDigits@ n == Sort@ DeleteCases[#, 0] &@ #]] &@ Flatten@ Map[IntegerDigits@ ConstantArray[#1, #2] & @@ # &, FactorInteger@ n]]]] (* Michael De Vlieger, Dec 10 2016 *)

A050219 Smaller of Smith brothers.

Original entry on oeis.org

728, 2964, 3864, 4959, 5935, 6187, 9386, 9633, 11695, 13764, 16536, 16591, 20784, 25428, 28808, 29623, 32696, 33632, 35805, 39585, 43736, 44733, 49027, 55344, 56336, 57663, 58305, 62634, 65912, 65974, 66650, 67067, 67728, 69279, 69835, 73615, 73616, 74168

Offset: 1

Eric W. Weisstein

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert Israel)
Eric Weisstein's World of Mathematics, Smith Brothers.

Cf. A006753, A050220.

issmith:= proc(n)
  if isprime(n) then return false fi;
  convert(convert(n,base,10),`+`) = add(t[2]*convert(convert(t[1],base,10),`+`),t=ifactors(n)[2])
end proc:
S:= select(issmith, {$4..10^5}):
sort(convert(S intersect map(`-`,S,1), list)); # Robert Israel, Jan 15 2018

smithQ[n_] := !PrimeQ[n] && Total[Flatten[IntegerDigits[Table[#[[1]], {#[[2]]}]& /@ FactorInteger[n]]]] == Total[IntegerDigits[n]];
Select[Range[10^5], smithQ[#] && smithQ[#+1]&] (* Jean-François Alcover, Jun 07 2020 *)

isone(n) = {if (!isprime(n), f = factor(n); sumdigits(n) == sum(k=1, #f~, f[k,2]*sumdigits(f[k,1])););}
isok(n) =  isone(n) && isone(n+1); \\ Michel Marcus, Jul 17 2015

from sympy import factorint
from itertools import count, islice
def sd(n): return sum(map(int, str(n)))
def smith():
    for k in count(1):
        f = factorint(k)
        if sum(f[p] for p in f) > 1 and sd(k) == sum(sd(p)*f[p] for p in f):
            yield k
def agen():
    prev = -1
    for s in smith():
        if s == prev + 1: yield prev
        prev = s
print(list(islice(agen(), 38))) # Michael S. Branicky, Dec 23 2022

Offset corrected by Arkadiusz Wesolowski, May 08 2012

A178213 Smith numbers of order 3.

Original entry on oeis.org

6606, 8540, 13086, 16866, 21080, 26637, 27468, 33387, 34790, 35364, 35377, 40908, 44652, 48154, 48860, 52798, 54814, 55055, 57726, 57894, 66438, 67297, 67356, 67594, 69549, 72465, 72598, 73026, 74371, 74785, 77485, 78745, 81546, 83175, 85927, 90174, 91208

Offset: 1

Paul Weisenhorn, Dec 19 2010

Composite numbers n not in A176670 such that the sum of the cubes of the digits of n equals the sum of the cubes of the digits of the prime factors of n (with multiplicity). A176670 lists composite numbers having the same digits as their prime factors (with multiplicity), excluding zero digits.

			6606 = 2*3*3*367 is composite; sum of cubes of the digits is 6^3+6^3+0^3+6^3 = 648. Sum of cubes of the digits of the prime factors 2, 3, 3, 367 (with multiplicity) is 2^3+3^3+3^3+3^3+6^3+7^3 = 648. The sums are equal, so 6606 is in the sequence.
21080 = 2*2*2*5*17*31 is composite; sum of cubes of the digits is 2^3+1^3+0^3+8^3+0^3 = 521. Sum of cubes of the digits of the prime factors 2, 2, 2, 5, 17, 31 (with multiplicity) is 2^3+2^3+2^3+5^3+1^3+7^3+3^3+1^3 = 521. The sums are equal, so 21080 is in the sequence.

Ely Golden and Donovan Johnson, Table of n, a(n) for n = 1..10000 (terms 1 to 1000 by Donovan Johnson)
Ely Golden, Smith number sequence generator optimized for order 3
Eric W. Weisstein, Smith Number

Cf. A006753 (Smith numbers), A174460, A176670, A178193, A178203, A178204.

fQ[n_] := Block[{id = Sort@ IntegerDigits@ n, fid = Sort@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@ n]}, While[ id[[1]] == 0, id = Drop[id, 1]]; While[ fid[[1]] == 0, fid = Drop[fid, 1]]; !PrimeQ@ n && id != fid && Plus @@ (id^3) == Plus @@ (fid^3)]; k = 2; lst = {}; While[k < 22002, If[fQ@ k, AppendTo[ lst, k]; Print@ k]; k++]; lst
With[{k = 3}, Select[Range[10^5], Function[n, And[Total@ Map[#^k &, IntegerDigits@ n] == Total@ Map[#^k &, Flatten@ IntegerDigits[#]], Not[Sort@ DeleteCases[#, 0] &@ IntegerDigits@ n == Sort@ DeleteCases[#, 0] &@#]] &@ Flatten@ Map[IntegerDigits@ ConstantArray[#1, #2] & @@ # &, FactorInteger@ n]]]] (* Michael De Vlieger, Dec 10 2016 *)

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

A104171 Reversible Smith numbers, i.e., Smith numbers whose reversal is also a Smith number.

Original entry on oeis.org

4, 22, 58, 85, 121, 202, 265, 319, 454, 535, 562, 636, 666, 913, 1111, 1507, 1642, 1881, 1894, 1903, 2461, 2583, 2605, 2614, 2839, 3091, 3663, 3852, 4162, 4198, 4369, 4594, 4765, 4788, 4794, 4954, 4974, 4981, 5062, 5386, 5458, 5539, 5674, 5818, 5926, 6295

Offset: 1

Shyam Sunder Gupta, Mar 10 2005

The palindromic Smith numbers (A098834) are a subset of the reversible Smith numbers.

			a(3) = 58 because 58 and its reverse 85 are Smith numbers.

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

Cf. A006753, A098834.

rev[n_] := FromDigits @ Reverse @ IntegerDigits[n]; digSum[n_] := Plus @@ IntegerDigits[n]; smithQ[n_] := CompositeQ[n] && Plus @@ (Last@#*digSum[First@#] & /@ FactorInteger[n]) == digSum[n]; Select[Range[6000], smithQ[#] && smithQ @ rev[#] &] (* Amiram Eldar, Aug 24 2020 *)

from sympy import factorint
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 ok(n): return smith(n) and smith(int(str(n)[::-1]))
print(list(filter(ok, range(6296)))) # Michael S. Branicky, Apr 22 2021

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 *)

A104391 3-Smith numbers.

Original entry on oeis.org

402, 510, 700, 1113, 1131, 1311, 2006, 2022, 2130, 2211, 2240, 3102, 3111, 3204, 3210, 3220, 4031, 4300, 4410, 5310, 6004, 6100, 6300, 7031, 7120, 9000, 10034, 10125, 10206, 10251, 10304, 10413, 10521, 10612, 10800, 11033, 11111, 11114, 11116, 11121, 11141

Offset: 1

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

			402 is a 3-Smith number because the sum of the digits of its prime factors, i.e., Sp(402) = Sp(2*3*67)= 2 + 3 + 6 + 7 = 18, which is equal to 3 times the digit sum of 402, i.e., 3*S(402) = 3*(4 + 0 + 2) = 18.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1500 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, A104390.

Select[Range[12000],Total[Flatten[IntegerDigits/@Table[#[[1]],{#[[2]]}]&/@ FactorInteger[#]]]/Total[IntegerDigits[#]]==3&] (* Harvey P. Dale, Feb 19 2013 *)

A178203 Smith numbers of order 5; composite numbers n such that sum of digits^5 equal sum of digits^5 of its prime factors without the numbers in A176670 that have the same digits as its prime factors (without the zero digits).

Original entry on oeis.org

414966, 443166, 454266, 1274664, 1371372, 1701856, 1713732, 1734616, 1771248, 1858436, 1858616, 2075664, 2624976, 3606691, 3771031, 3771301, 4266914, 4414866, 4461786, 4605146, 4670576, 4710739, 5209663, 5281767, 5434572, 5836565, 5861712, 5871968, 6046357

Offset: 1

Paul Weisenhorn, Dec 19 2010

			a(10) = 1858436 = 2*2*29*37*433;
1^5 + 3^5 + 4^5 + 5^5 + 6^5 + 2*8^5 = 3*2^5 + 3*3^5 + 4^5 + 7^5 + 9^5 = 77705.

Donovan Johnson, Table of n, a(n) for n = 1..1000
Patrick Costello, A new largest Smith number, Fibonacci Quarterly 40(4) (2002), 369-371.
Underwood Dudley, Smith numbers, Mathematics Magazine 67(1) (1994), 62-65.
S. S. Gupta, Smith Numbers, Mathematical Spectrum 37(1) (2004/5), 27-29.
S. S. Gupta, Smith Numbers.
Eric Weisstein's World of Mathematics, Smith number.
Wikipedia, Smith number.
A. Wilansky, Smith Numbers, Two-Year College Math. J. 13(1) (1982), p. 21.
Amin Witno, Another simple construction of Smith numbers, Missouri J. Math. Sci. 22(2) (2010), 97-101.
Amin Witno, Smith multiples of a class of primes with small digital sum, Thai Journal of Mathematics 14(2) (2016), 491-495.

Cf. A006753 (Smith numbers), A176670, A174460, A178213, A178193, A178204.

a(21) corrected by Donovan Johnson, Jan 02 2013

cp's OEIS Frontend

A278909 Binary Smith numbers: composite numbers n such that sum of bits of n = sum of bits of prime factors of n (counted with multiplicity).

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A067077 Numbers whose product of distinct prime factors is equal to its sum of digits.

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A174460 Smith numbers of order 2.

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A050219 Smaller of Smith brothers.

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A178213 Smith numbers of order 3.

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A098834 Palindromic Smith numbers.

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A104171 Reversible Smith numbers, i.e., Smith numbers whose reversal is also a Smith number.

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