A006753 -id:A006753 - cp's OEIS frontend

A050218 Sums of digits of Smith numbers A006753.

4, 4, 9, 13, 13, 13, 4, 13, 4, 13, 13, 13, 13, 13, 18, 13, 13, 15, 13, 15, 13, 13, 13, 13, 18, 21, 15, 13, 15, 15, 18, 15, 15, 18, 15, 13, 17, 18, 15, 22, 15, 15, 15, 22, 13, 15, 13, 22, 22, 15, 4, 13, 13, 13, 13, 15, 17, 18, 13, 15, 15, 13, 13, 22, 17, 18, 21, 22, 13, 15

Offset: 1

Eric W. Weisstein

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Smith Number.
Wikipedia, Smith number.

Cf. A006753, A050219, A050220.

Cf. A202388, A050223.

d[n_]:=IntegerDigits[n]; tr[n_]:=Transpose[FactorInteger[n]]; t={}; Do[If[!PrimeQ[n]&&(x=Total[d[n]])==Total[d@tr[n][[1]]*tr[n][[2]],2],AppendTo[t,x]],{n,4,1850}]; t (* Jayanta Basu, Jun 04 2013 *)

a(n) = A007953(A006753(n)).

Offset corrected by Reinhard Zumkeller, Dec 19 2011

A202388 Digital root of Smith numbers A006753.

Original entry on oeis.org

4, 4, 9, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 9, 4, 4, 6, 4, 6, 4, 4, 4, 4, 9, 3, 6, 4, 6, 6, 9, 6, 6, 9, 6, 4, 8, 9, 6, 4, 6, 6, 6, 4, 4, 6, 4, 4, 4, 6, 4, 4, 4, 4, 4, 6, 8, 9, 4, 6, 6, 4, 4, 4, 8, 9, 3, 4, 4, 6, 4, 9, 9, 4, 4, 9, 4, 9, 8, 9, 4, 4, 6, 9, 4, 4

Offset: 1

Reinhard Zumkeller, Dec 19 2011

a(n) = A010888(A006753(n)); range = {4,6,8,9}.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Smith Number
Wikipedia, Smith number

Cf. A050218, A202393.

d[n_]:=IntegerDigits[n]; dr[n_]:= NestWhile[Total[d[#]]&,n,#>9&]; tr[n_]:=Transpose[FactorInteger[n]]; t1=Select[Range[4,2.2*10^3],!PrimeQ[#]&&Total[d[#]]==Total[d@tr[#][[1]]*tr[#][[2]],2]&]; Table[dr[n],{n,t1}] (* t1 gives Smith numbers - Jayanta Basu, Jun 04 2013 *)

A059754 Smallest term in any sequence of n consecutive Smith numbers (A006753).

Original entry on oeis.org

4, 728, 73615, 4463535, 15966114, 2050918644, 164736913905, 8090674745553

Offset: 1

Shyam Sunder Gupta, Feb 11 2001

			The third number is 73615 since 73615, 73616, 73617 is the first example of 3 consecutive Smith numbers.

R. K. Guy, Unsolved Problems in Number Theory, Section B49.
Samuel Yates, Smith Numbers Congruent to 4 (Mod 9), Journal of Recreational Mathematics, Vol. 19(2), 1987.

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

Cf. A006753.

a(6) from Carlos Rivera, Dec 19 2003
a(7) from Jens Kruse Andersen. Max Alekseyev, Apr 21 2010
a(8) from Max Alekseyev, Oct 11 2010

A202387 Squarefree Smith numbers, cf. A006753.

Original entry on oeis.org

22, 58, 85, 94, 166, 202, 265, 274, 319, 346, 355, 382, 391, 438, 454, 483, 517, 526, 535, 562, 627, 634, 645, 654, 663, 690, 706, 762, 778, 861, 895, 913, 915, 922, 958, 985, 1086, 1111, 1165, 1219, 1255, 1282, 1507, 1581, 1626, 1633, 1642, 1678, 1795, 1822

Offset: 1

Reinhard Zumkeller, Dec 19 2011

Intersection of A006753 and A005117;
also squarefree hoax numbers: intersection of A019506 and A005117;
squarefree composite numbers m such that sum of digits of m = sum of digits of all prime factors of m.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Smith Number
Eric Weisstein's World of Mathematics, Hoax Number
Wikipedia, Smith number

Cf. A007953, A027746, A120944.

a202387 n = a202387_list !! (n-1)
a202387_list = [x | x <- a120944_list,
                    a007953 x == sum (map a007953 (a027746_row x))]

A382896 Smith sphenic numbers, i.e., Smith numbers (A006753) that are the product of three distinct prime numbers.

Original entry on oeis.org

438, 483, 627, 645, 654, 663, 762, 861, 915, 1086, 1581, 1626, 1842, 2067, 2265, 2373, 2409, 2679, 2751, 3138, 3246, 3345, 3615, 4173, 4191, 4209, 4974, 5253, 5298, 5397, 5946, 6054, 6315, 6531, 6567, 6585, 6603, 6693, 6702, 6855, 6981, 7026, 7089, 7287

Offset: 1

Shyam Sunder Gupta, Apr 08 2025

			438 is a term since 438 = 2 * 3 * 73, so it is the product of three distinct prime numbers (2, 3, and 73) and also a Smith number (4 + 3 + 8 = 2 + 3 + 7 + 3).

Shyam Sunder Gupta, Table of n, a(n) for n = 1..7523
Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.

Intersection of A006753 and A007304.

A277580 Numbers that are both Smith (A006753) and Lucas-Carmichael (A006972).

Original entry on oeis.org

8164079, 8421335, 21408695, 30071327, 47324639, 77350559, 103727519, 121538879, 134151479, 202767551, 239875559, 287432495, 306871487, 466861199, 560974259, 566019167, 574342145, 592557119, 594633599, 602758079, 677913599, 832477799

Offset: 1

Max Alekseyev, Oct 21 2016

Intersection of A006753 and A006972.

A357841 Smith numbers (A006753) for which the arithmetic derivative (A003415) is also a Smith number.

Original entry on oeis.org

4, 27, 85, 121, 166, 265, 517, 526, 634, 706, 778, 913, 985, 1633, 1822, 1966, 2173, 2218, 2326, 2434, 2605, 2785, 3505, 3802, 3865, 3973, 4306, 4369, 4765, 4918, 5248, 5674, 5818, 5926, 6178, 6385, 7186, 7726, 8185, 8257, 8653, 9193, 9301, 10201, 10489, 10606

Offset: 1

Marius A. Burtea, Oct 20 2022

			4 = A006753(1) and 4' = 4, so 4 is a term.
27 = A006753(3) and 27' = 27, so 27 is a term.
85  = A006753(5) and 85' = 22 = A006753(2), so 85 is a term.

Cf. A003415, A006753.

sm:=func; f:=func; [n:n in [2..10700]|sm(n) and sm(Floor(f(n)))];

digsum[n_] := Total@IntegerDigits[n]; smithQ[n_] := CompositeQ[n] && Plus @@ (Last[#]*digsum[First@#] & /@ FactorInteger[n]) == digsum[n]; d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[10^4], smithQ[#] && smithQ[d[#]] &] (* Amiram Eldar, Oct 21 2022 *)

A019506 Hoax numbers: composite numbers whose digit-sum equals the sum of the digit-sums of its distinct prime factors.

Original entry on oeis.org

22, 58, 84, 85, 94, 136, 160, 166, 202, 234, 250, 265, 274, 308, 319, 336, 346, 355, 361, 364, 382, 391, 424, 438, 454, 456, 476, 483, 516, 517, 526, 535, 562, 627, 634, 644, 645, 650, 654, 660, 663, 690, 702, 706, 732, 735, 762, 778, 855, 860

Offset: 1

Mario Velucchi (mathchess(AT)velucchi.it)

			22 = 2*11 and digit-sum(22) = 4 = digit-sum(2) + digit-sum(11).

Reinhard Zumkeller, 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.
Eric Weisstein's World of Mathematics, Hoax Number

Cf. A006753.

Cf. A002808, A027748, A001221, A007953, A050223, A202393, A202387.

a019506 n = a019506_list !! (n-1)
a019506_list = [x | x <- a002808_list,
                    a007953 x == sum (map a007953 (a027748_row x))]
-- Reinhard Zumkeller, Dec 19 2011

Select[Range[2,1000],!PrimeQ[#]&&Total[Flatten[IntegerDigits/@ Transpose[ FactorInteger[#]][[1]]]]==Total[IntegerDigits[#]]&] (* Harvey P. Dale, Feb 24 2013 *)

isok(m) = !isprime(m) && (sumdigits(m) == vecsum(apply(sumdigits, factor(m)[,1]))); \\ Michel Marcus, Feb 03 2022

A007953(a(n)) = sum(A007953(A027748(a(n),k)): k=1..A001221(a(n))) and A066247(a(n)) = 1. [Reinhard Zumkeller, Dec 19 2011]

A178193 Smith numbers of order 4.

Original entry on oeis.org

3777, 7773, 17418, 30777, 53921, 66111, 97731, 111916, 119217, 122519, 128131, 133195, 135488, 138878, 145229, 178814, 180174, 198581, 257376, 269636, 281179, 296396, 317686, 358256, 362996, 366514, 394114, 435777, 457377, 469552, 475856, 502960, 513833

Offset: 1

Paul Weisenhorn, Dec 19 2010

Composite numbers n not in A176670 such that the sum of the 4th power of the digits of n equals the sum of the 4th power 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.

			3777 = 3*1259 is composite; sum of 4th power of the digits is 3^4 + 7^4 + 7^4 + 7^4 = 7284. Sum of 4th power of the digits of the prime factors 3, 1259 is 3^4 + 1^4 + 2^4 + 5^4 + 9^4 = 7284. The sums are equal, so 3777 is in the sequence.
17418 = 2*3*2903 is composite; sum of 4th power of the digits is 1^4 + 7^4 + 4^4 + 1^4 + 8^4 = 6755. Sum of 4th power of the digits of the prime factors 2, 3, 2903 is 2^4 + 3^4 + 2^4 + 9^4 + 0^4 + 3^4 = 6755. The sums are equal, so 17418 is in the sequence.
269636 = 2*2*67409 is composite; sum of 4th power of the digits is 2^4 + 6^4 + 9^4 + 6^4 + 3^4 + 6^4 = 10546. Sum of 4th power of the digits of the prime factors 2, 2, 67409 (with multiplicity) is 2^4 + 2^4 + 6^4 + 7^4 + 4^4 + 0^4 + 9^4 = 10546. The sums are equal, so 269636 is in the sequence.

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, 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]]; id != fid && Plus @@ (id^4) == Plus @@ (fid^4)]; k = 1; lst = {}; While[k < 10^6, If[f Q@ k, AppendTo[lst, k]; Print@ k]; k++]; lst

A176670 Composite numbers having the same digits as their prime factors (with multiplicity), excluding zero digits.

Original entry on oeis.org

1111, 1255, 12955, 17482, 25105, 28174, 51295, 81229, 91365, 100255, 101299, 105295, 107329, 110191, 110317, 117067, 124483, 127417, 129595, 132565, 137281, 145273, 146137, 149782, 163797, 171735, 174082, 174298, 174793, 174982, 193117, 208174, 210181, 217894

Offset: 1

Paul Weisenhorn, Apr 23 2010

Subsequence of A006753 (Smith numbers).
These numbers still need a better name. - Ely Golden, Dec 25 2016
Terms of this sequence never have more zero digits than their prime factors. - Ely Golden, Jan 10 2017

			n = 25105 = 5*5021; both n and the factorization of n have digits 1, 2, 5, 5; sorted and excluding zeros.
n = 110191 = 101*1091; both n and the factorization of n have digits 1, 1, 1, 1, 9; sorted and excluding zeros.
n = 171735 = 3*5*107*107; both n and the factorization of n have digits 1, 1, 3, 5, 7, 7; sorted and excluding zeros.

Ely Golden, Table of n, a(n) for n = 1..10000 [Terms 1 through 2113 were computed by Paul Weisenhorn; and terms 2114 to 10000 by Ely Golden, Nov 30 2016]
Ely Golden, Smith number sequence generator optimized for A176670
Ely Golden, Proof that A280827(n)>=0 for all n>1
Eric W. Weisstein, Smith Number

Cf. A006753.

fQ[n_] := Block[{id = Sort@ IntegerDigits@ n, s = Sort@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@ n]}, While[ id[[1]] == 0, id = Drop[id, 1]]; While[ s[[1]] == 0, s = Drop[s, 1]]; n > 1 && ! PrimeQ@ n && s == id]; Select[ Range@ 200000, fQ]
Select[Range[2*10^5], Function[n, And[CompositeQ@ n, Sort@ DeleteCases[#, 0] &@ IntegerDigits@ n == Sort@ DeleteCases[#, 0] &@ Flatten@ Map[IntegerDigits@ ConstantArray[#1, #2] & @@ # &, FactorInteger@ n]]]] (* Michael De Vlieger, Dec 10 2016 *)

from sympy import factorint, flatten
def sd(n): return sorted(str(n).replace('0', ''))
def ok(n):
  f = factorint(n)
  return sum(f[p] for p in f) > 1 and sd(n) == sorted(flatten(sd(p)*f[p] for p in f))
print(list(filter(ok, range(220000)))) # Michael S. Branicky, Apr 22 2021

cp's OEIS Frontend

A050218 Sums of digits of Smith numbers A006753.

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A202388 Digital root of Smith numbers A006753.

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Mathematica

A059754 Smallest term in any sequence of n consecutive Smith numbers (A006753).

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A202387 Squarefree Smith numbers, cf. A006753.

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A382896 Smith sphenic numbers, i.e., Smith numbers (A006753) that are the product of three distinct prime numbers.

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A277580 Numbers that are both Smith (A006753) and Lucas-Carmichael (A006972).

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A357841 Smith numbers (A006753) for which the arithmetic derivative (A003415) is also a Smith number.

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Magma

Mathematica

A019506 Hoax numbers: composite numbers whose digit-sum equals the sum of the digit-sums of its distinct prime factors.

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A178193 Smith numbers of order 4.

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