A006753 -id:A006753 - cp's OEIS frontend

A098836 Deficient Smith numbers.

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 382, 391, 454, 483, 517, 526, 535, 562, 627, 634, 645, 663, 706, 729, 778, 825, 861, 895, 913, 915, 922, 958, 985, 1111, 1165, 1219, 1255, 1282, 1449, 1507, 1581, 1633, 1642, 1678, 1755, 1795

Offset: 1

Shyam Sunder Gupta, Oct 10 2004

			a(4) = 58 because 58 is a Smith number as well as a deficient number.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.

Intersection of A005100 and A006753.

sndnQ[n_]:=!PrimeQ[n]&&DivisorSigma[1,n]<2n&&Total[Flatten[ IntegerDigits/@ (Flatten[ Table[#[[1]],{#[[2]]}]&/@ FactorInteger[ n]])]]==Total[ IntegerDigits[ n]]; Select[Range[2,2000],sndnQ] (* Harvey P. Dale, Sep 10 2013 *)

A098837 Smith semiprimes.

Original entry on oeis.org

4, 22, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 382, 391, 454, 517, 526, 535, 562, 634, 706, 778, 895, 913, 922, 958, 985, 1111, 1165, 1219, 1255, 1282, 1507, 1633, 1642, 1678, 1795, 1822, 1858, 1894, 1903, 1921, 1966, 2038, 2155, 2173, 2182, 2218

Offset: 1

Shyam Sunder Gupta, Oct 10 2004

			a(3)=58 because 58 is a Smith number as well as a semiprime.

Charles R Greathouse IV, 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. A001358, A006753.

N:= 10000: # for terms <= N
P:= select(isprime, [2,seq(i,i=3..N/2,2)]):
nP:= nops(P):
sP:= map(p -> convert(convert(p,base,10),`+`), P):
Res:= {}:
for i from 1 to nP do
  for j from i to nP do
    n:= P[i]*P[j];
    if n > N then break fi;
    if convert(convert(n,base,10),`+`) = sP[i]+sP[j] then
      Res:= Res union {n}
    fi
od od:
sort(convert(Res,list)); # Robert Israel, Aug 24 2024

sspQ[n_]:=PrimeOmega[n]==2&&Total[Flatten[IntegerDigits/@(Table[#[[1]],#[[2]]]&/@FactorInteger[n])]]==Total[IntegerDigits[n]]; Select[Range[ 2220], sspQ] (* Harvey P. Dale, Jul 25 2019 *)

dsum(n)=my(s);while(n,s+=n%10;n\=10);s
list(lim)=my(v=List(),d); forprime(p=2, sqrt(lim), d=dsum(p); forprime(q=p, lim\p, if(d+dsum(q)==dsum(p*q),listput(v, p*q)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jan 03 2012

A098838 Smith cubic numbers.

Original entry on oeis.org

27, 729, 19683, 474552, 7077888, 7414875, 8489664, 62099136, 112678587, 236029032, 246491883, 257259456, 279726264, 345948408, 463684824, 567663552, 638277381, 721734273, 766060875, 988047936, 1177583616, 1412467848, 2131746903, 2493326016, 2714704875

Offset: 1

Shyam Sunder Gupta, Oct 10 2004

			a(1) = 27 because 27 is a Smith number as well as a cube.

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.

d[n_]:=IntegerDigits[n]; tr[n_]:=Transpose[FactorInteger[n]]; t={}; Do[If[!PrimeQ[n]&&Total[d[n]]==Total[d@tr[n][[1]]*tr[n][[2]],2],AppendTo[t,n]],{n,Range[2,1400]^3}]; t (* Jayanta Basu, Jun 04 2013 *)

A098839 Smith square numbers.

Original entry on oeis.org

4, 121, 576, 729, 6084, 10201, 17424, 18496, 36481, 51529, 100489, 124609, 184041, 195364, 410881, 559504, 674041, 695556, 732736, 887364, 896809, 966289, 988036, 1038361, 1190281, 1238769, 1726596, 1852321, 2166784, 2975625, 3407716, 3613801, 3663396

Offset: 1

Shyam Sunder Gupta, Oct 10 2004

			a(2) = 121 because 121 is a Smith number as well as a square.

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.

d[n_]:=IntegerDigits[n]; tr[n_]:=Transpose[FactorInteger[n]]; t={}; Do[If[!PrimeQ[n]&&Total[d[n]]==Total[d@tr[n][[1]]*tr[n][[2]],2],AppendTo[t,n]],{n,Range[2,1950]^2}]; t (* Jayanta Basu, Jun 04 2013 *)

A103124 1/5-Smith numbers.

Original entry on oeis.org

399996663, 666609999, 669969663, 690696969, 699966663, 2789929969, 3066999963, 3366339999, 3366999933, 3399696663, 3399996633, 3666699663, 3669933993, 3933969693, 6066690999, 6069996663, 6099996633, 6393996933, 6399636963, 6666009999, 6669669633, 6966939633

Offset: 1

Shyam Sunder Gupta, Mar 16 2005

			399996663 is a 5^(-1) Smith number because the digit sum of 399996663, i.e., S(399996663) = 3 + 9 + 9 + 9 + 9 + 6 + 6 + 6 + 3 = 60, which is equal to 5 times the sum of the digits of its prime factors, i.e., 5*Sp(399996663) = 5*Sp(3*11*101*120011) = 5*(3 + 1 + 1 + 1 + 0 + 1 + 1 + 2 + 0 + 0 + 1 + 1) = 60.

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, 25(1987), pp. 76-80.

Cf. A006753.

a(6)-a(22) from Donovan Johnson, Sep 20 2011

A104167 Numbers which when multiplied by any repunit prime Rp give a Smith number.

Original entry on oeis.org

1540, 1720, 2170, 2440, 5590, 6040, 7930, 8344, 8470, 8920, 23590, 24490, 25228, 29080, 31528, 31780, 33544, 34390, 35380, 39970, 40870, 42490, 42598, 43480, 44380, 45955, 46270, 46810, 46990, 47908, 48790, 49960, 51490, 51625, 52345, 52570, 53290, 57070

Offset: 1

Shyam Sunder Gupta, Mar 10 2005

Numbers in the sequence must have a digital root of 1.

If the definition is modified, considering only repunits greater than 11, other numbers have the same property: 3304, 12070, 11080, 11620, 16030, 21340, 22330, 24130, 24220. - Mauro Fiorentini, Jul 16 2015

			1720 is a number in the sequence because 1720*Rp is always a Smith number, where Rp is a Repunit prime. Let Rp=11, so 1720*11=18920, which is a Smith number as the sum of digits of 18920 is 1+8+9+2+0 = 20 and the sum of digits of prime factors of 18920 (i.e., 2*2*2*5*11*43) is also 20 (i.e., 2+2+2+5+1+1+4+3).

Giovanni Resta, Table of n, a(n) for n = 1..1000
Shyam Sunder Gupta, Smith Numbers.
Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157. See also Repunit Numbers, Ch. 11, 327-352.
Sham Oltikar, and Keith Wayland, Construction of Smith Numbers, Mathematics Magazine, vol. 56(1), 1983, pp. 36-37.
Carlos Rivera, Problem 108: Methods for generating Smith numbers, PrimePuzzles.Net.

Cf. A006753, A004022.

A104168 Smallest Smith number with n prime factors.

Original entry on oeis.org

4, 27, 636, 378, 729, 648, 576, 2688, 17496, 44928, 75776, 168960, 765952, 319488, 958464, 5537792, 5963776, 2883584, 5767168, 7077888, 279969792, 544997376, 778567680, 2579496960, 4567597056, 3875536896, 22749904896, 60699967488, 87509958656, 164886478848, 758296608768, 199715979264, 599147937792, 5295694675968, 446676598784, 2954937499648

Offset: 2

Shyam Sunder Gupta, Mar 10 2005 and May 03 2005

a(38) > 1.286e13. - Max Alekseyev, Oct 03 2024

			a(4) = 636 because 636 is the smallest Smith number with 4 prime factors.

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.
Shyam Sunder Gupta, Smith Numbers, Mathematical Spectrum 37(1) (2004/5), 27-29.
Shyam Sunder Gupta, Smith Numbers.
Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.
Eric Weisstein's World of Mathematics, Smith number.
Wikipedia, Smith number.
Albert 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, A104169.

a(28)-a(31) from Donovan Johnson, Jan 02 2013

a(32)-a(37) from Max Alekseyev, Oct 01 2024

A104169 Highly decomposable Smith numbers. A Smith number which sets a record for the number of prime factors (counting multiplicity) starting from first Smith number is called a highly decomposable Smith number.

Original entry on oeis.org

4, 27, 378, 576, 2688, 17496, 44928, 75776, 168960, 319488, 958464, 2883584, 5767168, 7077888, 279969792, 544997376, 778567680, 2579496960, 3875536896, 22749904896, 60699967488, 87509958656, 164886478848, 199715979264, 446676598784, 2954937499648

Offset: 1

Shyam Sunder Gupta, Mar 10 2005 and May 03 2005

It is conjectured that except for 27, all highly decomposable Smith numbers are even.

a(27) > 1.286e13. - Max Alekseyev, Oct 03 2024

			a(3) = 378 because the Smith number 378 has 5 prime factors which is > any Smith number < 378.

Shyam Sunder Gupta, Smith Numbers, Mathematical Spectrum, 37 (2004/5), 27-29.

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

Subsequence of A104168.

Cf. A006753.

Number m = A104168(k) in this sequence iff m = min_{n>=k} A104168(n).

a(20)-a(23) from Donovan Johnson, Jan 02 2013

a(24)-a(26) from Max Alekseyev, Sep 28 2024

A104170 Number of Smith numbers below 10^n.

Original entry on oeis.org

1, 6, 49, 376, 3294, 29928, 278411, 2632758, 25154060, 241882509, 2335807857, 22635291815, 219935518608

Offset: 1

Shyam Sunder Gupta, Mar 10 2005

			a(4)=376 because number of Smith number below 10^4 is 376.

S. S. Gupta, Smith Numbers, Mathematical Spectrum, 37 (2004/5), 27-29.

S. S. Gupta, Smith Numbers.

Cf. A006753.

a(11), a(12) from Max Alekseyev, Jan 20 2010

a(13) from Max Alekseyev, Oct 01 2010

A176385 The smallest number which when multiplied by the n-th repunit gives a Smith number.

Original entry on oeis.org

4, 2, 6, 56, 32, 97, 6, 95, 176, 4, 32, 309, 68, 68, 194, 616, 175, 96, 1540, 4, 816, 14, 1540, 95, 840, 32, 5, 437, 50, 10336, 95, 5, 995, 976, 175, 14, 40, 570, 1976, 995, 1400, 294, 1994, 176, 544, 507, 328, 392, 77, 11020, 18905, 18050, 9995, 779, 4, 805, 669

Offset: 1

Paul Weisenhorn, Apr 16 2010, Apr 23 2010

Smith numbers, A006753: the digits-sum equals the digits-sum of its prime factors.

Repunits: R(n)=(10^n-1)/9 = A002275(n).

			R(3)=111 multiplied by a(3)=6 yields z=666=2*3*3*37 = A006753(34): 6+6+6 = 2+3+3+3+7 = 18.

Paul Weisenhorn, Table of n, a(n) for n = 1..70

# digits-sum of primfactors of z=dsp(z)
for n from 2 to 70 do f(n):=1: test:=false:
while (f(n) < 420000) and (test=false) do
f(n):=f(n)+1: z:=f(n)*r(n): ds(z):=0:
dsp(z):=dsp(r(n))+dsp(f(n)):
while (z>0) do z:=iquo(z,10,'m'): ds(z):=ds(z)+m: end do:
if(ds(z)=dsp(z)) then test:=true: print(n,f(n)): end if:
end do: end do:

Keyword:base added by R. J. Mathar, Apr 24 2010

cp's OEIS Frontend

A098836 Deficient Smith numbers.

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A098837 Smith semiprimes.

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A098838 Smith cubic numbers.

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A098839 Smith square numbers.

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A103124 1/5-Smith numbers.

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A104167 Numbers which when multiplied by any repunit prime Rp give a Smith number.

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A104168 Smallest Smith number with n prime factors.

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A104169 Highly decomposable Smith numbers. A Smith number which sets a record for the number of prime factors (counting multiplicity) starting from first Smith number is called a highly decomposable Smith number.

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A104170 Number of Smith numbers below 10^n.

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