A006753 -id:A006753 - cp's OEIS frontend

A331464 Numbers k such that k and k + 1 are both binary Smith numbers (A278909).

1369, 1370, 1390, 1630, 1929, 2525, 2526, 2930, 3013, 3309, 3501, 3502, 3686, 3805, 3953, 3954, 4043, 4726, 4854, 5620, 5621, 5917, 6068, 6682, 6774, 6838, 7025, 7089, 7115, 7671, 7738, 7786, 8075, 9654, 9915, 10366, 10982, 11166, 11227, 11506, 11673, 11740, 11763

Offset: 1

Amiram Eldar, Jan 17 2020

			1369 is in the sequence since both 1369 and 1369 + 1 = 1370 are binary Smith numbers.

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

Cf. A006753, A050219, A050220, A278909, A331463, A331465.

binWt[n_] := Total @ IntegerDigits[n, 2]; binSmithQ[n_] := CompositeQ[n] && Plus @@ (Last@# * binWt[ First@# ] & /@ FactorInteger[n]) == binWt[n]; seq = {}; isSmith1 = binSmithQ[1]; Do[isSmith2 = binSmithQ[n]; If[isSmith1 && isSmith2, AppendTo[seq, n-1]]; isSmith1 = isSmith2, {n, 2, 12000}]; seq

A036924 Digit sum of composite even number equals digit sum of juxtaposition of its prime factors (counted with multiplicity).

Original entry on oeis.org

4, 22, 58, 94, 166, 202, 274, 346, 378, 382, 438, 454, 526, 562, 576, 588, 634, 636, 648, 654, 666, 690, 706, 728, 762, 778, 852, 922, 958, 1086, 1282, 1284, 1376, 1626, 1642, 1678, 1736, 1776, 1822, 1842, 1858, 1872, 1894, 1908, 1952, 1962, 1966, 2038

Offset: 1

Patrick De Geest, Jan 04 1999

Even Smith numbers. - Robert Israel, Aug 24 2024

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006753, A019506, A036925.

filter:= proc(n) local F;
  F:= ifactors(n)[2];
  convert(convert(n,base,10),`+`) = convert(map(t -> t[2]*convert(convert(t[1],base,10),`+`), F),`+`)
end proc:
select(filter, [seq(i,i=4..10000,2)]); # Robert Israel, Aug 24 2024

d[n_] := IntegerDigits[n]; co[n_,k_] := Nest[Flatten[d[{#,n}]]&, n, k-1]; t={}; Do[If[!PrimeQ[n] && Total[d[n]] == Total[Flatten[d[co@@@FactorInteger[n]]]], AppendTo[t,n]], {n,4,2040,2}]; t (* Jayanta Basu, Jun 04 2013 *)

Title made more precise by Sean A. Irvine, Nov 30 2020

A036925 Digit sum of composite odd number equals digit sum of juxtaposition of its prime factors (counted with multiplicity).

Original entry on oeis.org

27, 85, 121, 265, 319, 355, 391, 483, 517, 535, 627, 645, 663, 729, 825, 861, 895, 913, 915, 985, 1111, 1165, 1219, 1255, 1449, 1507, 1581, 1633, 1755, 1795, 1881, 1903, 1921, 1935, 2067, 2079, 2155, 2173, 2227, 2265, 2373, 2409, 2461, 2475, 2515, 2583

Offset: 1

Patrick De Geest, Jan 04 1999

Odd Smith numbers. - Robert Israel, Aug 25 2024

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006753, A019506, A036924.

filter:= proc(n) local F;
  if isprime(n) then return false fi;
  F:= ifactors(n)[2];
  convert(convert(n, base, 10), `+`) = convert(map(t -> t[2]*convert(convert(t[1], base, 10), `+`), F), `+`)
end proc:
select(filter, [seq(i,i=9..10000,2)]); # Robert Israel, Aug 25 2024

Title made more precise by Sean A. Irvine, Nov 30 2020

A060209 Dunckley sequence: number of bases in which the n-th composite number is a Smith number.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 1, 1, 1, 4, 1, 2, 3, 1, 2, 2, 2, 3, 1, 4, 1, 3, 3, 5, 1, 4, 3, 1, 3, 1, 1, 5, 6, 2, 2, 1, 1, 7, 1, 2, 2, 4, 6, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 2, 2, 2, 5, 3, 7, 3, 2, 4, 1, 1, 6, 3, 1, 4, 2, 3, 2, 3, 1, 1, 1, 5, 2, 4, 1, 5, 5, 1, 3, 2, 1, 5, 3, 2

Offset: 1

Alfred and Dominic Vella (dunckley(AT)thevellas.freeserve.co.uk), Mar 19 2001

			The first 4 composite numbers, 4, 6, 8, and 9, are not Smith numbers in any base, so a(n) = 0 for n = 1 to 4.
A002808(5) = 10 is a Smith number in one base, 4, so a(5) = 1.

A. Vella and D. Vella, On Smith and Dunckley Numbers, Mathematics Today (Bull. Inst. Math. Appl), Vol. 37, No. 2 (2001), 54-56.
A. Vella and D. Vella, More Properties of Dunckley Numbers (in preparation).

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

Cf. A002808, A006753.

digSum[n_, b_] := Plus @@ IntegerDigits[n, b]; smithCount[n_] := If[! CompositeQ[n], 0, Module[{c = 0, f = FactorInteger[n]}, p = f[[;; , 1]]; e = f[[;; , 2]]; Do[If[Total[e*(digSum[#, b] & /@ p)] == digSum[n, b], c++], {b, 2, n}]; c]]; smithCount /@ Select[Range[100], CompositeQ] (* Amiram Eldar, Aug 21 2020 *)

a(1) added and offset corrected by Amiram Eldar, Aug 21 2020

A067170 Numbers n such that sum of the cubes of the distinct prime factors of n equals the sum of the cubes of the digits of n.

Original entry on oeis.org

2, 3, 5, 7, 250, 735, 2500, 25000, 250000, 1858560, 2500000, 18585600, 25000000, 91990080, 185856000, 242121642, 250000000, 919900800, 1081088775, 1390120992, 1768635648, 1858560000, 2500000000, 5435938431, 7245987840, 9199008000, 9475854336, 17996666688, 18585600000, 24214634829, 25000000000

Offset: 1

Joseph L. Pe, Feb 18 2002

If 10*m is a term (e.g. m = 25, 185856, 9199008), then 10^k * m is a term for all k >= 1. Therefore this sequence is infinite. - Amiram Eldar, Sep 28 2019

The sum of cubes of digits of a k-digit number is at most 729*k. Therefore any term with at most k digits is p-smooth where p is the largest prime < (729*k)^(1/3). - David A. Corneth, Sep 28 2019

			The prime factors of 735 are 3,5,7, the sum of whose cubes = 495 = sum of the cubes of the digits of 735; so 735 is a term of the sequence.

David A. Corneth, Table of n, a(n) for n = 1..11790 (first 260 terms from Giovanni Resta, terms < 10^34)

Cf. A005064, A006753, A055012, A067184.

f[n_] := Module[{a, l, t, r}, a = FactorInteger[n]; l = Length[a]; t = Table[a[[i]][[1]], {i, 1, l}]; r = Sum[(t[[i]])^3, {i, 1, l}]]; g[n_] := Module[{b, m, s}, b = IntegerDigits[n]; m = Length[b]; s = Sum[(b[[i]])^3, {i, 1, m}]]; Select[Range[2, 10^6], f[ # ] == g[ # ] &]

sd(n) = my(d=digits(n)); sum(k=1, #d, d[k]^3); \\ A055012
sp(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^3); \\ A005064
isok(n) = sp(n) == sd(n); \\ Michel Marcus, Sep 28 2019

a(10)-a(14) from Amiram Eldar, Sep 28 2019
a(15)-a(18) from Michel Marcus, Sep 28 2019
a(20)-a(29) from David A. Corneth, Sep 28 2019
Missing a(19) from Giovanni Resta, Sep 28 2019

A067183 Product of the prime factors of n equals the product of the digits of n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 135, 175, 735, 1176, 1715, 131712

Offset: 1

Joseph L. Pe, Feb 18 2002

Terms are zeroless 7-smooth numbers (cf. A238985). - David A. Corneth, Sep 14 2022

			The prime factors of 1176 are 2,3,7 which have product = 42, the product of the digits of 1176, so 1176 is a term of the sequence.

Cf. A002473, A006753, A052382, A075048, A238985, A357132.

Do[ If[ Apply[ Times, Transpose[ FactorInteger[n]] [[1]] ] == Apply[ Times, IntegerDigits[n]], Print[n]], {n, 2, 2*10^7} ]
Select[Range[2,1000000],Times@@Transpose[FactorInteger[#]][[1]] == Times@@ IntegerDigits[#]&] (* Harvey P. Dale, Mar 19 2012 *)

is(n) = {if(n == 1, return(1)); my(f = factor(n, 7), d = digits(n)); if(f[#f~, 1] > 7, return(0)); vecprod(f[,1]) == vecprod(d)} \\ David A. Corneth, Sep 14 2022

Edited and extended by Robert G. Wilson v, Feb 19 2002

a(1)=1 inserted by Alois P. Heinz, Sep 14 2022

A067184 Numbers n such that sum of the squares of the prime factors of n equals the sum of the squares of the digits of n.

Original entry on oeis.org

2, 3, 5, 7, 250, 735, 792, 2500, 4992, 9075, 11760, 25000, 30625, 67914, 91476, 117600, 185625, 187278, 250000, 264992, 523908, 630784, 855360, 1082565, 1176000, 2395008, 2500000, 2546775, 2898350, 3608550, 3833280, 4299750, 4790016, 5899068, 8553600, 9243850

Offset: 1

Joseph L. Pe, Feb 18 2002

From David A. Corneth, Sep 28 2019: (Start)
If 10*m is in the sequence then so is 100*m.
The sum of squares of digits of a k-digit number is at most 81*k. Therefore any term with at most k digits is p-smooth where p is the largest prime < (81*k)^(1/2). (End)

			The prime factors of 4992 are 2,3,13, the sum of whose squares = 182 = sum of the squares of 4,9,9,2; so 4992 is a term of the sequence.

David A. Corneth, Table of n, a(n) for n = 1..14898 (terms < 10^20)

Cf. A006753, A067170.

f[n_] := Module[{a, l, t, r}, a = FactorInteger[n]; l = Length[a]; t = Table[a[[i]][[1]], {i, 1, l}]; r = Sum[(t[[i]])^2, {i, 1, l}]]; g[n_] := Module[{b, m, s}, b = IntegerDigits[n]; m = Length[b]; s = Sum[(b[[i]])^2, {i, 1, m}]]; Select[Range[2, 10^5], f[ # ] == g[ # ] &]
Select[Range[2,4300000],Total[Transpose[FactorInteger[#]][[1]]^2]== Total[ IntegerDigits[#]^2]&] (* Harvey P. Dale, Sep 01 2011 *)

a(16)-a(32) from Donovan Johnson, Sep 29 2009

A098840 Smith triangular numbers.

Original entry on oeis.org

378, 666, 861, 2556, 5253, 7503, 10296, 16653, 27261, 28920, 29890, 32896, 46056, 72771, 84255, 85905, 92235, 94395, 120786, 132870, 141778, 157641, 215496, 328455, 345696, 385881, 386760, 396495, 424581, 529935, 533028, 588070, 654940

Offset: 1

Shyam Sunder Gupta, Oct 10 2004

			a(1) = 378 because 378 is a Smith number as well as a triangular number.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 83-125. See also Smith Numbers, Ch. 4, 127-157.

Intersection of A000217 and A006753.

Rest[Select[Accumulate[Range[1500]],!PrimeQ[#]&&Total[IntegerDigits[#]] == Total[Flatten[ IntegerDigits/@Flatten[Table[#[[1]],{#[[2]]}]&/@ FactorInteger[ #]]]]&]] (* Harvey P. Dale, Oct 20 2012 *)

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

A105648 Smallest member of a set of Smith triples.

Original entry on oeis.org

73615, 209065, 225951, 283745, 305455, 342879, 656743, 683670, 729066, 747948, 774858, 879221, 954590, 1185547, 1262722, 1353955, 1369374, 1495718, 1622495, 1666434, 1790480, 2197579, 2299772, 2428854, 2561678, 2576441, 2580367, 2636516, 2665480, 2707580, 2741816

Offset: 1

Shyam Sunder Gupta, May 03 2005

If there are 3 consecutive numbers which are Smith numbers, these can be called a Smith triple.

			a(1) = 73615 because 73615 is the smallest of 3 consecutive integers which are Smith numbers, i.e., the three consecutive numbers 73615, 73616, 73617 are all Smith numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000
S. S. Gupta, Smith Numbers.

Cf. A006753, A050219, A059754, A105649, A105650.

digSum[n_] := Plus @@ IntegerDigits[n]; smithQ[n_] := CompositeQ[n] && Plus @@ (Last@#*digSum[First@#] & /@ FactorInteger[n]) == digSum[n]; sm = smithQ /@ Range[3]; seq = {}; Do[sm = Join[Rest[sm], {smithQ[k]}]; If[And @@ sm, AppendTo[seq, k - 2]], {k, 4, 10^6}]; seq (* Amiram Eldar, Aug 18 2020 *)

More terms from Amiram Eldar, Aug 18 2020

cp's OEIS Frontend

A331464 Numbers k such that k and k + 1 are both binary Smith numbers (A278909).

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A036924 Digit sum of composite even number equals digit sum of juxtaposition of its prime factors (counted with multiplicity).

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A036925 Digit sum of composite odd number equals digit sum of juxtaposition of its prime factors (counted with multiplicity).

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A060209 Dunckley sequence: number of bases in which the n-th composite number is a Smith number.

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A067170 Numbers n such that sum of the cubes of the distinct prime factors of n equals the sum of the cubes of the digits of n.

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A067183 Product of the prime factors of n equals the product of the digits of n.

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A067184 Numbers n such that sum of the squares of the prime factors of n equals the sum of the squares of the digits of n.

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A098840 Smith triangular numbers.

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