A050219 -id:A050219 - 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

A050220 Larger of Smith brothers.

Original entry on oeis.org

729, 2965, 3865, 4960, 5936, 6188, 9387, 9634, 11696, 13765, 16537, 16592, 20785, 25429, 28809, 29624, 32697, 33633, 35806, 39586, 43737, 44734, 49028, 55345, 56337, 57664, 58306, 62635, 65913, 65975, 66651, 67068, 67729, 69280, 69836, 73616, 73617, 74169

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, A050219.

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_] := n > 1 && !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 s
        prev = s
print(list(islice(agen(), 38))) # Michael S. Branicky, Dec 23 2022

Offset corrected by Arkadiusz Wesolowski, May 08 2012

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

Original entry on oeis.org

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

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

A105649 Smallest member of set of 4 consecutive numbers which are Smith numbers.

Original entry on oeis.org

4463535, 6356910, 8188933, 9425550, 11148564, 15966114, 15966115, 18542654, 21673542, 22821992, 23767287, 28605144, 36615667, 39227466, 47096634, 47395362, 48072396, 54054264, 55464835, 57484614, 57756450, 57761165, 58418508, 61843387, 62577157, 64572186, 65484066

Offset: 1

Shyam Sunder Gupta, May 03 2005

			a(1) = 4463535 because 4463535 is the smallest member of a set of 4 consecutive numbers which are Smith numbers i.e. four consecutive numbers 4463535, 4463536, 4463537, 4463538 are all Smith numbers.

Amiram Eldar, Table of n, a(n) for n = 1..1000
S. S. Gupta, Smith Numbers.

Cf. A006753, A050219, A059754, A105648, A105650.

digSum[n_] := Plus @@ IntegerDigits[n]; smithQ[n_] := CompositeQ[n] && Plus @@ (Last @#* digSum[First@#] & /@ FactorInteger[n]) == digSum[n]; sm = smithQ /@ Range[4]; seq = {}; Do[sm = Join[Rest[sm], {smithQ[k]}]; If[And @@ sm, AppendTo[seq, k - 3]], {k, 5, 10^7}]; seq (* Amiram Eldar, Aug 18 2020 *)

a(7) inserted and more terms added by Amiram Eldar, Aug 18 2020

A329935 Numbers k such that k and k+1 are both hoax numbers (A019506).

Original entry on oeis.org

84, 516, 644, 860, 2325, 3344, 4188, 4980, 5268, 5484, 6259, 6603, 6692, 6980, 7051, 7195, 8076, 8420, 9716, 10704, 11774, 12795, 12955, 12956, 13747, 14475, 14715, 14724, 16473, 17148, 17149, 17225, 17661, 19175, 21828, 22143, 22347, 24259, 24272, 24980

Offset: 1

Amiram Eldar, Nov 24 2019

Analogous to A050219 (smaller of Smith brothers) as A019506 (hoax numbers) is analogous to A006753 (Smith numbers).

			84 is in the sequence since 84 is a hoax number: 84 = 2^2 * 3 * 7 and 8 + 4 = 2 + 3 + 7 = 12, and 85 = 84 + 1 is also a hoax number: 85 = 5 * 17 and 8 + 5 = 5 + 1 + 7 = 13.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, page 27, entry 85.

Cf. A006753, A019506, A050219, A235766.

digitSum[n_]  := Total @ IntegerDigits[n]; hoaxQ[n_] := CompositeQ[n] && Total[ digitSum /@ FactorInteger[n][[;; , 1]] ] == digitSum[n]; seq = {}; isHoax1 = hoaxQ[1]; Do[isHoax2 = hoaxQ[n]; If[isHoax1 && isHoax2, AppendTo[seq, n-1]]; isHoax1 = isHoax2, {n, 2, 25000}]; seq

A331463 Numbers k such that k and k + 1 are both binary hoax numbers (A329936).

Original entry on oeis.org

8, 15, 49, 50, 252, 489, 699, 725, 755, 799, 951, 979, 980, 988, 989, 1023, 1134, 1350, 1351, 1370, 1390, 1599, 1629, 1630, 1660, 1690, 1694, 1763, 1854, 1908, 1929, 1939, 1940, 1960, 2006, 2015, 2166, 2312, 2358, 2645, 2700, 2779, 2787, 2862, 2923, 2930, 2988

Offset: 1

Amiram Eldar, Jan 17 2020

			8 is a term since both 8 and 8 + 1 = 9 are binary hoax numbers: 8 = 2^3 in binary representation is 1000 = 10^3 and 1 + 0 + 0 + 0 = 1 + 0, and 9 = 3^2 in binary representation is 1001 = 11^2 and 1 + 0 + 0 + 1 = 1 + 1.

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

Cf. A050219, A019506, A329935, A329936, A331464.

hoax:=func; [k:k in [2..3000]|hoax(k) and hoax(k+1)]; // Marius A. Burtea, Jan 17 2020

binWt[n_] := Total @ IntegerDigits[n, 2]; binHoaxQ[n_] := CompositeQ[n] && Total[binWt /@ FactorInteger[n][[;; , 1]]] == binWt[n]; seq = {}; isHoax1 = binHoaxQ[1]; Do[isHoax2 = binHoaxQ[n]; If[isHoax1 && isHoax2, AppendTo[seq, n-1]]; isHoax1 = isHoax2, {n, 2, 3000}]; seq

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A050218 Sums of digits of Smith numbers A006753.

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A050220 Larger of Smith brothers.

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

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A105648 Smallest member of a set of Smith triples.

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A105649 Smallest member of set of 4 consecutive numbers which are Smith numbers.

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A329935 Numbers k such that k and k+1 are both hoax numbers (A019506).

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Mathematica

A331463 Numbers k such that k and k + 1 are both binary hoax numbers (A329936).

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Magma

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