A037268 -id:A037268 - cp's OEIS frontend

A214949 Numerator of sum of reciprocals of all nonzero digits of n in decimal representation.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 3, 1, 5, 3, 7, 2, 9, 5, 11, 1, 4, 5, 2, 7, 8, 1, 10, 11, 4, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 1, 6, 7, 8, 9, 2, 11, 12, 13, 14, 1, 7, 2, 1, 5, 11, 1, 13, 7, 5, 1, 8, 9, 10, 11, 12, 13, 2, 15, 16

Offset: 0

Reinhard Zumkeller, Aug 02 2012

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A004719, A037264, A037268, A214958, A214959.

Cf. A214950 (denominators).

import Data.Ratio ((%), numerator)
a214949 = f 0 where
   f y 0 = numerator y
   f y x = f (y + if d == 0 then 0 else 1 % d) x'
           where (x',d) = divMod x 10

nsr[n_] := Numerator[Total[1/Select[IntegerDigits[n], # > 0 &]]]; nsr /@ Range[0, 79] (* Jayanta Basu, Jul 13 2013 *)

a(n) = my(d=digits(n)); numerator(sum(k=1, #d, if (d[k], 1/d[k]))); \\ Michel Marcus, Jan 26 2022

a(A037264(n)) = a(A037268(n)) = a(A214958(n)) = a(A214959(n)) = 1;

a(n) = a(A004719(n)).

A214950 Denominator of sum of reciprocals of all nonzero digits of n in decimal representation.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 2, 1, 6, 4, 10, 3, 14, 8, 18, 3, 3, 6, 3, 12, 15, 2, 21, 24, 9, 4, 4, 4, 12, 2, 20, 12, 28, 8, 36, 5, 5, 10, 15, 20, 5, 30, 35, 40, 45, 6, 6, 3, 2, 12, 30, 3, 42, 24, 18, 7, 7, 14, 21, 28, 35, 42

Offset: 0

Reinhard Zumkeller, Aug 02 2012

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A034708, A037268, A214957, A214959, A004719.

Cf. A214949 (numerators).

import Data.Ratio ((%), denominator)
a214950 = f 0 where
   f y 0 = denominator y
   f y x = f (y + if d == 0 then 0 else 1 % d) x'
           where (x',d) = divMod x 10

dsr[n_] := Denominator[Total[1/Select[IntegerDigits[n], # > 0 &]]]; dsr /@ Range[0, 76] (* Jayanta Basu, Jul 13 2013 *)

a(n) = my(d=digits(n)); denominator(sum(k=1, #d, if (d[k], 1/d[k]))); \\ Michel Marcus, Jan 26 2022

a(A034708(n)) = a(A037268(n)) = a(A214957(n)) = a(A214959(n)) = 1;

a(n) = a(A004719(n)).

A214959 Numbers for which the sum of reciprocals of nonzero digits = 1.

Original entry on oeis.org

1, 10, 22, 100, 202, 220, 236, 244, 263, 326, 333, 362, 424, 442, 623, 632, 1000, 2002, 2020, 2036, 2044, 2063, 2200, 2306, 2360, 2404, 2440, 2488, 2603, 2630, 2666, 2848, 2884, 3026, 3033, 3062, 3206, 3260, 3303, 3330, 3366, 3446, 3464, 3602, 3620, 3636

Offset: 1

Reinhard Zumkeller, Aug 02 2012

Intersection of A214957 and A214958: A214949(a(n))*A214950(a(n)) = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A037268 (subsequence).

import Data.Ratio ((%), numerator, denominator)
a214959 n = a214959_list !! (n-1)
a214959_list = [x | x <- [0..], f x 0] where
   f 0 v = numerator v == 1 && denominator v == 1
   f u v | d > 0     = f u' (v + 1 % d)
         | otherwise = f u' v  where (u',d) = divMod u 10

SumReciprocalsDigits:=func; [n: n in [1..3636] | IsOne(SumReciprocalsDigits(n))]; // Bruno Berselli, Aug 02 2012

idnQ[n_]:=Total[1/Select[IntegerDigits[n],#>0&]]==1; Select[Range[ 4000],idnQ] (* Harvey P. Dale, Dec 08 2012 *)

A037265 Integers arising from A037264.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 2, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane

The remaining terms are all 1.

T. D. Noe, Table of n, a(n) for n=1..1232 (complete sequence)

Cf. A037264.

Cf. A037268

More terms from Erich Friedman.

A091784 Numbers n with digits in nondecreasing order such that sum of the reciprocal of digits is an integer.

Original entry on oeis.org

1, 11, 22, 111, 122, 236, 244, 333, 1111, 1122, 1236, 1244, 1333, 2222, 2488, 2666, 3366, 3446, 4444, 11111, 11122, 11236, 11244, 11333, 12222, 12488, 12666, 13366, 13446, 14444, 22236, 22244, 22333, 26999, 28888, 33999, 34688, 36666, 44488, 44666, 55555, 111111, 111122

Offset: 1

Amarnath Murthy, Feb 17 2004

236 is a member and 263, 326, 362, 623, 632 which are digit permutations of 236 are not included (unlike A037268). Subsidiary sequences: (1) Sum of the reciprocals of all n-digit members. (2) Let the terms with reciprocal sum n be arranged in nondecreasing order. (i) The n-th term in the above sequence (2). (ii) The number of digits in this term of (i).

Subsequence of A009994. - David A. Corneth, Sep 05 2016

			236 is a member as 1/2 + 1/3 +1/6 = 1.

Harvey P. Dale and David A. Corneth, Table of n, a(n) for n = 1..10001 (First 237 terms from Harvey P. Dale)

Cf. A009994, A037268, A034708.

Do[l = IntegerDigits[n]; If[Intersection[l, {0}] == {} && IntegerQ[Plus @@ Map[(1/#)&, l]] && Sort[l] == l, Print[n]], {n, 1, 10^5}] (* Ryan Propper, Aug 27 2005 *)
Select[Range[50000],Min[Differences[IntegerDigits[#]]]>=0&&IntegerQ[ Total[ 1/IntegerDigits[#]]]&] (* Harvey P. Dale, Aug 22 2016 *)

is(n)=my(d=digits(n), v=vecsort(d),s); if(d==v, s=sum(i=1,#d,1/d[i]); s==s\1, 0) \\ David A. Corneth, Sep 06 2016

getNDigitTerms(n)=my(v=List(),t); forvec(x=vector(8,i,[0,n]), my(u=vector(n,i,1),X=concat(x,n)); for(i=2,9, for(j=X[i-1]+1, X[i],u[j]=i)); if(denominator(sum(i=1,#u,1/u[i]))==1, listput(v,fromdigits(u))),1); Set(v) \\ Charles R Greathouse IV, Sep 06 2016

More terms from Ryan Propper, Aug 27 2005

Name corrected by David A. Corneth, Sep 05 2016

A309654 Numbers k such that k multiplied by the sum of reciprocals of digits is the digit reversal of k.

Original entry on oeis.org

1, 22, 333, 424, 864, 3663, 4444, 6336, 39993, 46664, 48484, 55555, 64646, 66366, 84448, 88288, 93939, 362436, 488884, 666666, 848848, 884488, 6699966, 6886886, 6969696, 7777777, 8686868, 8866688, 8884888, 9669669, 9993999, 18181818, 26666664, 36484836

Offset: 1

Stéphane Rézel, Aug 11 2019

Integers with at least one 0 digit cannot be terms. There is no other palindrome after 999999999, but is the sequence complete? The non-palindromic numbers in the sequence are 864, 362436, 18181818, 26666664, 36484836, 48363648 and 1666516665, in which zero, seven and nine do not appear as a digit. These non-palindromes are multiples of 3 and have a multiple of 6 as the sum of their digits.

The sequence is finite because the sum of the reciprocals of the digits of every zeroless number greater than 10^81-1 exceeds 9, while the ratio R(n)/n is always smaller than 9. a(43) > 10^13, if it exists. - Giovanni Resta, Aug 12 2019

			864 is in the sequence because 864 * (1/8 + 1/6 + 1/4) = 468, the digit reversal of 864.

Stéphane Rézel, Table of n, a(n) for n = 1..42

Cf. A037268.

[k:k in [1..4000000]| not 0 in Set(Intseq(k))  and k*(&+[1/Intseq(k)[i]:i in [1..#Intseq(k)]]) eq Seqint(Reverse(Intseq(k)))]; // Marius A. Burtea, Aug 11 2019

Select[Range[365*10^5],#*Total[1/IntegerDigits[#]]==IntegerReverse[#]&]//Quiet (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 09 2020 *)

isok(k) = my(d=digits(k)); if (vecmin(d), k*sum(i=1, #d, 1/d[i]) == fromdigits(Vecrev(d))); \\ Michel Marcus, Aug 11 2019

A091783 Numbers with digits in nondecreasing order such that sum of the reciprocals of the digits is 1.

Original entry on oeis.org

1, 22, 236, 244, 333, 2488, 2666, 3366, 3446, 4444, 26999, 28888, 33999, 34688, 36666, 44488, 44666, 55555, 366999, 368888, 446999, 448888, 466688, 666666, 3999999, 4688999, 4888888, 6666999, 6668888, 7777777, 66999999, 68888999, 88888888, 999999999

Offset: 1

Amarnath Murthy, Feb 17 2004

236 is a member but 263, 326, 362, 623, 632 which are digit permutations of 236 are not included (unlike A037268).

By definition, this is a subsequence of A052382 (zeroless numbers). - Michel Marcus, Jul 06 2015

			236 is a member as 1/2 + 1/3 + 1/6 = 1.

Cf. A037268, A052382.

F:= proc(t,ns) option remember;
   local n0, k,r;
   if ns = [] then
      if t = 0 then return {[]}
      else return {}
      fi;
   fi;
   n0:= ns[1];
   `union`(seq(map(r -> [k,op(r)], procname(t - k/n0, ns[2..-1])),k=0..floor(t*n0)));
end proc:
g:= proc(t) local L,i; L:= [seq(i$t[i],i=1..9)]; add(L[i]*10^(nops(L)-i),i=1..nops(L)) end proc:
sort(convert(map(g,F(1,[$1..9])),list)); # Robert Israel, Jul 06 2015

lista(nn) = {for (n=1, nn, d = digits(n); if (vecmin(d) && (vecsort(d)==d) && (sum(k=1, #d, 1/d[k])==1), print1(n, ", ")););} \\ Michel Marcus, Jul 06 2015

More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Apr 17 2004

Incorrect term 39999 corrected to 33999 by Thomas Oléron Evans, Jul 06 2015

A239685 Prime numbers for which the sum of reciprocals of nonzero digits equals 1.

Original entry on oeis.org

263, 2063, 4463, 4643, 6203, 20063, 26003, 60443, 62003, 64403, 68483, 69929, 86843, 88463, 88643, 92699, 200063, 260003, 260999, 296099, 296909, 400643, 406403, 406883, 446003, 449699, 460403, 464003, 464999, 468803, 488603, 494699, 496499, 496949, 499649

Offset: 1

Michel Lagneau, Mar 24 2014

Primes in A214959.

Property of the sequence: a(n) == 3 or 9 (mod 10). If n contains nonzero digits, each number > 263 contains at least two identical digits, and the subsequence of the corresponding sum of reciprocals of digits (primes in A037268) is finite.

			2063 is in the sequence because 1/2 + 1/6 + 1/3 = 1.

Cf. A037268, A214959.

with(numtheory):nn:=500000:for m from 1 to nn do:n:=ithprime(m):y:=convert(n,base,10):n2:=nops(y):s:=0:for i from 1 to n2 do: if y[i]<>0 then s:=s+1/y[i]:else fi:od:if s=1 then printf(`%d, `,n):else fi:od:

Select[Prime[Range[42000]],Total[1/Select[IntegerDigits[#],#!=0&]]==1&] (* Harvey P. Dale, May 31 2019 *)

cp's OEIS Frontend

A214949 Numerator of sum of reciprocals of all nonzero digits of n in decimal representation.

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A214950 Denominator of sum of reciprocals of all nonzero digits of n in decimal representation.

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A214959 Numbers for which the sum of reciprocals of nonzero digits = 1.

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A037265 Integers arising from A037264.

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A091784 Numbers n with digits in nondecreasing order such that sum of the reciprocal of digits is an integer.

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A309654 Numbers k such that k multiplied by the sum of reciprocals of digits is the digit reversal of k.

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Magma

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PARI

A091783 Numbers with digits in nondecreasing order such that sum of the reciprocals of the digits is 1.

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Keywords

Comments

Examples

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Maple

PARI

Extensions

A239685 Prime numbers for which the sum of reciprocals of nonzero digits equals 1.

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