A034708 -id:A034708 - cp's OEIS frontend

A037268 Sum of reciprocals of digits = 1.

1, 22, 236, 244, 263, 326, 333, 362, 424, 442, 623, 632, 2488, 2666, 2848, 2884, 3366, 3446, 3464, 3636, 3644, 3663, 4288, 4346, 4364, 4436, 4444, 4463, 4634, 4643, 4828, 4882, 6266, 6336, 6344, 6363, 6434, 6443, 6626, 6633, 6662, 8248, 8284, 8428, 8482, 8824

Offset: 1

N. J. A. Sloane

This sequence has 1209 terms.

Intersection of A037264 and A034708: A214949(a(n))*A214950(a(n))*A168046(a(n)) = 1. - Reinhard Zumkeller, Aug 02 2012

Nathaniel Johnston, Table of n, a(n) for n = 1..1209 (full sequence)

Cf. A020473, A037264, A038034.

Subsequence of A214959.

a037268 n = a037268_list !! (n-1)
a037268_list = filter ((== 1) . a168046) $
                      takeWhile (<= 999999999) a214959_list
-- Reinhard Zumkeller, Aug 02 2012

A037268 := proc(n) option remember: local d,k: if(n=1)then return 1: fi: for k from procname(n-1)+1 do d:=convert(k,base,10): if(not member(0,d) and add(1/d[j],j=1..nops(d))=1)then return k: fi: od: end: seq(A037268(n),n=1..50); # Nathaniel Johnston, May 28 2011

Select[Range[10000],Total[1/(IntegerDigits[#]/.(0->1))]==1&] (* Harvey P. Dale, Jul 23 2025 *)

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

from fractions import Fraction
def ok(n):
  sn = str(n)
  return False if '0' in sn else sum(Fraction(1, int(d)) for d in sn) == 1
def aupto(limit): return [m for m in range(1, limit+1) if ok(m)]
print(aupto(8824)) # Michael S. Branicky, Jan 22 2021

More terms from Christian G. Bower, Jun 15 1998

Two missing terms inserted by Nathaniel Johnston, May 28 2011

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)).

A214957 Numbers for which the sum of reciprocals of nonzero digits is an integer.

Original entry on oeis.org

0, 1, 10, 11, 22, 100, 101, 110, 111, 122, 202, 212, 220, 221, 236, 244, 263, 326, 333, 362, 424, 442, 623, 632, 1000, 1001, 1010, 1011, 1022, 1100, 1101, 1110, 1111, 1122, 1202, 1212, 1220, 1221, 1236, 1244, 1263, 1326, 1333, 1362, 1424, 1442, 1623, 1632

Offset: 1

Reinhard Zumkeller, Aug 02 2012

A214950(a(n)) = 1.

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

Cf. A034708 (subsequence).

a214957 n = a214957_list !! (n-1)
a214957_list = [x | x <- [0..], a214950 x == 1]

Join[{0},Select[Range[2000],IntegerQ[Total[1/DeleteCases[ IntegerDigits[ #],0]]]&]] (* Harvey P. Dale, Sep 21 2014 *)

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

A266815 Primes whose sum of reciprocal of digits is a prime.

Original entry on oeis.org

11, 2441, 3313, 3331, 4241, 4421, 12163, 12613, 13313, 13331, 16231, 16363, 16633, 21163, 21613, 26113, 31663, 32233, 32323, 32611, 33113, 33223, 33311, 48281, 48821, 61231, 61363, 62131, 62311, 63211, 63361, 88241, 112121, 114643, 116443, 122263, 123323, 126223

Offset: 1

Paolo P. Lava, Feb 12 2016

Subset of A034708.

			11: 1/1 + 1/1 = 2;
2441: 1/2 + 1/4 + 1/4 + 1/1 = 2;
3313: 1/3 + 1/3 + 1/1 + 1/3 = 2; etc.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A000040, A034708.

P:=proc(q) local a,k,n,ok;
for n from 1 to q do if isprime(n) then ok:=1; a:=0; for k from 0 to ilog10(n) do
if trunc(n/10^k) mod 10>0 then a:=a+1/(trunc(n/10^k) mod 10) else ok:=0; break; fi; od;
if ok=1 and type(a,integer) then if isprime(a) then print(n) fi; fi; fi; od; end: P(10^9);
# Alternative:
N:= 8: # to get all terms of up to N digits
S1:= proc(t,k,N)
    option remember;
    if t = 0 then {[]}
    elif k = 0 then {}
    else
      `union`(seq(map(p -> [op(p),k$m], procname(t - m*2520/k, k-1,N-m)),
         m = 0 .. min(N, floor(t*k/2520))))
    fi
end proc:
targets:= 2520*select(isprime,[$2..N]):
Dlists:= select(p -> convert(p,`+`) mod 3 <> 0, `union`(seq(S1(t,9,N),t=targets))):
g:= proc(L) local i,m;
   m:= nops(L);
   op(select(isprime, map(t -> add(t[i]*10^(i-1),i=1..m), combinat:-permute(L))));
end proc:
sort(convert(map(g, Dlists),list)); # Robert Israel, Feb 12 2016

Select[Prime@ Range@ 12000, If[MemberQ[#, 0], False, PrimeQ@ Total[1/#]] &@ IntegerDigits@ # &] (* Michael De Vlieger, Feb 12 2016 *)

isok(n) = if (isprime(n), my(d = digits(n)); vecmin(d) && (denominator(s=sum(k=1, #d, 1/d[k])) == 1) && isprime(s)) \\ Michel Marcus, Feb 12 2016

A064779 Primes such that the sum of their digits and the sum of the reciprocals of their digits is also prime.

Original entry on oeis.org

11, 2441, 4241, 4421, 12163, 12613, 13313, 13331, 16231, 16363, 16633, 21163, 21613, 26113, 31663, 32233, 32323, 32611, 33113, 33223, 33311, 48281, 48821, 61231, 61363, 62131, 62311, 63211, 63361, 88241

Offset: 1

Santi Spadaro, Oct 19 2001

Zero, five, and seven never appear as a digit of any of the terms of this sequence. - Harvey P. Dale, Jul 17 2013

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A034708.

f[ n_ ] := 1/n a[ n_ ] := Apply[ Plus, Map[ f, IntegerDigits[ n ] ] ] b[ n_ ] := Apply[ Plus, IntegerDigits[ n ] ] Select[ Range[ 100000 ], FreeQ[ IntegerDigits[ # ], 0 ] && PrimeQ[ a[ # ] ] && PrimeQ[ b[ # ] ] && PrimeQ[ # ] & ]
sdpQ[n_]:=Module[{idn=IntegerDigits[n]},Min[idn]>0&&And@@PrimeQ[{Total[ idn], Total[ 1/idn]}]]; Select[Prime[Range[10000]],sdpQ] (* Harvey P. Dale, Jul 17 2013 *)

A354466 Numbers k such that the decimal expansion of the sum of the reciprocals of the digits of k starts with the digits of k in the same order.

Original entry on oeis.org

1, 13, 145, 153, 1825, 15789, 16666, 21583, 216666, 2416666, 28428571, 265833333, 3194444444, 3333333333, 9111111111, 35333333333, 3166666666666, 3819444444444, 26666666666666, 34166666666666, 527857142857142, 3944444444444444, 6135714285714285, 615833333333333333

Offset: 1

Metin Sariyar, Jun 01 2022

The sequence is infinite because all numbers of the form 10^(10^n-6) + 6*(10^(10^n-6)-1)/9, (n>0) are terms.
All terms are zeroless since 1/0 is undefined.
If n gives a sum < 1 then that sum is taken as 0.xyz.. but n does not start with 0, so not a term.

			28428571 is a term because 1/2 + 1/8 + 1/4 + 1/2 + 1/8 + 1/5 + 1/7 + 1/1 = 2.8428571...
825 is not a term since 1/8 + 1/2 + 1/5 = 0.825.

Kevin Ryde, Table of n, a(n) for n = 1..382
Michael S. Branicky, Python program
Kevin Ryde, PARI/GP Code

Cf. A009994, A034708, A337904.

Do[If[FreeQ[IntegerDigits[n], 0]&&Floor[Total[1/IntegerDigits[n]]*10^(IntegerLength[n]-IntegerLength[Floor[Total[1/IntegerDigits[n]]]])]==n&&Floor[Total[1/IntegerDigits[n]]]>0, Print[n]], {n, 1, 216666}]

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a(12)-a(24) from Michael S. Branicky, Jun 03 2022

A182452 Numbers for which the sum of reciprocals of square of digits is an integer.

Original entry on oeis.org

1, 11, 111, 1111, 2222, 11111, 12222, 21222, 22122, 22212, 22221, 111111, 112222, 121222, 122122, 122212, 122221, 211222, 212122, 212212, 212221, 221122, 221212, 221221, 222112, 222121, 222211, 222336, 222363, 222633, 223236, 223263, 223326, 223362, 223623

Offset: 1

Michel Lagneau, Apr 29 2012

			223623 is in the sequence because 1/2^2 + 1/2^2 + 1/3^2 + 1/6^2 + 1/2^2 + 1/3^2 = 1 is an integer.

Cf. A034708.

T:=array(1..10):for n from 1 to 10^7 do:T:=convert(n,base,10):n1:=nops(T): s:=0:j:=0:for k from 1 to n1 do:if T[k]<>0 then s:=s+evalf(1/T[k]^2):else j:=1:fi: od: if j=0 and s=floor(s) then printf(`%d, `,n):else fi:od:

f[ n_ ] := 1/n^2; a[ n_ ] := Apply[ Plus, Map[ f, IntegerDigits[ n ] ] ] ; Select[ Range[ 1000 ], FreeQ[ IntegerDigits[ # ], 0 ] && IntegerQ[ a [ # ] ] & ]

A275666 Multisets of numbers such that the sum of reciprocals is 1 and each element e occurs at most lpf(e) - 1 times.

Original entry on oeis.org

2, 3, 6, 2, 5, 5, 10, 3, 5, 5, 6, 10, 2, 4, 6, 12, 3, 3, 4, 12, 4, 5, 5, 6, 10, 12, 2, 7, 7, 7, 14, 3, 6, 7, 7, 7, 14, 4, 6, 7, 7, 7, 12, 14, 5, 5, 7, 7, 7, 10, 14, 2, 3, 10, 15, 2, 4, 10, 12, 15, 2, 5, 6, 15, 15, 3, 3, 5, 15, 15, 3, 3, 6, 10, 15, 3, 5, 5, 5, 15, 3, 4

Offset: 1

David A. Corneth, Aug 23 2016

lpf(n) gives the least prime factor of n (A020639(n)).
The multisets are ordered primarily by largest element, then secondly by length, then thirdly by smallest distinct element.
Sets are placed in nondecreasing order. The separation between two sets is where an element a(n) > a(n + 1).
Let M be the largest element of a multiset. No primes >= (M + 1)/2 are in that multiset.
This sequence may be used to find multisets that have the sum of reciprocals an integer, along with the following operations.
In a set we can replace a number by twice its double. For example, in (2, 3, 6), we can replace the 6 by two twelves, giving {2, 3, 12, 12}. The sum of reciprocals is still 1. However 12 occurs twice in the set, more than lpf(12) - 1 = 1.
{3, 4, 4, 6} isn't included as we could replace two fours by one 2 to get {2, 3, 6}. To exclude such possibilities, each element e is in a multiset at most lpf(e) - 1.
We can also take the union of two sets, for example the sets {2, 3, 6} and {2, 5, 5, 10} giving {2, 2, 3, 5, 5, 6, 10} which has sum of reciprocals 1 + 1 = 2.
We can put 1 in a set, so from {2, 3, 6} we can find {1, 2, 3, 6}.
Using any such operations, we get out of the sequence but more terms of A034708 can be found and A020473 may be investigated further.

			{2, 3, 6}
{2, 5, 5, 10}
{3, 5, 5, 6, 10}
{2, 4, 6, 12}
{3, 3, 4, 12}
{4, 5, 5, 6, 10, 12}
{2, 7, 7, 7, 14}
{3, 6, 7, 7, 7, 14}
{4, 6, 7, 7, 7, 12, 14}
{5, 5, 7, 7, 7, 10, 14}
{2, 3, 10, 15}
{2, 4, 10, 12, 15}
{2, 5, 6, 15, 15}
{3, 3, 5, 15, 15}
{3, 3, 6, 10, 15}
{3, 5, 5, 5, 15}
...
{3, 3, 4, 12} comes before {2, 7, 7, 7, 14} because the largest element of the first is less than the one from the second.
{2, 5, 5, 10} comes before {3, 5, 5, 6, 10} because they both have the largest element 10 but the latter has more elements.
{2, 4, 10, 12, 15} comes before {2, 5, 6, 15, 15} because they both have the largest element 15 and the same number of elements but the first smallest different element, 4 resp. 5, is less for the first.

Cf. A020473, A020639, A034708, A091784.

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A037268 Sum of reciprocals of digits = 1.

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

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A214957 Numbers for which the sum of reciprocals of nonzero digits is an integer.

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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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A266815 Primes whose sum of reciprocal of digits is a prime.

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A064779 Primes such that the sum of their digits and the sum of the reciprocals of their digits is also prime.

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A354466 Numbers k such that the decimal expansion of the sum of the reciprocals of the digits of k starts with the digits of k in the same order.

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