A059708 -id:A059708 - cp's OEIS frontend

A059717 Start with decimal expansion of n; if all digits have the same parity, stop; otherwise write down the number formed by the even digits and the number formed by the odd digits and add them; repeat.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 3, 13, 5, 15, 7, 17, 9, 19, 20, 3, 22, 5, 24, 7, 26, 9, 28, 11, 3, 31, 5, 33, 7, 35, 9, 37, 11, 39, 40, 5, 42, 7, 44, 9, 46, 11, 48, 13, 5, 51, 7, 53, 9, 55, 11, 57, 13, 59, 60, 7, 62, 9, 64, 11, 66, 13, 68, 15, 7, 71, 9, 73, 11, 75

Offset: 0

N. J. A. Sloane, Feb 08 2001

a(A011557(n)) = 1; a(A059708(n)) = A059708(n). [Reinhard Zumkeller, Jul 05 2011]

			For example, 59708 -> (0)8 + 597 = 605 -> 60 + 5 = 65 -> 6 + 5 = 11, stop, so a(59708) = 11.

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

Cf. A059707, A059708.

import Data.List (unfoldr)
a059717 n = if u == n || v == n then n else a059717 (u + v) where
   (u,v) = foldl (\(x,y) d -> if odd d then (10*x+d,y) else (x,10*y+d))
        (0,0) $ reverse $ unfoldr
        (\z -> if z == 0 then Nothing else Just $ swap $ divMod z 10) n
-- Reinhard Zumkeller, Nov 16 2011 (corrected), Jul 05 2011

f[n_] := (id = IntegerDigits[n]; oddDigits = Select[id, OddQ]; evenDigits = Select[id, EvenQ]; Which[ oddDigits == {}, FromDigits[ evenDigits ], evenDigits == {}, FromDigits[ oddDigits ], True, FromDigits[ evenDigits ] + FromDigits[ oddDigits ]]); a[n_] := FixedPoint[f, n]; Table[a[n], {n, 0, 75}] (* Jean-François Alcover, May 31 2013 *)

A228709 Numbers having in decimal representation at least one pair of consecutive digits with the same parity.

Original entry on oeis.org

11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 31, 33, 35, 37, 39, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 60, 62, 64, 66, 68, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 91, 93, 95, 97, 99, 100, 102, 104, 106, 108, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120

Offset: 1

Reinhard Zumkeller, Aug 31 2013

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

Cf. A030141 (complement), A228710, A228722, A228723.

Cf. subsequences: A059708, A010785 (apart from first 10 terms).

a228709 n = a228709_list !! (n-1)
a228709_list = filter ((== 0) . a228710) [0..]

A228710(a(n)) = 0.

A059707 If all digits have the same parity, stop; otherwise write down the number formed by the even digits and the number formed by the odd digits and multiply them; repeat.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 2, 13, 4, 15, 6, 17, 8, 19, 20, 2, 22, 6, 24, 0, 26, 4, 28, 8, 0, 31, 6, 33, 2, 35, 8, 37, 24, 39, 40, 4, 42, 2, 44, 20, 46, 28, 48, 8, 0, 51, 0, 53, 20, 55, 0, 57, 40, 59, 60, 6, 62, 8, 64, 0, 66, 42, 68, 20, 0, 71, 4, 73, 28, 75, 42

Offset: 0

N. J. A. Sloane, Feb 07 2001

a(A059708(n)) = A059708(n). - Reinhard Zumkeller, Jun 15 2012

			89 -> 8*9 = 72 -> 7*2 = 14 -> 1*4 = 4, stop, so a(89) = 4.
33278 -> 28*337 = 9436 -> 46*93 = 4278 -> 42*78 -> 2996 -> 26*99 = 2574 -> 24*57 = 1368 -> 68*13 = 884, stop, so a(33278) = 884.

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

Cf. A059708, A059717.

import Data.List (unfoldr)
a059707 n = if u == n || v == n then n else a059707 (u * v) where
   (u,v) = foldl (\(x,y) d -> if odd d then (10*x+d,y) else (x,10*y+d))
        (0,0) $ reverse $ unfoldr
        (\z -> if z == 0 then Nothing else Just $ swap $ divMod z 10) n
-- Reinhard Zumkeller, Jun 15 2012

f[n_] := (id = IntegerDigits[n]; oddDigits = Select[id, OddQ]; evenDigits = Select[id, EvenQ]; Which[oddDigits == {}, FromDigits[evenDigits], evenDigits == {}, FromDigits[oddDigits], True, FromDigits[evenDigits] * FromDigits[oddDigits]]); a[n_] := FixedPoint[f, n]; Table[a[n], {n, 0, 76}] (* Jean-François Alcover, May 16 2013 *)
sp[n_]:=Module[{idn=IntegerDigits[n],e,o},e=Select[idn,EvenQ];o= Select[ idn,OddQ];If[Min[Length[o],Length[e]]>0,FromDigits[o] FromDigits[e], n]]; Table[FixedPoint[sp,i],{i,0,80}] (* Harvey P. Dale, Jun 05 2014 *)

a(50) corrected by Reinhard Zumkeller, Jun 15 2012

A374097 a(n) = A196563(n)*A196564(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Paolo Xausa, Jun 28 2024

More than the usual number of terms are shown in order to distinguish this sequence from A180160, from which it first differs at n = 100.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A055642, A059708 (positions of zeros), A180160, A196563, A196564.

A374097[n_] := #*(IntegerLength[n] - #) & [Total[Mod[IntegerDigits[n], 2]]];
Array[A374097, 120, 0]

A306520 Numbers k with property that the arithmetic mean of any subset of its digits is an integer.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 31, 33, 35, 37, 39, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 60, 62, 64, 66, 68, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 91, 93, 95, 97, 99, 111, 117, 135, 153, 159, 171, 177, 195

Offset: 1

R. J. Cano, Feb 21 2019

This sequence is different from A061383. Here digits in k must have all the same parity, otherwise the average of at least a pair of digits wouldn't be an integer. Note that for every 2-digit term in A061383 both digits have the same parity. But not every number whose digits have all the same parity (sequence A059708) belongs here.

			17 is in this sequence because the set of digits (1,7) has an integer average: 4.
159 and 195 are in this sequence because the sets of digits (1,5), (1,9), (5,9), and (1,5,9) all have integer averages, respectively: 3, 5, 7, and 5.

Harvey P. Dale, Table of n, a(n) for n = 1..177 (all terms up to 1 million)

Cf. A061383, A059708, A010785, A165165.

Select[Range[0,200],AllTrue[Mean/@Subsets[IntegerDigits[#],{2, IntegerLength[ #]}],IntegerQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 09 2020 *)

firstTerms_vec(n)={my(v=vector(n),c,t,w:list,h);for(i=1,+oo,w=List();forsubset(i,k,listput(w,k));listpop(w,1);forvec(j=vector(i,z,[(z==1)&&(i>1),9]),h=j[1]%2;for(l=2,#j,if((j[l]%2)!=h,next(2)));for(k=1,#w,t=vecextract(j,w[k]);if(vecsum(t)%(#w[k]),next(2)));v[c++]=fromdigits(j);if(c==n,return(v))))}

isok(m,{B=10})={my(w=digits(m,B));forsubset(#w,y,if(y!=Vecsmall([]),if(vecsum(vecextract(w,y))%(#y),return(0)),next));1}

Apparently a(158+n) = A010785(35+n).

cp's OEIS Frontend

A059717 Start with decimal expansion of n; if all digits have the same parity, stop; otherwise write down the number formed by the even digits and the number formed by the odd digits and add them; repeat.

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A228709 Numbers having in decimal representation at least one pair of consecutive digits with the same parity.

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A059707 If all digits have the same parity, stop; otherwise write down the number formed by the even digits and the number formed by the odd digits and multiply them; repeat.

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A374097 a(n) = A196563(n)*A196564(n).

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A306520 Numbers k with property that the arithmetic mean of any subset of its digits is an integer.

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