A036301 -id:A036301 - cp's OEIS frontend

A340470 Two adjacent integers sum up to a term of A036301.

0, 112, 9, 103, 18, 94, 27, 85, 36, 76, 45, 67, 54, 58, 63, 49, 72, 40, 81, 31, 90, 22, 99, 13, 108, 4, 117, 17, 95, 26, 86, 35, 77, 44, 68, 53, 59, 62, 50, 71, 41, 80, 32, 89, 23, 98, 14, 107, 5, 116, 198, 138, 73, 39, 82, 30, 91, 21, 100, 12, 109, 3, 118, 16, 96, 25

Offset: 1

Eric Angelini and Carole Dubois, Jan 08 2021

This is the lexicographically earliest sequence of distinct nonnegative terms with this property.

			a(1) + a(2) = 0 + 112 = 112 (a term of A036301);
a(2) + a(3) = 112 + 9 = 121 (a term of A036301);
a(3) + a(4) = 9 + 103 = 112;
a(4) + a(5) = 103 + 18 = 121; etc.

Carole Dubois, Table of n, a(n) for n = 1..10002

Cf. A036301 (even digits sum = odd digits sum).

f:=func; a:=[0]; for n in [2..80] do k:=1; while k in a or not f(k+a[n-1]) do k:=k+1; end while; Append(~a,k); end for; a; // Marius A. Burtea, Jan 12 2021

A364863 Number of iterations of x -> x + min { k in A036301 | k > x } until an element of A036301 is reached, or -1 if this never happens, starting with n.

Original entry on oeis.org

0, 21

Offset: 0

M. F. Hasler, Aug 11 2023

The question whether the iteration always reaches an element of A036301 was raised on the SeqFan list in 2018, with "closest" instead of "next larger" (element of A036301). In that case one has 0 < n < 56 as a trivial counterexample. It is still open to our knowledge.

The first unknown term is currently a(2). Starting with x =2 we reach x = 336917039990529107004169 after 72 iterations.

			a(0) = 0 because n = 0 is an element of A036301 and therefore no iteration is required to reach such an element.
The smallest nonzero element of A036301 is 112. Therefore, all smaller positive numbers 0 < n < 112 go to n + 112 under the first iteration of
  f: x -> x + min { k in A036301 | k > x }.
Under iterations of f, 1 -> 113 -> 234 -> 548 -> 1109 -> 2229 -> 4460 -> 8931 -> 17865 -> 35872 -> 71875 -> 143891 -> 287898 -> 575804 -> 1151810 -> 2303826 -> 4607657 -> 9215347 -> 18430735 -> 36861480 -> 73723189 -> 147446477 which is the first element of A036301 to be reached, after a(1) = 21 iterations.

Eric Angelini, Balanced/Unbalanced numbers (and iterations), personal blog "Cinquante signes" on blogspot,com, June 23, 2023
Eric Angelini, Balanced/Unbalanced numbers (and iterations), personal blog "Cinquante signes" on blogspot,com, June 23, 2023 [Cached copy]
M. F. Hasler and N. J. A. Sloane, in reply to E. Angelini, Re: Balanced / Unbalanced numbers, Dec. 10, 2018

Cf. A036301, A071650.

a(n) = for(k=0,oo, A071650(n) || return(k); n+=next_A036301(n))

a(n) = 0 iff A071650(n) = 0, i.e., for all n in A036301.

A340574 a(1) = 112. a(n) is the smallest number k with the sum of the even digits equal to the sum of the odd digits (A036301), which is not an earlier term, for which k*a(n - 1) has the sum of the even digits equal to the sum of the odd digits (A036301).

Original entry on oeis.org

112, 583, 1120, 781, 1452, 615, 1627, 1694, 1506, 1403, 1078, 1043, 538, 121, 4983, 1087, 1708, 1304, 314, 385, 341, 134, 187, 1340, 718, 2123, 358, 1021, 1102, 211, 835, 2110, 1322, 3265, 2558, 561, 1034, 871, 2167, 3085, 1232, 1245, 413, 2716, 1201, 1012, 336

Offset: 1

Marius A. Burtea, Jan 12 2021

Sequence only with terms in A036301 for which a(n)*a(n+1) is a term in A036301.
a(1) = 112 = A036301(2) is the first nonzero term of A036301.
The sequence is infinite.

			a(1) = 112, a(2) = 583 = A036301(22) and a(1)*a(2) = 112*583 = 65296 is a term in A036301 because 6 + 2 + 6 = 14 = 5 + 9.
a(2) = 583, a(3) = 1120 = A036301(41) and a(2)*a(3) = 583*1120 = 652960 is a term in A036301 because 6 + 2 + 6 + 0 = 14 = 5 + 9.

Cf. A036301, A340470.

f:=func; a:=[112]; for n in [2..50] do k:=1; while k in a or not f(k) or not f(k*a[n-1]) do k:=k+1; end while; Append(~a,k); end for; a;

isokd(n) = my(d=digits(n)); sum(k=1, #d, d[k]*(d[k] % 2)) == sum(k=1, #d, d[k]*(1-d[k]%2)); \\ A036301
nextk(va, n) = {my(ok = 0, k = 1); while (! (isokd(k) && isokd(k*va[n-1]) && !#select(x->(x==k), va)), k++); k;}
lista(nn) = {my(va = vector(nn)); va[1] = 112; for (n=2, nn, my(k = nextk(va, n)); va[n] = k;); va;} \\ Michel Marcus, Jan 14 2021

A124176 Consider the map f that sends m to m + (sum of odd digits of m) - (sum of even digits of m). Sequence gives numbers m such that f^(k)(m) = m for some k.

Original entry on oeis.org

0, 11, 13, 17, 18, 25, 28, 54, 55, 64, 65, 112, 121, 134, 137, 143, 148, 155, 156, 165, 166, 173, 178, 184, 187, 198, 200, 209, 211, 216, 231, 233, 234, 237, 244, 245, 270, 275, 280, 285, 314, 336, 341, 358, 363, 385, 396, 402, 407, 410, 413, 429, 431, 432

Offset: 1

Eric Angelini, Dec 04 2006

Terms computed by Barry and Theunis de Jong.

Subsequence A036301 lists fixed points of the map f = A304439. - M. F. Hasler, May 18 2018

			11 and 13 loop on themselves, but 12 doesn't:
11 -> 13 -> 17 -> 25 -> 28 -> 18 -> 11
12 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 -> 11
13 -> 17 -> 25 -> 28 -> 18 -> 11 -> 13.

Eric Angelini, Dec 04 2006, Table of n, a(n) for n = 1..172

Cf. A124177, A036301, A304439; A071648, A071649, A071650.

is(n,S=List())=until(setsearch(Set(S),n=A304439(n)),listput(S,n));n==S[1] \\ M. F. Hasler, May 18 2018

A124177 Consider the map f that sends m to m + (sum of even digits of m) - (sum of odd digits of m). Sequence gives numbers m such that f^(k)(m) = m for some k.

Original entry on oeis.org

0, 22, 26, 27, 34, 35, 44, 49, 52, 63, 66, 78, 79, 81, 88, 99, 104, 107, 108, 112, 115, 121, 126, 133, 134, 143, 144, 151, 156, 165, 178, 187, 211, 224, 229, 232, 233, 283, 290, 314, 336, 341, 358, 363, 385, 413, 431, 467, 470, 489, 492, 516, 538, 561, 583, 615

Offset: 1

Eric Angelini, Dec 04 2006

Terms computed by Theunis de Jong.

Subsequence A036301 lists fixed points of the map f = A304440. - M. F. Hasler, May 18 2018

			26 and 27 loop on themselves, but 28 doesn't.
26 -> 34 -> 35 -> 27 -> 22 -> 26
27 -> 22 -> 26 -> 34 -> 35 -> 27
28 -> 38 -> 43 -> 44 -> 52 -> 49 -> 44.

Eric Angelini, Table of n, a(n) for n = 1..182
Eric Angelini, Self-loopers.

Cf. A124176, A036301, A304439, A304440, A071650, A071648, A071649.

is(n,S=List())={until(setsearch(Set(S),n=A304440(n)),listput(S,n));n==S[1]} \\ M. F. Hasler, May 18 2018

A341002 Numbers whose sum of even digits and sum of odd digits differ by 1.

Original entry on oeis.org

1, 10, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 98, 100, 102, 120, 201, 203, 210, 223, 225, 230, 232, 245, 247, 252, 254, 267, 269, 274, 276, 289, 296, 298, 302, 304, 320, 322, 340, 403, 405, 425, 427, 430, 447, 449, 450, 452, 469, 472, 474, 494, 496, 504

Offset: 1

Eric Angelini and Carole Dubois, Feb 02 2021

Carole Dubois, Table of n, a(n) for n = 1..5001

Cf. A036301 (sums are equal), A341002 to A341010 (sums differ by 1 to 9).

Select[Range[1000], Abs[Plus @@ Select[(d = IntegerDigits[#]), OddQ] - Plus @@ Select[d, EvenQ]] == 1 &] (* Amiram Eldar, Feb 02 2021 *)

A071650 Difference between sums of odd and even digits of n.

Original entry on oeis.org

1, -2, 3, -4, 5, -6, 7, -8, 9, 1, 2, -1, 4, -3, 6, -5, 8, -7, 10, -2, -1, -4, 1, -6, 3, -8, 5, -10, 7, 3, 4, 1, 6, -1, 8, -3, 10, -5, 12, -4, -3, -6, -1, -8, 1, -10, 3, -12, 5, 5, 6, 3, 8, 1, 10, -1, 12, -3, 14, -6, -5, -8, -3, -10, -1, -12, 1, -14, 3, 7, 8, 5, 10

Offset: 1

Reinhard Zumkeller, May 28 2002

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A036301, A071648, A071649.

Table[Total[Select[IntegerDigits[n],OddQ]]-Total[Select[ IntegerDigits[ n],EvenQ]],{n,80}] (* Harvey P. Dale, Jul 27 2020 *)

a(n) = {my(d=digits(n), s = 0); for (k=1, #d, if (d[k] % 2, s += d[k], s -= d[k]);); s;} \\ Michel Marcus, Aug 05 2017

A071650(n)=-vecsum(apply(t->(-1)^t*t,digits(n))) \\ M. F. Hasler, Dec 09 2018

a(n) = A071649(n) - A071648(n);

a(A036301(n)) = 0.

A341010 Numbers whose sum of even digits and sum of odd digits differ by 9.

Original entry on oeis.org

9, 90, 117, 128, 135, 146, 153, 164, 171, 182, 218, 281, 315, 333, 348, 351, 366, 384, 416, 438, 461, 483, 513, 531, 568, 586, 614, 636, 641, 658, 663, 685, 711, 788, 812, 821, 834, 843, 856, 865, 878, 887, 900, 1017, 1028, 1035, 1046, 1053, 1064, 1071, 1082, 1107, 1129, 1170

Offset: 1

Eric Angelini and Carole Dubois, Feb 02 2021

Carole Dubois, Table of n, a(n) for n = 1..5001

Cf. A036301 (sums are equal), A341002 to A341010 (sums differ by 1 to 9).

Select[Range[1200], Abs[Plus @@ Select[(d = IntegerDigits[#]), OddQ] - Plus @@ Select[d, EvenQ]] == 9 &] (* Amiram Eldar, Feb 02 2021 *)

A304439 Add to n the sum of its odd digits minus the sum of its even digits.

Original entry on oeis.org

0, 2, 0, 6, 0, 10, 0, 14, 0, 18, 11, 13, 11, 17, 11, 21, 11, 25, 11, 29, 18, 20, 18, 24, 18, 28, 18, 32, 18, 36, 33, 35, 33, 39, 33, 43, 33, 47, 33, 51, 36, 38, 36, 42, 36, 46, 36, 50, 36, 54, 55, 57, 55, 61, 55, 65, 55, 69, 55, 73, 54, 56, 54, 60, 54, 64, 54, 68, 54, 72

Offset: 0

M. F. Hasler, May 18 2018

Subsequence A036301 lists fixed points of this map, the first nontrivial one being 112. It is a subsequence of A124176 (and A124177) which considers iterations of this map, more precisely, numbers which are in a cyclic orbit for iterations of this map.

Cf. A304440, A071650, A071648, A071649, A036301, A124176, A124177.

soded[n_]:=Module[{idn=IntegerDigits[n]},n+Total[Select[idn,OddQ]]-Total[ Select[idn,EvenQ]]]; Array[soded,70,0] (* Harvey P. Dale, Aug 12 2021 *)

A304439(n)=n-vecsum(apply(t->t*(-1)^t,digits(n)))

a(n) = n + A071650(n).

A304440 Add to n the sum of its even digits minus the sum of its odd digits.

Original entry on oeis.org

0, 0, 4, 0, 8, 0, 12, 0, 16, 0, 9, 9, 13, 9, 17, 9, 21, 9, 25, 9, 22, 22, 26, 22, 30, 22, 34, 22, 38, 22, 27, 27, 31, 27, 35, 27, 39, 27, 43, 27, 44, 44, 48, 44, 52, 44, 56, 44, 60, 44, 45, 45, 49, 45, 53, 45, 57, 45, 61, 45, 66, 66, 70, 66, 74, 66, 78, 66, 82, 66, 63

Offset: 0

M. F. Hasler, May 18 2018

A036301 lists fixed points of this map, the first nonzero one being 112. It is also a subsequence of A124177 (and A124176) which lists numbers which are in a cyclic orbit under iterations of this map.

Cf. A304439 (variant: + even - odd digits), A071650 (odd - even digits), A071648, A071649, A036301 (fixed points), A124177, A124176.

nseo[n_]:=Module[{idn=IntegerDigits[n]},n+Total[Select[idn,EvenQ]]-Total[Select[idn,OddQ]]]; Array[nseo,80,0] (* Harvey P. Dale, Dec 26 2023 *)

A304440(n)=n+vecsum(apply(t->t*(-1)^t,digits(n)))

a(n) = n - A071650(n).

cp's OEIS Frontend

A340470 Two adjacent integers sum up to a term of A036301.

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A364863 Number of iterations of x -> x + min { k in A036301 | k > x } until an element of A036301 is reached, or -1 if this never happens, starting with n.

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A340574 a(1) = 112. a(n) is the smallest number k with the sum of the even digits equal to the sum of the odd digits (A036301), which is not an earlier term, for which k*a(n - 1) has the sum of the even digits equal to the sum of the odd digits (A036301).

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A124176 Consider the map f that sends m to m + (sum of odd digits of m) - (sum of even digits of m). Sequence gives numbers m such that f^(k)(m) = m for some k.

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A124177 Consider the map f that sends m to m + (sum of even digits of m) - (sum of odd digits of m). Sequence gives numbers m such that f^(k)(m) = m for some k.

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A341002 Numbers whose sum of even digits and sum of odd digits differ by 1.

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A071650 Difference between sums of odd and even digits of n.

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PARI

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A341010 Numbers whose sum of even digits and sum of odd digits differ by 9.

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A304439 Add to n the sum of its odd digits minus the sum of its even digits.