A071649 -id:A071649 - cp's OEIS frontend

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.

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

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.

A071648 Sum of even decimal digits of n.

Original entry on oeis.org

0, 2, 0, 4, 0, 6, 0, 8, 0, 0, 0, 2, 0, 4, 0, 6, 0, 8, 0, 2, 2, 4, 2, 6, 2, 8, 2, 10, 2, 0, 0, 2, 0, 4, 0, 6, 0, 8, 0, 4, 4, 6, 4, 8, 4, 10, 4, 12, 4, 0, 0, 2, 0, 4, 0, 6, 0, 8, 0, 6, 6, 8, 6, 10, 6, 12, 6, 14, 6, 0, 0, 2, 0, 4, 0, 6, 0, 8, 0, 8, 8, 10, 8, 12, 8, 14, 8, 16, 8, 0, 0, 2

Offset: 1

Reinhard Zumkeller, May 28 2002

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007953, A071649, A071650.

Array[Total@ Select[IntegerDigits@ #, EvenQ] &, 101] (* Michael De Vlieger, Dec 09 2018 *)

a(n)=my(d=digits(n)); sum(i=1,#d,if(d[i]%2,,d[i])) \\ Charles R Greathouse IV, Apr 04 2014

A071648(n)=vecsum(select(d->!bittest(d,0),digits(n))) \\ Nearly twice as fast. - M. F. Hasler, Dec 09 2018

a(n) = A007953(n) - A071649(n). (Corrected by M. F. Hasler, Dec 09 2018)

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).

A341012 The cumulative sum of the even digits so far in the sequence and the cumulative sum of the odd digits so far differ by n for all a(n)s.

Original entry on oeis.org

1, 10, 16, 7, 23, 32, 45, 54, 67, 76, 89, 98, 100, 203, 225, 230, 247, 252, 269, 274, 296, 302, 320, 405, 427, 449, 450, 472, 494, 504, 522, 540, 607, 629, 670, 692, 706, 724, 742, 760, 809, 890, 908, 926, 944, 962, 980, 1000, 1112, 1121, 1134, 1143, 1156, 1165, 1178

Offset: 1

Eric Angelini and Carole Dubois, Feb 02 2021

This is the lexicographically earliest sequence of distinct integers > 0 having this property.

			Say that the current sequence is S, the cumulative sum at any moment of the even digits of S is E, the cumulative sum at any moment of the odd digits of S is O and the absolute difference |E-O| is D. We would then have:
S = 1, 10, 16, 7, 23, 32, 45, 54, 67, 76, 89, 98,...
E = 0   0   6  6   8  10  14  18  24  30  38  46
O = 1   2   3 10  13  16  21  26  33  40  49  58
D = 1   2   3  4   5   6   7   8   9  10  11  12 <-- this is = n.

Carole Dubois, Table of n, a(n) for n = 1..5000

Cf. A071648, A071649.

Cf. A341002 (numbers whose sum of even digits and sum of odd digits differ by 1).

A158085 Starting at a(1)=2, a(n) is the smallest prime larger than a(n-1) such that the sum of odd digits of a(n) is not smaller than the sum of odd digits of a(n-1).

Original entry on oeis.org

2, 3, 5, 7, 17, 19, 37, 59, 79, 97, 179, 197, 199, 379, 397, 577, 599, 797, 977, 997, 1979, 1997, 1999, 5779, 7759, 7993, 9199, 9397, 9739, 9973, 13799, 13997, 13999, 17599, 17959, 17977, 19597, 19759, 19777, 19979

Offset: 1

Juri-Stepan Gerasimov, Mar 12 2009

"Odd digits" means odd-valued digits (not digits in odd-indexed positions).

			The sequence of the sums of odd digits is 0, 3=3, 5=5, 7=7, 1+7=8, 1+9=10, 3+7=10, 5+9=14, 7+9=16, 9+7=16, 1+7+9=17, 1+9+7=17, 1+9+9=19 and so on. - _R. J. Mathar_, Feb 02 2015

A158085 := proc(n)
    option remember;
    if n =1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isprime(a) then
                if A071649(a) >= A071649(procname(n-1)) then
                    return a;
                end if;
            end if;
        end do:
end if; # R. J. Mathar, Feb 02 2015

spl[n_]:=Module[{sod=Total[Select[IntegerDigits[n],OddQ]],p1= NextPrime[ n]}, While[ Total[ Select[ IntegerDigits[ p1],OddQ]]Harvey P. Dale, Nov 15 2018 *)

A071649(a(n)) >= A071649(a(n-1)). - R. J. Mathar, Feb 02 2015

Corrected (997 inserted, 1699 removed, 9199 to 9739 inserted) by R. J. Mathar, May 19 2010

A340015 a(n) is the least even number not used earlier and equal to the sum of the odd digits of the terms up to and including a(n), if such a number exists; otherwise, a(n) is the least odd number not occurring earlier.

Original entry on oeis.org

0, 1, 3, 4, 5, 10, 7, 18, 9, 30, 11, 13, 15, 42, 17, 19, 60, 21, 23, 64, 25, 76, 27, 92, 29, 102, 31, 33, 114, 116, 118, 35, 130, 134, 138, 37, 154, 39, 174, 41, 43, 45, 184, 194, 47, 49, 51, 53, 224, 55, 57, 246, 59, 260, 61, 63, 264, 65, 276, 67, 292, 69, 304, 71, 316, 73, 332, 338, 75, 358, 77, 79, 81, 83, 85, 87, 404, 89, 414, 91

Offset: 0

Eric Angelini and Carole Dubois, Dec 26 2020

From M. F. Hasler, Dec 06 2022: (Start)
From the definition it is immediate that any even term is equal to the sum of all odd digits in the sequence up to that term.
Also, the subsequences of terms of given parity are both strictly increasing: The odd terms give exactly the sequence of all odd numbers, A005408, and any even number not occurring before a given even a(n) (e.g., 2, 6, 8, 12, 14, 16, ...) will never occur in the sequence.
The search space to check whether an even number can extend the sequence is bounded because using a number with more digits can increase the sum of digits by at most 9 per digit, while the number itself becomes (roughly) 10 times larger with each additional digit.
We have the following properties:
1) If the sum of all odd digits up to a(n) has only even digits, then a(n+1) equals that sum.
2) An even term a(n) can never be immediately followed by a term a(n+1) with only even digits.
3) An even term a(n) can be followed by another even term a(n+1) if the sum of the odd digits of a(n+1) is equal to a(n+1) - a(n), as for example at (..., 114, 116, 118, ...) and (..., 130, 134, 138, ...).
4) If a(n) is even and s = (sum of the odd digits of a(n)) can be added to a(n) without changing any of a(n)'s odd digits and leaving a(n)'s even digits even, then a(n+1) <= a(n) + s. (There may be a smaller solution a(n+1) whose sum of odd digits is smaller than s.) (End)

			The 1st nonzero even term is 4 and 4 is the sum of the odd digits so far, 1 and 3;
The 2nd even term is 10 and 10 is the sum of 1+3+5+1 (the last 1 being the 1 of 10 itself);
The 3rd even term is 18 and 10 is the sum of 1+3+5+1+7+1 (the last 1 being the 1 of 18 itself);
The 4th even term is 30 and 30 is the sum of 1+3+5+1+7+1+9+3 (the last 3 being the 3 of 30 itself); etc.

Carole Dubois, Table of n, a(n) for n = 0..5001
Eric Angelini, Cumulative sum of odd digits, "Cinquante Signes" on blogspot.com, Dec 06 2022
Eric Angelini, Cumulative sum of odd digits, "Cinquante Signes" on blogspot.com, Dec 06 2022 [Cached copy]

Cf. A338741, A338742, A338743, A338744, A338745, A338746.

Cf. A005408 (odd numbers), A071649 (sum of odd decimal digits of n).

def A357051_first(N=100):
    S = []; used_even = set(); next_odd = 1; sod = 0 # sum of odd digits (so far)
    for n in range(N):
        x = sod + sod % 2; lim = sod + 9*len(str(x)); sodx = A071649(x)
        while x < lim:
            if x == sod + sodx and x not in used_even:
                used_even |= { x } ; break
            x += 2
            if x % 10 == 0:
                sodx = A071649(x)
                if sodx == 1: lim += 9
        else: x = next_odd; next_odd += 2; sodx = A071649(x)
        S += [ x ] ; sod += sodx
    return S
# M. F. Hasler, Dec 06 2022

A226017 Primes such that sum of odd digits is also prime.

Original entry on oeis.org

3, 5, 7, 11, 23, 43, 47, 67, 83, 101, 113, 131, 137, 139, 151, 157, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 263, 283, 311, 313, 317, 331, 337, 353, 359, 373, 379, 397, 443, 463, 467, 487, 557, 571, 577, 593, 599, 607, 643, 647, 683, 719, 733, 739

Offset: 1

Jayanta Basu, May 23 2013

Primes such that A104260(n) is prime.

			181 is a member since sum of odd digits=2.

Cf. A071649, A104260.

soQ[n_]:=PrimeQ[Total[Select[IntegerDigits[n],OddQ[#] &]]]; Select[Prime[Range[132]],soQ[#] &]

cp's OEIS Frontend

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

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Mathematica

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A071648 Sum of even decimal digits of n.

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Mathematica

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

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

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A341012 The cumulative sum of the even digits so far in the sequence and the cumulative sum of the odd digits so far differ by n for all a(n)s.

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A158085 Starting at a(1)=2, a(n) is the smallest prime larger than a(n-1) such that the sum of odd digits of a(n) is not smaller than the sum of odd digits of a(n-1).

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