A054683 -id:A054683 - cp's OEIS frontend

A269740 Index where A054683(n) appears in A270264.

1, 5, 10, 25, 3, 7, 12, 73, 35, 4, 9, 16, 78, 43, 13, 17, 80, 54, 47, 15, 19, 88, 62, 48, 22, 93, 69, 92, 31, 24, 95, 81, 98, 63, 125, 83, 115, 67, 283, 127, 102, 122, 138, 296, 107, 140, 147, 320, 100, 6, 20, 33, 129, 116, 8, 23, 37, 133, 126, 26, 38, 136, 128

Offset: 1

N. J. A. Sloane, Mar 15 2016

Conjectured to be a permutation of the positive numbers excluding 2.

This conjecture is equivalent to Polignac's conjecture. - Chai Wah Wu, Mar 15 2016

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A054683, A270264.

More terms from Chai Wah Wu, Mar 16 2016

A007953 Digital sum (i.e., sum of digits) of n; also called digsum(n).

Original entry on oeis.org

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

Offset: 0

R. Muller

Do not confuse with the digital root of n, A010888 (first term that differs is a(19)).
Also the fixed point of the morphism 0 -> {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, 1 -> {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, 2 -> {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, etc. - Robert G. Wilson v, Jul 27 2006
For n < 100 equal to (floor(n/10) + n mod 10) = A076314(n). - Hieronymus Fischer, Jun 17 2007
It appears that a(n) is the position of 10*n in the ordered set of numbers obtained by inserting/placing one digit anywhere in the digits of n (except a zero before 1st digit). For instance, for n=2, the resulting set is (12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82, 92) where 20 is at position 2, so a(2) = 2. - Michel Marcus, Aug 01 2022
Also the total number of beads required to represent n on a Russian abacus (schoty). - P. Christopher Staecker, Mar 31 2023
a(n) / a(2n) <= 5 with equality iff n is in A169964, while a(n) / a(3n) is unbounded, since if n = (10^k + 2)/3, then a(n) = 3*k+1, a(3n) = 3, so a(n) / a(3n) = k + 1/3 -> oo when k->oo (see Diophante link). - Bernard Schott, Apr 29 2023
Also the number of symbols needed to write number n in Egyptian numerals for n < 10^7. - Wojciech Graj, Jul 10 2025

			a(123) = 1 + 2 + 3 = 6, a(9875) = 9 + 8 + 7 + 5 = 29.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Krassimir Atanassov, On the 16th Smarandache Problem, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 1, 36-38.
Krassimir Atanassov, On Some of the Smarandache's Problems, 1999.
Jean-Luc Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, Vol. 18 (2011), #P178.
F. M. Dekking, The Thue-Morse Sequence in Base 3/2, J. Int. Seq., Vol. 26 (2023), Article 23.2.3.
Diophante, A1762, Des chiffres à la moulinette (in French).
Ernesto Estrada and Puri Pereira-Ramos, Spatial 'Artistic' Networks: From Deconstructing Integer-Functions to Visual Arts, Complexity, Vol. 2018 (2018), Article ID 9893867.
A. O. Gel'fond, Sur les nombres qui ont des propriétés additives et multiplicatives données (French) Acta Arith., Vol. 13 (1967/1968), pp. 259-265. MR0220693 (36 #3745)
Christian Mauduit and András Sárközy, On the arithmetic structure of sets characterized by sum of digits properties J. Number Theory, Vol. 61, No. 1 (1996), pp. 25-38. MR1418316 (97g:11107)
Christian Mauduit and András Sárközy, On the arithmetic structure of the integers whose sum of digits is fixed, Acta Arith., Vol. 81, No. 2 (1997), pp. 145-173. MR1456239 (99a:11096)
Kerry Mitchell, Spirolateral-Type Images from Integer Sequences, 2013.
Kerry Mitchell, Spirolateral image for this sequence . [taken, with permission, from the Spirolateral-Type Images from Integer Sequences article]
Jan-Christoph Puchta and Jürgen Spilker, Altes und Neues zur Quersumme, Mathematische Semesterberichte, Vol. 49 (2002), pp. 209-226.
Jan-Christoph Puchta and Jürgen Spilker, Altes und Neues zur Quersumme.
Maxwell Schneider and Robert Schneider, Digit sums and generating functions, arXiv:1807.06710 [math.NT], 2018.
Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
Vladimir Shevelev, Compact integers and factorials, Acta Arith., Vol. 126, No. 3 (2007), pp. 195-236 (cf. pp. 205-206).
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.
Wikipedia, Digit sum.
Index entries for Colombian or self numbers and related sequences

Cf. A003132, A055012, A055013, A055014, A055015, A010888, A007954, A031347, A055017, A076313, A076314, A054899, A138470, A138471, A138472, A000120, A004426, A004427, A054683, A054684, A069877, A179082-A179085, A108971, A169964, A179987, A179988, A180018, A180019, A217928, A216407, A037123, A074784, A231688, A231689, A225693, A254524 (ordinal transform).
Bisections: A004092, A004155.
For n + digsum(n) see A062028.

a007953 n | n < 10 = n
          | otherwise = a007953 n' + r where (n',r) = divMod n 10
-- Reinhard Zumkeller, Nov 04 2011, Mar 19 2011

[ &+Intseq(n): n in [0..87] ];  // Bruno Berselli, May 26 2011

A007953 := proc(n) add(d,d=convert(n,base,10)) ; end proc: # R. J. Mathar, Mar 17 2011

Table[Sum[DigitCount[n][[i]] * i, {i, 9}], {n, 50}] (* Stefan Steinerberger, Mar 24 2006 *)
Table[Plus @@ IntegerDigits @ n, {n, 0, 87}] (* or *)
Nest[Flatten[# /. a_Integer -> Array[a + # &, 10, 0]] &, {0}, 2] (* Robert G. Wilson v, Jul 27 2006 *)
Total/@IntegerDigits[Range[0,90]] (* Harvey P. Dale, May 10 2016 *)
DigitSum[Range[0, 100]] (* Requires v. 14 *) (* Paolo Xausa, May 17 2024 *)

a(n)=if(n<1, 0, if(n%10, a(n-1)+1, a(n/10))) \\ Recursive, very inefficient. A more efficient recursive variant: a(n)=if(n>9, n=divrem(n, 10); n[2]+a(n[1]), n)

a(n, b=10)={my(s=(n=divrem(n, b))[2]); while(n[1]>=b, s+=(n=divrem(n[1], b))[2]); s+n[1]} \\ M. F. Hasler, Mar 22 2011

a(n)=sum(i=1, #n=digits(n), n[i]) \\ Twice as fast. Not so nice but faster:

a(n)=sum(i=1,#n=Vecsmall(Str(n)),n[i])-48*#n \\ M. F. Hasler, May 10 2015
/* Since PARI 2.7, one can also use: a(n)=vecsum(digits(n)), or better: A007953=sumdigits. [Edited and commented by M. F. Hasler, Nov 09 2018] */

a(n) = sumdigits(n); \\ Altug Alkan, Apr 19 2018

def A007953(n):
    return sum(int(d) for d in str(n)) # Chai Wah Wu, Sep 03 2014

def a(n): return sum(map(int, str(n))) # Michael S. Branicky, May 22 2021

(0 to 99).map(.toString.map(.toInt - 48).sum) // Alonso del Arte, Sep 15 2019

"Recursive version for general bases. Set base = 10 for this sequence."
digitalSum: base
| s |
base = 1 ifTrue: [^self].
(s := self // base) > 0
  ifTrue: [^(s digitalSum: base) + self - (s * base)]
  ifFalse: [^self]
"by Hieronymus Fischer, Mar 24 2014"

A007953(n): String(n).compactMap{$0.wholeNumberValue}.reduce(0, +) // Egor Khmara, Jun 15 2021

a(A051885(n)) = n.
a(n) <= 9(log_10(n)+1). - Stefan Steinerberger, Mar 24 2006
From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(10n+i) = a(n) + i for 0 <= i <= 9.
a(n) = n - 9*(Sum_{k > 0} floor(n/10^k)) = n - 9*A054899(n). (End)
From Hieronymus Fischer, Jun 17 2007: (Start)
G.f. g(x) = Sum_{k > 0, (x^k - x^(k+10^k) - 9x^(10^k))/(1-x^(10^k))}/(1-x).
a(n) = n - 9*Sum_{10 <= k <= n} Sum_{j|k, j >= 10} floor(log_10(j)) - floor(log_10(j-1)). (End)
From Hieronymus Fischer, Jun 25 2007: (Start)
The g.f. can be expressed in terms of a Lambert series, in that g(x) = (x/(1-x) - 9*L[b(k)](x))/(1-x) where L[b(k)](x) = sum{k >= 0, b(k)*x^k/(1-x^k)} is a Lambert series with b(k) = 1, if k > 1 is a power of 10, else b(k) = 0.
G.f.: g(x) = (Sum_{k > 0} (1 - 9*c(k))*x^k)/(1-x), where c(k) = Sum_{j > 1, j|k} floor(log_10(j)) - floor(log_10(j-1)).
a(n) = n - 9*Sum_{0 < k <= floor(log_10(n))} a(floor(n/10^k))*10^(k-1). (End)
From Hieronymus Fischer, Oct 06 2007: (Start)
a(n) <= 9*(1 + floor(log_10(n))), equality holds for n = 10^m - 1, m > 0.
lim sup (a(n) - 9*log_10(n)) = 0 for n -> oo.
lim inf (a(n+1) - a(n) + 9*log_10(n)) = 1 for n -> oo. (End)
a(n) = A138530(n, 10) for n > 9. - Reinhard Zumkeller, Mar 26 2008
a(A058369(n)) = A004159(A058369(n)); a(A000290(n)) = A004159(n). - Reinhard Zumkeller, Apr 25 2009
a(n) mod 2 = A179081(n). - Reinhard Zumkeller, Jun 28 2010
a(n) <= 9*log_10(n+1). - Vladimir Shevelev, Jun 01 2011
a(n) = a(n-1) + a(n-10) - a(n-11), for n < 100. - Alexander R. Povolotsky, Oct 09 2011
a(n) = Sum_{k >= 0} A031298(n, k). - Philippe Deléham, Oct 21 2011
a(n) = a(n mod b^k) + a(floor(n/b^k)), for all k >= 0. - Hieronymus Fischer, Mar 24 2014
Sum_{n>=1} a(n)/(n*(n+1)) = 10*log(10)/9 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

More terms from Hieronymus Fischer, Jun 17 2007

Edited by Michel Marcus, Nov 11 2013

A054684 Numbers whose sum of digits is odd.

Original entry on oeis.org

1, 3, 5, 7, 9, 10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 111, 113, 115, 117, 119, 120, 122, 124, 126, 128, 131

Offset: 1

Odimar Fabeny, Apr 19 2000

Union of A179083 and A179085; A179081(a(n)) = 1. - Reinhard Zumkeller, Jun 28 2010

Equivalently, integers with an odd number of odd digits. - Bernard Schott, Nov 06 2022

			1, 3, 5, 7, 9, 10(1), 12(3), 14(5), 16(7), 18(9), 21(3) and so on.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for 10-automatic sequences.
Index entries for Colombian or self numbers and related sequences

Cf. A054683, A137233 (number of n-digits terms).
Cf. A356929 (even number of even digits).
A294601 (exactly one odd decimal digit) is a subsequence.

[seq(`if`(convert(convert(2*n-1,base,10),`+`)::odd, 2*n-1, 2*n-2), n=1..501)];

Select[Range[200],OddQ[Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Nov 27 2021 *)

is(n)=my(d=digits(n));sum(i=1,#d,d[i])%2 \\ Charles R Greathouse IV, Aug 09 2013

isok(m) = sumdigits(m) % 2; \\ Michel Marcus, Nov 06 2022

a(n) = n=2*(n-1); n + !(sumdigits(n)%2); \\ Kevin Ryde, Nov 07 2022

def ok(n): return sum(map(int, str(n)))&1
print([k for k in range(132) if ok(k)]) # Michael S. Branicky, Nov 06 2022

a(n) = n * 2 - 1 for the first 5 numbers; a(n) = n * 2 for the second 5 numbers.
From Robert Israel, Jun 27 2017: (Start)
a(n) = 2*n-2 if floor((n-1)/5) is in the sequence, 2*n-1 if not.
G.f. g(x) satisfies g(x) = (1-x)*(1+x+x^2+x^3+x^4)^2*g(x^10)/x^9 + x^2*(2+x^4+3*x^5-x^9+3*x^10)/((1-x)*(1+x^5))^2.
(End)

More terms from James Sellers, Apr 19 2000

A179082 Even numbers having an even sum of digits in their decimal representation.

Original entry on oeis.org

0, 2, 4, 6, 8, 20, 22, 24, 26, 28, 40, 42, 44, 46, 48, 60, 62, 64, 66, 68, 80, 82, 84, 86, 88, 110, 112, 114, 116, 118, 130, 132, 134, 136, 138, 150, 152, 154, 156, 158, 170, 172, 174, 176, 178, 190, 192, 194, 196, 198, 200, 202, 204, 206, 208, 220, 222, 224, 226

Offset: 1

Reinhard Zumkeller, Jun 28 2010

a(n) = A014263(n) for n <= 25;
intersection of A005843 and A054683: A059841(a(n))*(1-A179081(a(n)))=1;
complement of A179083 with respect to A005843;
complement of A179084 with respect to A054683;
a(n) mod 2 = 0 and A007953(a(n)) mod 2 = 0.

Select[Range[0,250,2],EvenQ[Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Mar 19 2012 *)

A179081 Parity of sum of digits of n.

Original entry on oeis.org

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, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1

Offset: 0

Reinhard Zumkeller, Jun 28 2010

a(n) = A000035(A007953(n));
characteristic function of numbers with an odd sum of digits in their decimal representation:
a(A054684(n)) = 1; a(A054683(n)) = 0;
a(A179083(n)) = a(A179085(n)) = 1;
a(A179082(n)) = a(A179084(n)) = 0.

			a(789) = (7+8+9) mod 2 = 0;
a(790) = (7+9+0) mod 2 = 0;
a(791) = (7+9+1) mod 2 = 1.

R. Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for characteristic functions

Array[Mod[Total[IntegerDigits[#]],2]&,110,0] (* Harvey P. Dale, Feb 20 2013 *)

a(n) = vecsum(digits(n)) % 2 \\ Jeppe Stig Nielsen, Jan 06 2018

a(n) = if n<10 then (n mod 2) else a([n/10]) XOR (n mod 2).

A119449 Primes with even digit sum.

Original entry on oeis.org

2, 11, 13, 17, 19, 31, 37, 53, 59, 71, 73, 79, 97, 101, 103, 107, 109, 127, 149, 163, 167, 181, 211, 233, 239, 251, 257, 271, 277, 293, 307, 347, 349, 367, 383, 389, 419, 431, 433, 439, 457, 479, 491, 499, 503, 509, 521, 523, 541, 547, 563, 569, 587, 613, 617

Offset: 1

Zak Seidov, May 20 2006

On average, there are as many prime numbers for which the sum of decimal digits is even as prime numbers for which it is odd [A119450]. This hypothesis, first made in 1968, has recently been proved by researchers from the Institut de Mathematiques de Luminy. - Jonathan Vos Post, May 13 2010

Also primes such that absolute value of difference between largest digit and the sum of all the other digits is an even integer. This is in accordance with hypothesis of Alexandre Gelfond, proved by C. Mauduit and J. Rivat, see the link below. - Osama Abuajamieh, Feb 10 2017.

T. D. Noe, Table of n, a(n) for n = 1..10000
Christian Mauduit and Joël Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Annals Math., 171 (2010), 1591-1646.
ScienceDaily, Sum of Digits of Prime Numbers Is Evenly Distributed: New Mathematical Proof of Hypothesis, May 13, 2010.

Intersection of A054683 and A000040.

Primes with odd digit sum A119450.

Select[Prime@ Range@ 113, EvenQ@ Total@ IntegerDigits@ # &] (* Michael De Vlieger, Feb 11 2017 *)

isok(n) = isprime(n) && (sumdigits(n) % 2 == 0); \\ Michel Marcus, Oct 10 2013

A179084 Odd numbers having an even sum of digits in their decimal representation.

Original entry on oeis.org

11, 13, 15, 17, 19, 31, 33, 35, 37, 39, 51, 53, 55, 57, 59, 71, 73, 75, 77, 79, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 121, 123, 125, 127, 129, 141, 143, 145, 147, 149, 161, 163, 165, 167, 169, 181, 183, 185, 187, 189, 211, 213, 215, 217, 219, 231, 233, 235

Offset: 1

Reinhard Zumkeller, Jun 28 2010

Intersection of A005408 and A054683: A000035(a(n))*(1-A179081(a(n)))=1;
complement of A179085 with respect to A005408;
complement of A179082 with respect to A054683;
a(n) mod 2 = 1 and A007953(a(n)) mod 2 = 0.

A119894 Numbers k such that 2^k, 3^k, 5^k, 7^k and 11^k have even digit sum.

Original entry on oeis.org

64, 90, 106, 168, 181, 185, 229, 242, 369, 407, 447, 470, 481, 503, 552, 568, 583, 612, 648, 657, 683, 684, 742, 758, 804, 811, 852, 863, 896, 915, 924, 928, 1000, 1004, 1068, 1103, 1113, 1126, 1182, 1402, 1410, 1412, 1420, 1428, 1473, 1483, 1484, 1546, 1566

Offset: 1

Zak Seidov, May 26 2006

Cf. A054683, A118734, A118867, A119895, A119896, A119897.

edsQ[n_]:=And@@EvenQ[Total[IntegerDigits[#]]&/@(Prime[Range[5]]^n)]; Select[Range[1600],edsQ]  (* Harvey P. Dale, Apr 21 2011 *)

A119897 Numbers k such that 2^k, 3^k, 5^k, 7^k, 11^k, 13^k, 17^k and 19^k have even digit sum.

Original entry on oeis.org

552, 657, 811, 1412, 1655, 2390, 2504, 2721, 2803, 2902, 3002, 3060, 3135, 3393, 3660, 4414, 4500, 4547, 4750, 4787, 4824, 5036, 5539, 5906, 6782, 7728, 8650, 9263, 9873, 9953, 10367, 10643, 10684, 10723, 11369, 11647, 11867, 11954

Offset: 1

Zak Seidov, May 26 2006

Harvey P. Dale, Table of n, a(n) for n = 1..1600

Cf. A054683, A118734, A118867, A119894, A119895, A119896.

edsQ[n_]:=AllTrue[Prime[Range[8]]^n,EvenQ[Total[IntegerDigits[#]]]&]; Select[ Range[12000],edsQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 10 2017 *)

More terms from Harvey P. Dale, May 10 2017

A067510 Powers of 6 with digit sum divisible by 6.

Original entry on oeis.org

6, 1296, 279936, 1679616, 10077696, 60466176, 13060694016, 78364164096, 2821109907456, 16926659444736, 101559956668416, 3656158440062976, 789730223053602816, 4738381338321616896, 28430288029929701376

Offset: 1

Amarnath Murthy, Feb 11 2002

Robert Israel, Table of n, a(n) for n = 1..630

Cf. A067499, A067500, A067501, A067502, A067503, A067504, A067505, A067506, A067507, A067508, A067509.

Intersection of A000400 and A054683.

select(n -> convert(convert(n,base,10),`+`)::even, [seq(6^i, i=1..100)]); # Robert Israel, Apr 20 2020

Offset changed by Robert Israel, Apr 20 2020

cp's OEIS Frontend

A269740 Index where A054683(n) appears in A270264.

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A007953 Digital sum (i.e., sum of digits) of n; also called digsum(n).

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A054684 Numbers whose sum of digits is odd.

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A179082 Even numbers having an even sum of digits in their decimal representation.

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A179081 Parity of sum of digits of n.

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A119449 Primes with even digit sum.

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