A037123 -id:A037123 - cp's OEIS frontend

A231689 a(n) = Sum_{i=0..n} digsum(i)^4, where digsum(i) = A007953(i).

0, 1, 17, 98, 354, 979, 2275, 4676, 8772, 15333, 15334, 15350, 15431, 15687, 16312, 17608, 20009, 24105, 30666, 40666, 40682, 40763, 41019, 41644, 42940, 45341, 49437, 55998, 65998, 80639, 80720, 80976, 81601, 82897, 85298, 89394, 95955, 105955, 120596, 141332, 141588, 142213, 143509, 145910, 150006, 156567, 166567, 181208, 201944, 230505, 231130, 232426, 234827, 238923

Offset: 0

N. J. A. Sloane, Nov 13 2013

Amiram Eldar, Table of n, a(n) for n = 0..10001
Jean Coquet, Power sums of digital sums, J. Number Theory, Vol. 22, No. 2 (1986), pp. 161-176.
P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, On the moments of the sum-of-digits function, PDF, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993, pp. 263-271.
Robert E. Kennedy and Curtis N. Cooper, An extension of a theorem by Cheo and Yien concerning digital sums, Fibonacci Quarterly, Vol. 29, No. 2 (1991), pp. 145-149.
J.-L. Mauclaire and Leo Murata, On q-additive functions. I, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 6 (1983), pp. 274-276.
J.-L. Mauclaire and Leo Murata, On q-additive functions. II, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 9 (1983), pp. 441-444.
Harald Riede, Asymptotic estimation of a sum of digits, Fibonacci Quarterly, Vol. 36, No. 1 (1998), pp. 72-75.
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag., Vol. 41, No. 1 (1968), pp. 21-25.

Cf. A007953, A037123, A074784, A231688.

Maple
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See A037123.
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Accumulate[Table[Total[IntegerDigits[n]]^4,{n,0,60}]] (* Harvey P. Dale, May 12 2014 *)

a(n) = sum(i=0, n, sumdigits(i)^4); \\ Michel Marcus, Sep 20 2017

A053064 Alternately append n to end or beginning of previous term.

Original entry on oeis.org

1, 12, 312, 3124, 53124, 531246, 7531246, 75312468, 975312468, 97531246810, 1197531246810, 119753124681012, 13119753124681012, 1311975312468101214, 151311975312468101214, 15131197531246810121416, 1715131197531246810121416, 171513119753124681012141618

Offset: 1

Felice Russo, Feb 25 2000

A055642(a(n)) = A058183(n); A007953(a(n)) = A037123(n); A010879(a(n)) = A010879(2*floor(n/2)) = A010879(A131055(n)); A000030(a(n)) = A000030(ceiling(n/2)). - Reinhard Zumkeller, Oct 10 2008

Felice Russo, A set of new Smarandache functions, sequences and conjectures in number theory, American Research Press 2000

R. Zumkeller, Table of n, a(n) for n = 1..100

Cf. A007908, A053063.

More terms from Sean A. Irvine, Dec 05 2021

A256379 Cumulative sum for n in base 10 when alternately adding and subtracting each digit of a particular value.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 45, 44, 44, 43, 39, 36, 30, 25, 17, 10, 0, 2, 1, 1, 6, 8, 15, 19, 28, 34, 45, 42, 44, 39, 39, 38, 30, 27, 17, 12, 0, 4, 1, 7, 6, 6, 15, 17, 28, 32, 45, 40, 44, 37, 39, 30, 30, 29, 17, 14, 0, 6, 1, 9, 6, 16, 15, 15, 28, 30, 45, 38, 44, 35, 39, 28, 30, 17, 17, 16, 0, 8, 1, 11, 6, 18, 15, 29, 28, 28, 45, 36, 44, 33, 39, 26, 30

Offset: 1

Anthony Sand, Mar 27 2015

For n = 1..9, the function is encountering each digit for the first time, therefore each is added to the cumulative sum. At n = 9, the sum is 45. At n = 10, the sum is 44, because the digit 1 is encountered for the second time and is therefore subtracted. At n = 11, the sum is again 44, because 1 is added and then subtracted. At n = 12, the sum is 43, because 1 is added and 2, encountered for the second time, is subtracted.

0 <= a(n) <= 45 for all n; a(n) = a(n mod 20) for odd n. - Danny Rorabaugh, Mar 31 2015

Anthony Sand, Table of n, a(n) for n = 1..1000

Cf. A037123, A167232, A256100, A256851 (first differences).

f[n_] := Block[{g, r = PadRight[Range@ 9, 10]}, g[x_] := Boole[OddQ /@ DigitCount[x]]; Total[r Boole[OddQ /@ Total[g /@ Range@ n]]]]; Array[f, 120] (* Michael De Vlieger, Mar 29 2015 *)

{ nmx=1000; b=10; dig=vector(b); for(i=1,b,dig[i]=1); n=0; s=0; while(n

a(n) = Sum_{k=1..n} M(k), with M(k) := Sum_{m=1..r(k)} (-1)^(a(k,m) + 1)*digit(k,m), where a(k,m) = A256100(k,m) read as an array with row length r(k) (number of digits of k), and digit(k,m) is the m-th digit of k. - Wolfdieter Lang, Apr 08 2015

A271626 Numbers n such that the sum of the digits of the numbers from 0 to n is a square.

Original entry on oeis.org

0, 1, 8, 17, 19, 27, 46, 62, 91, 99, 145, 152, 304, 359, 472, 513, 571, 684, 720, 799, 913, 1204, 1232, 1413, 1771, 2599, 2907, 3059, 3509, 3769, 3887, 4158, 4507, 4787, 5071, 6209, 7399, 7739, 8059, 8486, 9566, 10709, 11545, 12139, 13284, 13573, 14607, 15417

Offset: 1

Paolo P. Lava, Apr 11 2016

			0 = 0^2 and 1 = 1^2;
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 = 6^2;
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 1 + 0 + 1 + 1 + 1 + 2 + 1 + 3 + 1 + 4 + 1 + 5 + 1 + 6 + 1 + 7 = 81 = 9^2.

Paolo P. Lava, Table of n, a(n) for n = 1..500

Cf. A001108, A037123, A271629.

with(numtheory): P:=proc(q) local a,b,c,k,n; a:=0; for n from 0 to q do b:=0; c:=n;
for k from 1 to ilog10(n)+1 do b:=b+(c mod 10); c:=trunc(c/10); od; a:=a+b;
if a=trunc(sqrt(a))*trunc(sqrt(a)) then print(n); fi; od; end: P(10^6);

Select[Range[0, 16000], IntegerQ@ Sqrt@ Total@ Map[Total@ IntegerDigits@ # &, Range[0, #]] &] (* Michael De Vlieger, Apr 11 2016 *)

isok(n) = issquare(sum(k=1, n, sumdigits(k))); \\ Michel Marcus, Apr 11 2016

A071121 a(n) = a(n-1) + sum of decimal digits of n^3.

Original entry on oeis.org

1, 9, 18, 28, 36, 45, 55, 63, 81, 82, 90, 108, 127, 144, 162, 181, 198, 216, 244, 252, 270, 289, 306, 324, 343, 369, 396, 415, 441, 450, 478, 504, 531, 550, 576, 603, 622, 648, 675, 685, 711, 738, 766, 792, 810, 838, 855, 873, 901, 909, 927, 946, 981, 1008

Offset: 1

Labos Elemer, May 27 2002

N. Agronomof, Question 4420, L'Intermédiaire des Math. 21 (1914), 147.

Cf. A037123, A000788, A051351, A071317.

Partial sums of A004164.

s=0; Do[s=s+Apply[Plus, IntegerDigits[n^3]]; Print[s], {n, 1, 128}]
nxt[{n_,a_}]:={n+1,a+Total[IntegerDigits[(n+1)^3]]}; NestList[nxt,{1,1},60][[;;,2]] (* Harvey P. Dale, Aug 30 2025 *)

A256851 First-order differences for the cumulative sum of the digits of the integers when alternately adding and subtracting each digit of a particular value.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, -1, 0, -1, -4, -3, -6, -5, -8, -7, -10, 2, -1, 0, 5, 2, 7, 4, 9, 6, 11, -3, 2, -5, 0, -1, -8, -3, -10, -5, -12, 4, -3, 6, -1, 0, 9, 2, 11, 4, 13, -5, 4, -7, 2, -9, 0, -1, -12, -3, -14, 6, -5, 8, -3, 10, -1, 0, 13, 2, 15, -7, 6, -9, 4, -11, 2, -13, 0, -1, -16, 8, -7, 10, -5, 12, -3, 14, -1, 0, 17, -9, 8, -11, 6, -13, 4, -15

Offset: 1

Anthony Sand, Apr 11 2015

The sequence was suggested by Wolfdieter Lang and represents a(n) - a(n-1) for the sequence A256379, which alternately adds and subtracts each digit of a particular value in the integers.

			a(0) = 0, therefore a(1) - a(0) = 1 - 0 = 1.
For n = 1..9, the function is encountering each digit for the first time, therefore a(9) = 45.
For n = 10, the function encounters the digit 1 for the second time and subtracts it. Therefore a(10) = 44 and a(10) - a(9) = -1.

Anthony Sand, Table of n, a(n) for n = 1..1000

Cf. A167232, A037123, A256379.

{ nmx=1000; b=10; dig=vector(b); si=0; for(i=1,b,dig[i]=1); n=0; s=0; while(n

a(n) = Sum_{m=1..r(n)} (-1)^(a(n,m) + 1)*digit(n,m), where a(n,m) = A256100(n,m) read as an array with row length r(n) (number of digits of n), and digit(n,m) is the m-th digit of n (see the formula for A256379). - Wolfdieter Lang, Apr 15 2015

A271629 Squares that are the sum of the digits of the numbers from 0 to n, for some n.

Original entry on oeis.org

0, 1, 36, 81, 100, 144, 289, 441, 784, 900, 1225, 1296, 3025, 3600, 5041, 5625, 6400, 8100, 8649, 10000, 11881, 15625, 15876, 18225, 23716, 36100, 41616, 44100, 50625, 55225, 57600, 62001, 67600, 72900, 78400, 99225, 122500, 129600, 136900, 145161, 168921, 189225

Offset: 1

Paolo P. Lava, Apr 11 2016

			0 = 0^2; 1 = 1^2; 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 = 6^2.

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

Cf. A037123, A271626.

with(numtheory): P:=proc(q) local a,b,c,k,n; a:=0; for n from 0 to q do b:=0; c:=n;
for k from 1 to ilog10(n)+1 do b:=b+(c mod 10); c:=trunc(c/10); od; a:=a+b;
if a=trunc(sqrt(a))*trunc(sqrt(a)) then print(a); fi; od; end: P(10^6);

Select[Accumulate@ Map[Total@ IntegerDigits@ # &, Range[0, 10^4]], IntegerQ@ Sqrt@ # &] (* Michael De Vlieger, Apr 11 2016 *)

lista(nn) = for (n=0, nn, if (issquare(s=sum(k=1, n, sumdigits(k))), print1(s, ", "))); \\ Michel Marcus, Apr 11 2016

A071421 a(n) = a(n-1) + sum of decimal digits of n^n.

Original entry on oeis.org

1, 5, 14, 27, 38, 65, 90, 127, 172, 173, 214, 268, 326, 378, 477, 565, 663, 771, 898, 929, 1046, 1194, 1340, 1493, 1644, 1798, 1987, 2150, 2317, 2380, 2564, 2769, 2976, 3190, 3450, 3720, 3991, 4256, 4562, 4674, 4982, 5297, 5610, 5935, 6241, 6593, 6967

Offset: 1

Labos Elemer, May 27 2002

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

Cf. A037123, A000788, A051351.

s=0; Do[s=s+Apply[Plus, IntegerDigits[n^n]]; Print[s], {n, 1, 128}]
nxt[{n_,a_}]:={n+1,a+Total[IntegerDigits[(n+1)^(n+1)]]}; NestList[nxt,{1,1},50][[All,2]] (* Harvey P. Dale, Dec 11 2016 *)

A071422 a(n) = a(n-1) + sum of decimal digits of sigma(n), the sum of divisors of n.

Original entry on oeis.org

1, 4, 8, 15, 21, 24, 32, 38, 42, 51, 54, 64, 69, 75, 81, 85, 94, 106, 108, 114, 119, 128, 134, 140, 144, 150, 154, 165, 168, 177, 182, 191, 203, 212, 224, 234, 245, 251, 262, 271, 277, 292, 300, 312, 327, 336, 348, 355, 367, 379, 388, 405, 414, 417, 426, 429

Offset: 1

Labos Elemer, May 27 2002

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

Cf. A037123, A000788, A051351.

Partial sums of A067342.

s=0; Do[s=s+Apply[Plus, IntegerDigits[DivisorSigma[1, n]]]; Print[s], {n, 1, 128}]
nxt[{n_,a_}]:={n+1,a+Total[IntegerDigits[DivisorSigma[1,n+1]]]}; Transpose[ NestList[nxt,{1,1},60]][[2]] (* Harvey P. Dale, Jan 25 2013 *)

A075359 Sum of the digits of the next n numbers.

Original entry on oeis.org

1, 5, 15, 25, 20, 39, 49, 53, 72, 91, 113, 141, 160, 137, 102, 121, 161, 207, 253, 203, 168, 223, 290, 354, 202, 287, 369, 388, 293, 393, 502, 362, 444, 571, 440, 531, 649, 509, 672, 655, 659, 813, 670, 881, 441, 505, 473, 537, 556, 605, 633, 697, 692, 801

Offset: 1

Amarnath Murthy, Sep 19 2002

			a(5) = digit sum of 11,12,13,14 and 15 = 20.

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

Cf. A037123.

Total[Flatten[IntegerDigits/@#]]&/@With[{nn=60},TakeList[Range[(nn(nn+1))/2],Range[ nn]]] (* Harvey P. Dale, Jul 11 2023 *)

a(n) = A037123(n*(n+1)/2)-A037123(n*(n-1)/2). - David Wasserman, Jan 16 2005

More terms from David Wasserman, Jan 16 2005

cp's OEIS Frontend

A231689 a(n) = Sum_{i=0..n} digsum(i)^4, where digsum(i) = A007953(i).

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A053064 Alternately append n to end or beginning of previous term.

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A256379 Cumulative sum for n in base 10 when alternately adding and subtracting each digit of a particular value.

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A271626 Numbers n such that the sum of the digits of the numbers from 0 to n is a square.

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Maple

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A071121 a(n) = a(n-1) + sum of decimal digits of n^3.

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Mathematica

A256851 First-order differences for the cumulative sum of the digits of the integers when alternately adding and subtracting each digit of a particular value.

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PARI

Formula

A271629 Squares that are the sum of the digits of the numbers from 0 to n, for some n.

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Maple

Mathematica

PARI

A071421 a(n) = a(n-1) + sum of decimal digits of n^n.

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