A055264 -id:A055264 - cp's OEIS frontend

A055263 a(n) = Sum of digits of (n + a(n-1)).

0, 1, 3, 6, 1, 6, 3, 1, 9, 9, 10, 3, 6, 10, 6, 3, 10, 9, 9, 10, 3, 6, 10, 6, 3, 10, 9, 9, 10, 12, 6, 10, 6, 12, 10, 9, 9, 10, 12, 6, 10, 6, 12, 10, 9, 9, 10, 12, 6, 10, 6, 12, 10, 9, 9, 10, 12, 15, 10, 15, 12, 10, 9, 9, 10, 12, 15, 10, 15, 12, 10, 9, 9, 10, 12, 15, 10, 15, 12, 10, 9, 9, 10

Offset: 0

Henry Bottomley, May 08 2000

If n=0 or 8 mod 9, then a(n)=0 mod 9; if n=1, 4 or 7 mod 9, then a(n)=1 mod 9; if n=2 or 6 mod 9, then a(n)=3 mod 9; if n=3 or 5 mod 9, then a(n)=6 mod 9.

			a(13)=10 because a(12)=6, 13 + 6 = 19 and 1 + 9 = 10.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A055262, A055264.

nxt[{n_,a_}]:={n+1,Total[IntegerDigits[n+a+1]]}; Transpose[NestList[nxt,{0,0},90]][[2]] (* Harvey P. Dale, Aug 11 2016 *)

a(n) = A007953(A055262(n)) = A007953(n + a(n-1)).

More terms from Paolo P. Lava, Jul 31 2007

A062688 Smallest triangular number with digit sum n (or 0 if no such number exists).

Original entry on oeis.org

1, 0, 3, 0, 0, 6, 0, 0, 36, 28, 0, 66, 0, 0, 78, 0, 0, 378, 496, 0, 1596, 0, 0, 8385, 0, 0, 5778, 5995, 0, 8778, 0, 0, 47895, 0, 0, 67896, 58996, 0, 196878, 0, 0, 468996, 0, 0, 887778, 1788886, 0, 4896885, 0, 0, 5897895, 0, 0, 13999986, 15997996, 0, 38997696

Offset: 1

Erich Friedman, Jul 04 2001

From Jon E. Schoenfield, Dec 04 2021: (Start)
a(n) = 0 iff n == (2,4,5,7,8) mod 9.
Nonzero terms are not nondecreasing; e.g., a(9)=36 > a(10)=28.
(End)

			66 is the smallest triangular number with digit sum 12, so a(12)=66.

Jon E. Schoenfield, Table of n, a(n) for n = 1..168

Cf. A000217, A007953, A055264, A068808.

(With[{tbl={#,Total[IntegerDigits[#]]}&/@Accumulate[Range[9000]]},Table[SelectFirst[ tbl,#[[2]] ==n&],{n,60}]]/.Missing["NotFound"]->{0,0})[[;;,1]] (* Harvey P. Dale, Aug 21 2024 *)

a(n) = if (vecsearch([2,4,5,7,8], n % 9), return (0)); my(k=1); while (sumdigits(k*(k+1)/2) != n, k++); k*(k+1)/2; \\ Michel Marcus, Dec 12 2021

A350667 Numbers congruent to 1, 3, and 8 modulo 9: positions of 1 in A159955.

Original entry on oeis.org

1, 3, 8, 10, 12, 17, 19, 21, 26, 28, 30, 35, 37, 39, 44, 46, 48, 53, 55, 57, 62, 64, 66, 71, 73, 75, 80, 82, 84, 89, 91, 93, 98, 100, 102, 107, 109, 111, 116, 118, 120, 125, 127, 129, 134, 136, 138, 143, 145, 147

Offset: 0

Wolfdieter Lang, Jan 29 2022

This sequence, together with A350666 and A350668, gives a 3-set partition of the nonnegative integers.

This sequence {a(n)}A347834,%20that%20are%20modulo%206%20periodic%20with%20period%20length%203,%20namely%20%7BA347834(a(n),%20m)%20mod%206%7D">{n>=0}, gives the indices of the row sequences of array A = A347834, that are modulo 6 periodic with period length 3, namely {A347834(a(n), m) mod 6}{m>=0} = {repeat(1, 5, 3)}.

			Rows of array {A347834(a(n), m)}_{m>=0}, with modulo 6 congruence:
n = 0: row 1: {1, 5, 21, 85, 341, 1365, 5461, ...} mod 6 = {repeat(1, 5, 3)},
n = 1: row 3: {7, 29, 117, 469, 1877, 7509, ...} mod 6 = {repeat(1, 5, 3)},
...

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A049310, A055264, A159955, A347834, A350666, A350668.

Select[Range[0, 150], MemberQ[{1, 3, 8}, Mod[#, 9]] &] (* Amiram Eldar, Jan 29 2022 *)

A159955(a(n)) = 1.
Trisection: a(3*k) = 1 + 9*k, a(3*k+1) = 3 + 9*k, and a(3*k+3) = 8 + 9*k, for k >= 0.
G.f.: (1 + 2*x + 5*x^2 + x^3)/((1 - x)*(1 - x^3)).
a(n) = 1 + 3*n - 2*sin(2*n*Pi/3)/sqrt(3). - Stefano Spezia, Jan 30 2022
a(n) = 1 + 3*n - S(n-1,-1), with S(-1, x) = 0, with the Chebyshev S polynomials from A049310. From the g.f., or from the previous formula (see also Spezia's formula in A350666).

A375241 Nontriangular numbers with digital root in {1, 3, 6, 9}.

Original entry on oeis.org

9, 12, 18, 19, 24, 27, 30, 33, 37, 39, 42, 46, 48, 51, 54, 57, 60, 63, 64, 69, 72, 73, 75, 81, 82, 84, 87, 90, 93, 96, 99, 100, 102, 108, 109, 111, 114, 117, 118, 123, 126, 127, 129, 132, 135, 138, 141, 144, 145, 147, 150, 154, 156, 159, 162, 163, 165, 168, 172

Offset: 1

Stefano Spezia, Aug 07 2024

Except for 0, the triangular numbers (A000217) have digital root in {1, 3, 6, 9} (A055264), but the reverse is not always true since there are nontriangular numbers (A014132) with digital root in the same set.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 190.

Stefano Spezia, Table of n, a(n) for n = 1..10000

Intersection of A014132 and A055264. Complement of A375242 in A014132.

Cf. A000217, A010888.

A010888[n_]:=If[n>0,n - 9*Floor[(n-1)/9],0]; Select[Range[0,200], !OddQ[Sqrt[1+8#]] && MemberQ[{1,3,6,9},A010888[#]] &]

A375242 Numbers with digital root in {2, 4, 5, 7, 8}.

Original entry on oeis.org

2, 4, 5, 7, 8, 11, 13, 14, 16, 17, 20, 22, 23, 25, 26, 29, 31, 32, 34, 35, 38, 40, 41, 43, 44, 47, 49, 50, 52, 53, 56, 58, 59, 61, 62, 65, 67, 68, 70, 71, 74, 76, 77, 79, 80, 83, 85, 86, 88, 89, 92, 94, 95, 97, 98, 101, 103, 104, 106, 107, 110, 112, 113, 115, 116

Offset: 1

Stefano Spezia, Aug 07 2024

Also nontriangular numbers with digital root in {2, 4, 5, 7, 8}.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 190.

Stefano Spezia, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Complement of A055264. Complement of A375241 in A014132.

Cf. A010888.

A010888[n_]:=If[n>0,n - 9*Floor[(n-1)/9],0]; Select[Range[0,130], MemberQ[{2,4,5,7,8},A010888[#]] &]

list(lim)=my(v=List()); forstep(n=2,lim\1,[2, 1, 2, 1, 3], listput(v,n)); Vec(v) \\ Charles R Greathouse IV, Aug 07 2024

is(n)=!setsearch([0,1,3,6],n%9) \\ Charles R Greathouse IV, Aug 07 2024

G.f.: x*(2 + 2x + x^2 + 2*x^3 + x^4 + x^5)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).

a(n) ~ 9*n/5.

A067181 Duplicate of A062688.

Original entry on oeis.org

0, 1, 0, 3, 0, 0, 6, 0, 0, 36, 28, 0, 66, 0, 0, 78, 0, 0, 378, 496, 0, 1596, 0, 0, 8385, 0, 0, 5778, 5995, 0, 8778, 0, 0, 47895, 0, 0, 67896, 58996, 0, 196878, 0, 0, 468996, 0, 0, 887778, 1788886, 0, 4896885, 0, 0, 5897895, 0, 0, 13999986, 15997996, 0, 38997696

Offset: 0

Amarnath Murthy, Jan 09 2002

dead

Cf. A055264.

a(n) = A062688(n), n > 0. - R. J. Mathar, Sep 30 2008

More terms from Sascha Kurz, Mar 23 2002

cp's OEIS Frontend

A055263 a(n) = Sum of digits of (n + a(n-1)).

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A062688 Smallest triangular number with digit sum n (or 0 if no such number exists).

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A350667 Numbers congruent to 1, 3, and 8 modulo 9: positions of 1 in A159955.

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A375241 Nontriangular numbers with digital root in {1, 3, 6, 9}.

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A375242 Numbers with digital root in {2, 4, 5, 7, 8}.

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A067181 Duplicate of A062688.

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