A028898 -id:A028898 - cp's OEIS frontend

A081502 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 3x+y.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 21, 22, 23, 24, 25, 26, 27, 28, 29

Offset: 0

N. J. A. Sloane, Apr 22 2003

Eswaran observes that n is divisible by 7 iff repeated application of a ends at the number 7.

a(n) is divisible by 7 iff n is divisible by 7: e.g., a(7) = a(14) = a(21) = 7, a(28) = a(35) = a(42) = 14 etc. - Zak Seidov, Mar 19 2014

R. Eswaran, Test of divisibility of the number 7, Abstracts Amer. Math. Soc., 23 (No. 2, 2002), #974-00-5, p. 275.

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

Different from A028898 for n>=100 (e.g. a(111) = 34, A029989(111) = 13).

Cf. A081503, A081594, A081595, A081596, A081597, A081598, A081599, A081600.

A081502 := proc(n)
    local x,y ;
    y := modp(n,10) ;
    x := iquo(n,10) ;
    3*x+y ;
end proc:
seq(A081502(n),n=0..120) ; # R. J. Mathar, Oct 03 2014

Table[n - 7 * Floor[n / 10], {n, 0, 100}] (* Joshua Oliver, Dec 04 2019 *)

a(n) = 3*(n\10) + (n % 10); \\ Michel Marcus, Mar 19 2014

a(n) = [3,1]*divrem(n,10); \\ Kevin Ryde, Dec 04 2019

G.f.: -x*(6*x^9-x^8-x^7-x^6-x^5-x^4-x^3-x^2-x-1) / (x^11-x^10-x+1). - Colin Barker, Mar 19 2014

a(n) = n-7*floor(n/10). - Wesley Ivan Hurt, May 12 2016

A381607 For any nonnegative integer n with ternary expansion Sum_{k >= 0} t_k * 3^k, a(n) = Sum_{k >= 0} t_k * A000045(2*k + 2).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 21, 22, 23, 24, 21, 22, 23, 24, 25, 26, 27, 28, 29, 29, 30, 31, 32, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 42, 43, 44, 45, 42, 43, 44, 45, 46, 47, 48, 49, 50, 50, 51, 52, 53, 54

Offset: 0

Rémy Sigrist, Mar 01 2025

Every nonnegative integer appears in the sequence, a finite number of times.

			42 = 3^3 + 3^2 + 2*3^1, so a(42) = A000045(8) + A000045(6) + 2*A000045(4) = 21 + 8 + 2*3 = 35.

Rémy Sigrist, Table of n, a(n) for n = 0..6560

See A022290 for a similar sequence.

Cf. A000045, A028898, A381579.

a(n) = { my (t = Vecrev(digits(n, 3))); sum(k = 1, #t, t[k] * fibonacci(2*k)); }

a(A028898(A381579(n))) = n.

A360506 Read A360505(n) as if it were a base-3 string and write it in base 10.

Original entry on oeis.org

1, 7, 34, 358, 4003, 43369, 456712, 4708240, 47754961, 1339156591, 39693785002, 1169411930926, 34213667699203, 995038950807565, 28790341783585180, 829295063367580492, 23793774263808446005, 680307709052882601259, 19390954850541496025998

Offset: 1

N. J. A. Sloane, Feb 17 2023

This has the same relationship to A360505 as A048435 does to A360502.
The primes in A048435 are in A360503. What are the primes in the present sequence?
Answer: The first primes are a(2) = 7, a(5) = 4003, a(13) = 34213667699203, a(57) and a(109). See A360507. - Rémy Sigrist, Feb 18 2023

			A360505(4) = 111021 and 111021_3 = 358_10 = a(4).

Winston de Greef, Table of n, a(n) for n = 1..407
Index entries for sequences related to Most Wanted Primes video

Cf. A048435, A028898, A360502, A360503, A360505, A360507.

a(n) = fromdigits(concat([digits(k, 3) | k <- Vecrev([1..n])]), 3) \\ Rémy Sigrist, Feb 18 2023

from sympy.ntheory import digits
def a(n): return int("".join("".join(map(str, digits(k, 3)[1:])) for k in range(n, 0, -1)), 3)
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Feb 19 2023

# faster version for initial segment of sequence
from sympy.ntheory import digits
from itertools import count, islice
def agen(s=""): yield from (int(s:="".join(map(str, digits(n, 3)[1:]))+s, 3) for n in count(1))
print(list(islice(agen(), 20))) # Michael S. Branicky, Feb 19 2023

from itertools import count, islice
def A360506_gen(): # generator of terms
    a, b, c = 3, 1, 0
    for i in count(1):
        if i >= a:
            a *= 3
        c += i*b
        yield c
        b *= a
A360506_list = list(islice(A360506_gen(),30)) # Chai Wah Wu, Nov 08 2023

a(n) = A028898(A360505(n)). - Rémy Sigrist, Feb 18 2023

More terms from Rémy Sigrist, Feb 18 2023

A081599 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 8x+y.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 56, 57, 58, 59, 60, 61

Offset: 0

N. J. A. Sloane, Apr 22 2003

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

Cf. A081502. Different from A028898.

k:=8; [n-(10-k)*Floor(n/10): n in [0..100]]; // Bruno Berselli, Jun 24 2014

A081599 := proc(n)
    local x,y ;
    x := floor(n/10) ;
    y := modp(n,10) ;
    8*x+y ;
end proc:
seq(A081599(n),n=0..100) ; # R. J. Mathar, May 25 2023

Table[n-2*Floor[n/10],{n,0,80}] (* Harvey P. Dale, Nov 07 2017 *)

my(n, x, y); vector(200, n, y=(n-1)%10; x=(n-1-y)\10; 8*x+y) \\ Colin Barker, Jun 24 2014

G.f.: -x*(x^9 -x^8 -x^7 -x^6 -x^5 -x^4 -x^3 -x^2 -x -1) / ((x -1)^2*(x +1)*(x^4 -x^3 +x^2 -x +1)*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Jun 24 2014

a(n) = n - 2*floor(n/10). - Bruno Berselli, Jun 24 2014

A083291 Triangular array read by rows: T(n,k) = k*floor(n/10) + n mod 10, 0<=k<=n.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Apr 23 2003

A010879(n)=T(n,0);
A076314(0)=T(0,0), A076314(n)=T(n,1) for n>0;
A028897(n)=T(n,n) for n<=1, A028897(n)=T(n,2) for n>1;
A028898(n)=T(n,n) for n<=2, A028898(n)=T(n,3) for n>2;
A028899(n)=T(n,n) for n<=3, A028899(n)=T(n,4) for n>3;
A028900(n)=T(n,n) for n<=4, A028900(n)=T(n,5) for n>4;
A028901(n)=T(n,n) for n<=5, A028901(n)=T(n,6) for n>5;
A028902(n)=T(n,n) for n<=6, A028902(n)=T(n,7) for n>6;
A028903(n)=T(n,n) for n<=7, A028903(n)=T(n,8) for n>7;
A028904(n)=T(n,n) for n<=8, A028904(n)=T(n,9) for n>8;
T(n,n) = n for n<=9, T(n,10) = n for n>9;
A083292(n) = T(n,n).

			From _Paolo Xausa_, May 22 2024: (Start)
Triangle begins:
   [0] 0;
   [1] 1, 1;
   [2] 2, 2, 2;
   [3] 3, 3, 3, 3;
   [4] 4, 4, 4, 4, 4;
   [5] 5, 5, 5, 5, 5, 5;
   [6] 6, 6, 6, 6, 6, 6, 6;
   [7] 7, 7, 7, 7, 7, 7, 7, 7;
   [8] 8, 8, 8, 8, 8, 8, 8, 8, 8;
   [9] 9, 9, 9, 9, 9, 9, 9, 9, 9, 9;
  [10] 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
  ... (End)

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).

Table[k*Floor[n/10] + Mod[n, 10], {n, 0, 10}, {k, 0, n}]//Flatten (* Paolo Xausa, May 22 2024 *)

Offset changed to 0 by Paolo Xausa, May 22 2024

A381608 Nonnegative integers whose ternary expansion does not contain pairs of 2's only separated by (zero or more) 1's, with offset 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93

Offset: 0

Rémy Sigrist, Mar 01 2025

The ternary expansion of a(n) equals the decimal expansion of A381579(n).

			The ternary expansion of 20, "211", has no pairs of 2's, so 20 belongs to the sequence.The ternary expansion of 21, "212", has a pair of 2s only separated by 1's, so 21 does not belong to the sequence.

Rémy Sigrist, Table of n, a(n) for n = 0..6560

Cf. A028898, A381579, A381607.

a(n) = { for (k = 1, oo, my (f = fibonacci(2*k)); if (f >= n, my (v = 0); while (n, while (n >= f, n -= f; v += 3^(k-1);); f = fibonacci(2*k--);); return (v););); }

a(n) = A028898(A381579(n)).

A381607(a(n)) = n.

cp's OEIS Frontend

A081502 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 3x+y.

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A381607 For any nonnegative integer n with ternary expansion Sum_{k >= 0} t_k * 3^k, a(n) = Sum_{k >= 0} t_k * A000045(2*k + 2).

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A360506 Read A360505(n) as if it were a base-3 string and write it in base 10.

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A081599 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 8x+y.

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A083291 Triangular array read by rows: T(n,k) = k*floor(n/10) + n mod 10, 0<=k<=n.

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A381608 Nonnegative integers whose ternary expansion does not contain pairs of 2's only separated by (zero or more) 1's, with offset 0.

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