cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 41-44 of 44 results.

A352317 Numbers m such that A352688(m) = 1.

Original entry on oeis.org

3, 9, 10, 12, 18, 19, 21, 27, 28, 30, 36, 37, 39, 45, 46, 48, 54, 55, 57, 63, 64, 66, 72, 73, 75, 81, 82, 84, 90, 91, 93, 99, 100, 102, 108, 109, 111, 117, 118, 120, 126, 127, 129, 135, 136, 138, 144, 145, 147, 153, 154, 156, 162, 163, 165, 171, 172, 174, 180, 181, 183, 189, 190, 192, 198, 199
Offset: 1

Views

Author

Bernard Schott, Apr 14 2022

Keywords

Comments

Equivalently: numbers m such that the sum of digits (A007953) of the integers from 1 to A331786(m) is not divisible by m.
Numbers m such that the first run of A331786(m) consecutive numbers whose sum of digits (A007953) is not divisible by m begins at 1.
A331786(m) is the largest possible number of consecutive integers whose sum of digits is not divisible by m.
For this sequence here, A352689(m) = A331786(m).

Examples

			For m = 10, the sum of digits of the integers from 1 up to A331786(10) = 18 is not divisible by 10; then for 19, sod(19) = 10 is divisible by 10, hence 10 is a term.

Links

Crossrefs

Cf. A007953, A331786, A352688, A352689.
A008591 \ {0} and A017173 \ {1} are subsequences.

Programs

  • PARI
    a88(n) = my(t=gcd(n%9, 9)); if(t<9, 10^lift(Mod(-1, n/t)/(9/t)) - 10^(n\9)*(n%9-t+1) + 1, 1); \\ A352688
isok(m) = a88(m) == 1; \\ Michel Marcus, Apr 15 2022

Extensions

More terms from Michel Marcus, Apr 15 2022

A361203 a(n) = n*A010888(n).

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 10, 22, 36, 52, 70, 90, 112, 136, 162, 19, 40, 63, 88, 115, 144, 175, 208, 243, 28, 58, 90, 124, 160, 198, 238, 280, 324, 37, 76, 117, 160, 205, 252, 301, 352, 405, 46, 94, 144, 196, 250, 306, 364, 424, 486, 55, 112, 171, 232
Offset: 0

Views

Author

Stefano Spezia, Apr 20 2023

Keywords

Comments

Every run of increasing terms ends with a positive multiple of 81, and except for the first run, it starts with a term of A017173 which is a fixed point for this sequence (see 4th formula).

Links

Crossrefs

Cf. A010888, A017173, A057147.

Programs

  • Mathematica
    a[n_]:=n(n - 9*Floor[(n-1)/9]); Join[{0},Array[a,58]]
  • Python
    def A361203(n): return n*(1 + (n - 1) % 9) # Chai Wah Wu, Apr 23 2023

Formula

G.f.: x*(1 + 4*x + 9*x^2 + 16*x^3 + 25*x^4 + 36*x^5 + 49*x^6 + 64*x^7 + 81*x^8 + 8*x^9 + 14*x^10 + 18*x^11 + 20*x^12 + 20*x^13 + 18*x^14 + 14*x^15 + 8*x^16)/((1 - x)^2*(1 + x + x^2)^2*(1 + x^3 + x^6)^2).
a(n) = 2*a(n-9) - a(n-18) for n > 17.
a(n) = n*(n - 9*floor((n-1)/9)) for n > 0.
a(A017173(n)) = A017173(n).

A367856 Table T(n,k), read by downward antidiagonals: T(n,k) = floor((3*T(n,k-1)+2)/2) starting with T(n,0) = 3*n.

Original entry on oeis.org

0, 1, 3, 2, 5, 6, 4, 8, 10, 9, 7, 13, 16, 14, 12, 11, 20, 25, 22, 19, 15, 17, 31, 38, 34, 29, 23, 18, 26, 47, 58, 52, 44, 35, 28, 21, 40, 71, 88, 79, 67, 53, 43, 32, 24, 61, 107, 133, 119, 101, 80, 65, 49, 37, 27, 92, 161, 200, 179, 152, 121, 98, 74, 56, 41, 30
Offset: 0

Views

Author

Philippe Deléham, Dec 03 2023

Keywords

Comments

Permutation of nonnegative numbers.

Examples

			Square array starts:
  0,  1,  2,  4,   7,  11,  17,  26,  40,   61, ...
  3,  5,  8, 13,  20,  31,  47,  71, 107,  161, ...
  6, 10, 16, 25,  38,  58,  88, 133, 200,  301, ...
  9, 14, 22, 34,  52,  79, 119, 179, 269,  404, ...
 12, 19, 29, 44,  67, 101, 152, 229, 344,  517, ...
 15, 23, 35, 53,  80, 121, 182, 274, 412,  619, ...
 18, 28, 43, 65,  98, 148, 223, 335, 503,  755, ...
 21, 32, 49, 74, 112, 169, 254, 382, 574,  862, ...
 24, 37, 56, 85, 128, 193, 290, 436, 655,  983, ...
 27, 41, 62, 94, 142, 214, 322, 484, 727, 1091, ...
 ...

Links

Crossrefs

Cf. A001651, A006999, A008585, A017173, A017221.

Programs

Formula

T(n,0) = 3*n = A008585(n).
T(2*n,1) = 9*n+1 = A017173(n).
T(2*n+1,1) = 9*n+5 = A017221(n).
T(0,k) = A006999(k).
T(2^k+n, k) = 3^(k+1) + T(n, k).

Extensions

More terms from Paolo Xausa, Apr 03 2024

A360145 Triangle read by rows where row n is the largest (or middle or n-th) column of the reverse pyramid summation of order n described in A359087.

Original entry on oeis.org

1, 2, 4, 3, 7, 19, 4, 10, 28, 78, 5, 13, 37, 105, 301, 6, 16, 46, 132, 382, 1108, 7, 19, 55, 159, 463, 1351, 3951, 8, 22, 64, 186, 544, 1594, 4680, 13758, 9, 25, 73, 213, 625, 1837, 5409, 15945, 47049, 10, 28, 82, 240, 706, 2080, 6138, 18132, 53610, 158616, 11, 31, 91, 267, 787, 2323, 6867, 20319, 60171, 178299, 528619
Offset: 1

Views

Author

Bernard Schott, Jan 27 2023

Keywords

Comments

The integer that is at the k-th row of the middle column of this pyramid of order n will be noted T(n,k).
Each row has n terms.

Examples

			Triangle begins:
   n=1:  1;
   n=2:  2,  4;
   n=3:  3,  7, 19;
   n=4:  4, 10, 28,  78;
   n=5:  5, 13, 37, 105, 301;
   n=6:  6, 16, 46, 132, 382, 1108;
   ...
For n=5, the reverse pyramid summation is as follows and row 5 here is the middle column 5,13,37,...
  1    2    3    4    5    4    3    2    1
       6    9   12   13   12    9    6
           27   34   37   34   27
                98  105   98
                    301

Crossrefs

Cf. A132894, A359087 (right diagonal).
Columns k=1..3: A000027, A016777, A017173.

Programs

  • PARI
    f(v) = if (#v == 1, v, vector(#v-2, i, v[i]+v[i+1]+v[i+2]));
row(n) = my(u = concat([1..n], Vecrev([1..n-1])), v=u, w = vector(n)); for (i=1, n, w[i] = v[#v\2+1]; v = f(v);); w; \\ Michel Marcus, Jan 30 2023

Formula

T(n,1) = n.
T(n,2) = 3n - 2.
T(n,3) = 9n - 8.
T(n,4) = 27n - 30.
T(n,5) = 81n - 104.
T(n,n) = A359087(n).
T(n,k) = 3^(k-1)*n - 2*A132894(k-1) for 1 <= k <= n (conjectured).
Previous Showing 41-44 of 44 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.