A334318 -id:A334318 - cp's OEIS frontend

A333405 Number of integers in base n having exactly three distinct digits such that the number formed by the consecutive subsequence of the initial j digits is divisible by j for all j in {1,2,3}.

0, 0, 0, 0, 5, 6, 18, 33, 50, 67, 115, 134, 206, 258, 340, 398, 537, 598, 778, 891, 1086, 1209, 1487, 1614, 1950, 2148, 2504, 2716, 3181, 3398, 3938, 4245, 4810, 5135, 5835, 6166, 6958, 7398, 8220, 8682, 9665, 10134, 11226, 11823, 12950, 13573, 14887, 15518

Offset: 0

Alois P. Heinz, May 04 2020

			a(4) = 5: 102, 120, 123, 201, 321 (written in base 4).
a(5) = 6: 132, 201, 204, 314, 402, 421 (written in base 5).
a(6) = 18: 103, 120, 123, 140, 143, 203, 240, 243, 320, 340, 403, 420, 423, 503, 520, 523, 540, 543 (written in base 6).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,0,2,-2,-2,2,0,0,-1,1,1,-1).

Column k=3 of A334318.

G.f.: (2*x^11+5*x^9+10*x^8+21*x^6+2*x^5+5*x^4+14*x^3+7*x^2+x+5)*x^4 / ((x^2+x+1)^2 *(x^2-x+1)^2 *(x+1)^3 *(x-1)^4).

A333469 Number of integers in base n having exactly four distinct digits such that the number formed by the consecutive subsequence of the initial j digits is divisible by j for all j in {1,2,3,4}.

Original entry on oeis.org

0, 0, 0, 0, 2, 1, 8, 34, 58, 98, 168, 275, 428, 586, 849, 1193, 1647, 2017, 2679, 3454, 4410, 5283, 6676, 7900, 9838, 11396, 13758, 15994, 19216, 21493, 25450, 29026, 33854, 37636, 43724, 48369, 55884, 61374, 69831, 76803, 87269, 94285, 106337, 116062, 129862

Offset: 0

Alois P. Heinz, May 04 2020

			a(4) = 2: 1230, 3210 (written in base 4).
a(5) = 1: 3140 (written in base 5).
a(6) = 6: 1032, 1204, 1432, 3204, 4032, 5032, 5204, 5432 (written in base 6).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,0,0,0,0,0,0,0,3,-3,-3,3,0,0,0,0,0,0,0,0,-3,3,3,-3,0,0,0,0,0,0,0,0,1,-1,-1,1).

Column k=4 of A334318.

LinearRecurrence[{1,1,-1,0,0,0,0,0,0,0,0,3,-3,-3,3,0,0,0,0,0,0,0,0,-3,3,3,-3,0,0,0,0,0,0,0,0,1,-1,-1,1},{0,0,0,0,2,1,8,34,58,98,168,275,428,586,849,1193,1647,2017,2679,3454,4410,5283,6676,7900,9838,11396,13758,15994,19216,21493,25450,29026,33854,37636,43724,48369,55884,61374,69831,76803},110] (* Harvey P. Dale, Oct 06 2023 *)

G.f.: -(6*x^35 -4*x^33 +41*x^32 +11*x^31 +87*x^30 -46*x^29 +40*x^28 +165*x^27 +126*x^26 -40*x^25 +293*x^24 +120*x^23 +94*x^22 +181*x^21 +296*x^20 +150*x^19 +299*x^18 +56*x^17 +243*x^16 +324*x^15 +193*x^14 +29*x^13 +185*x^12 +186*x^11 +110*x^10 +51*x^9 +83*x^8 +67*x^7 +46*x^6 +14*x^5 +17*x^4 +27*x^3 +5*x^2 -x +2)*x^4 / ((x^2+1)^3 *(x^2+x+1)^3 *(x^2-x+1)^3 *(x^4-x^2+1)^3 *(x+1)^4 *(x-1)^5).

A334320 Number of even integers in base n with exactly two distinct digits.

Original entry on oeis.org

0, 0, 1, 1, 5, 6, 13, 15, 25, 28, 41, 45, 61, 66, 85, 91, 113, 120, 145, 153, 181, 190, 221, 231, 265, 276, 313, 325, 365, 378, 421, 435, 481, 496, 545, 561, 613, 630, 685, 703, 761, 780, 841, 861, 925, 946, 1013, 1035, 1105, 1128, 1201, 1225, 1301, 1326, 1405

Offset: 0

Alois P. Heinz, Apr 22 2020

			a(4) = 5: 10, 12, 20, 30, 32 (written in base 4).
a(7) = 15: 13, 15, 20, 24, 26, 31, 35, 40, 42, 46, 51, 53, 60, 62, 64 (written in base 7).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Column k=2 of A334318.

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

a(n) = A334318(n,2).

A334319 Number of integers m in base n with distinct digits such that the number formed by the consecutive subsequence of the initial j digits is divisible by j for all j in {1,...,length(m)}.

Original entry on oeis.org

1, 3, 4, 16, 18, 54, 112, 225, 397, 707, 1566, 2664, 6960, 9213, 23066, 39980, 100239, 116229, 455539, 465054, 1157163, 2064246, 6735123, 4609476, 22943866, 27085154, 64108419, 76354062, 420698429, 180541932, 1833215296, 1057775180, 3361833346, 5293490772, 14955881506, 7186246508

Offset: 1

Alois P. Heinz, Apr 22 2020

Row sums of A334318.

b:= proc(n, s, w) option remember; `if`(s={}, 0, (k-> add((t->
      `if`(t=0, 1, `if`(irem(t, k)=0, b(n, s minus {j}, t)
          +1, 0)))(w*n+j), j=s)))(1+n-nops(s))
    end:
a:= n-> b(n, {$0..n-1}, 0):
seq(a(n), n=1..18);

a(27)-a(36) from Giovanni Resta, May 04 2020

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A333405 Number of integers in base n having exactly three distinct digits such that the number formed by the consecutive subsequence of the initial j digits is divisible by j for all j in {1,2,3}.

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A333469 Number of integers in base n having exactly four distinct digits such that the number formed by the consecutive subsequence of the initial j digits is divisible by j for all j in {1,2,3,4}.

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A334320 Number of even integers in base n with exactly two distinct digits.

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A334319 Number of integers m in base n with distinct digits such that the number formed by the consecutive subsequence of the initial j digits is divisible by j for all j in {1,...,length(m)}.

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