A243658 -id:A243658 - cp's OEIS frontend

A242350 Multiply a(n-1) by 2 and drop all 0's.

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 124, 248, 496, 992, 1984, 3968, 7936, 15872, 31744, 63488, 126976, 253952, 5794, 11588, 23176, 46352, 9274, 18548, 3796, 7592, 15184, 3368, 6736, 13472, 26944, 53888, 17776, 35552, 7114, 14228, 28456, 56912, 113824

Offset: 1

J. Lowell, May 11 2014

Sequence enters a loop having period 36 at index 491: a(491) = a(527) = 366784, min and max being 28714 and 11772544. Starting with 3 instead of 1 gives another cycle. - Tom Edgar and Michel Marcus, May 13 2014

Zeroless analog of powers of 2. - N. J. A. Sloane, Jun 11 2014

			Term after 512 is 124 because 512*2=1024, and 1024 becomes 124 if all 0's are taken out.

Michel Marcus, Table of n, a(n) for n = 1..600
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A052382, A004719, A243063, A243657, A243658.

NestList[FromDigits[Select[IntegerDigits[2 #],#!=0&]]&,1,50] (* Harvey P. Dale, Oct 22 2018 *)

dropz(n)=d = digits(n); s = 0; for (i=1, #d, if (d[i], s = 10*s + d[i]);); s;
lista(nn) = a = 1; for (i=1, nn, print1(a, ", "); a = dropz(2*a);) \\ Michel Marcus, May 12 2014

More terms from Michel Marcus, May 12 2014

A243657 Zeroless factorials: a(0)=1; thereafter a(n) = noz(n*a(n-1)), where noz(n) = A004719(n) omits the zeros from n.

Original entry on oeis.org

1, 1, 2, 6, 24, 12, 72, 54, 432, 3888, 3888, 42768, 513216, 667188, 934632, 141948, 2271168, 3869856, 6965748, 132349212, 264698424, 555866694, 1222967268, 28128247164, 67577931936, 16894482984, 439256557584, 1185992754768, 332779713354, 965611687266, 289683561798, 89819415738

Offset: 0

N. J. A. Sloane, Jun 11 2014

Zeroless analog of factorial numbers.

A very crude calculation suggests a(n) may grow like n^10. I don't really believe that, but certainly a(n) grows very slowly in comparison with n!. - N. J. A. Sloane, Sep 03 2014

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Cf. A000142, A004719, A242350, A243063, A243658.

noz:=proc(n) local a,t1,i,j; a:=0; t1:=convert(n,base,10); for i from 1 to nops(t1) do j:=t1[nops(t1)+1-i]; if j <> 0 then a := 10*a+j; fi; od: a; end;
t1:=[1]; for n from 1 to 50 do t1:=[op(t1),noz(n*t1[n])]; od: t1;

nxt[{n_,a_}]:={n+1,FromDigits[DeleteCases[IntegerDigits[a(n+1)],0]]}; NestList[nxt,{0,1},40][[;;,2]] (* Harvey P. Dale, Feb 13 2024 *)

from itertools import count, islice
def noz(n): return int(str(n).replace("0", ""))
def agen(): # generator of terms
    yield (an:=1)
    yield from (an:=noz(n*an) for n in count(1))
print(list(islice(agen(), 32))) # Michael S. Branicky, Jul 02 2024

A243063 Numbers generated by a Fibonacci-like sequence in which zeros are suppressed.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 61, 438, 499, 937, 1436, 2373, 389, 2762, 3151, 5913, 964, 6877, 7841, 14718, 22559, 37277, 59836, 97113, 156949, 25462, 182411, 27873, 21284, 49157, 7441, 56598, 6439, 6337, 12776, 19113, 31889, 512, 3241

Offset: 1

Anthony Sand, Jun 09 2014

Let x(1) = 1, x(2) = 1, then begin the sequence x(i) = no-zero(x(i-2) + x(i-1)), where the function no-zero(n) removes all zero digits from n.

The sequence behaves like a standard Fibonacci sequence until step 15, where x = no-zero(233 + 377) = no-zero(610) = 61. At step 16, x = 377 + 61 = 438. The sequence then proceeds until step 927, where x = no-zero(224 + 377) = no-zero(601) = 61. Therefore at step 928, x = 377 + 61 = 438 and the sequence repeats.

			x(3) = x(1) + x(2) = 1 + 1 = 2.
x(4) = x(2) + x(3) = 1 + 2 = 3.
x(15) = no-zero(x(13) + x(14)) = no-zero(233 + 377) = no-zero(610) = 61.
x(16) = 377 + 61 = 438.

Anthony Sand, Table of n, a(n) for n = 1..927
Index entries for linear recurrences with constant coefficients, order 912.

Cf. A000045, A004719, A242350, A243657, A243658, A306773.

noz:=proc(n) local a,t1,i,j; a:=0; t1:=convert(n,base,10); for i from 1 to nops(t1) do j:=t1[nops(t1)+1-i]; if j <> 0 then a := 10*a+j; fi; od: a; end; # A004719
t1:=[1,1]; for n from 3 to 100 do t1:=[op(t1),noz(t1[n-1]+t1[n-2])]; od: t1; # N. J. A. Sloane, Jun 11 2014

Nest[Append[#, FromDigits@ DeleteCases[IntegerDigits[Total@ #[[-2 ;; -1]] ], ?(# == 0 &)]] &, {1, 1}, 45] (* _Michael De Vlieger, Jun 27 2020 *)
nxt[{a_,b_}]:={b,FromDigits[DeleteCases[IntegerDigits[a+b],0]]}; NestList[nxt,{1,1},50][[All,1]] (* Harvey P. Dale, Sep 12 2022 *)

x(i) = no-zero(x(i-2) + x(i-1)). For example, no-zero(233 + 377) = no-zero(610) = 61.

A373169 Square array read by ascending antidiagonals: T(n,k) = noz(T(n,k-1) + (k-1)*(n-2) + 1), with T(n,1) = 1, n >= 2, k >= 1, where noz(n) = A004719(n).

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 9, 1, 5, 1, 6, 12, 16, 6, 6, 1, 7, 15, 22, 25, 12, 7, 1, 8, 18, 28, 35, 36, 19, 8, 1, 9, 21, 34, 45, 51, 49, 27, 9, 1, 1, 24, 4, 55, 66, 7, 64, 36, 1, 1, 11, 18, 46, 29, 81, 91, 29, 81, 46, 2, 1, 12, 3, 43, 75, 6, 112, 12, 54, 1, 57, 3

Offset: 2

Paolo Xausa, May 27 2024

Row n is the zeroless analog of the positive n-gonal numbers.

			The array begins:
  n\k|  1  2   3   4   5    6    7    8    9   10  ...
  ----------------------------------------------------
   2 |  1, 2,  3,  4,  5,   6,   7,   8,   9,   1, ... = A177274
   3 |  1, 3,  6,  1,  6,  12,  19,  27,  36,  46, ... = A243658 (from n = 1)
   4 |  1, 4,  9, 16, 25,  36,  49,  64,  81,   1, ... = A370812
   5 |  1, 5, 12, 22, 35,  51,   7,  29,  54,  82, ... = A373171
   6 |  1, 6, 15, 28, 45,  66,  91,  12,  45,  82, ... = A373172
   7 |  1, 7, 18, 34, 55,  81, 112, 148, 189, 235, ...
   8 |  1, 8, 21,  4, 29,   6,  43,  86, 135,  19, ...
   9 |  1, 9, 24, 46, 75, 111, 154,  24,  81, 145, ...
  10 |  1, 1, 18, 43, 76, 117, 166, 223, 288, 361, ...
  ...      |                                     \______ A373170 (main diagonal)
        A004719 (from n = 2)

Paolo Xausa, Table of n, a(n) for n = 2..11326 (first 150 antidiagonals, flattened).
Wikipedia, Polygonal number.

Cf. rows 2..6: A177274, A243658, A370812, A373171, A373172.
Cf. A373170 (main diagonal).
Cf. A004719, A057145.

noz[n_] := FromDigits[DeleteCases[IntegerDigits[n], 0]];
A373169[n_, k_] := A373169[n, k] = If[k == 1, 1, noz[A373169[n, k-1] + (k-1)*(n-2) + 1]];
Table[A373169[n - k + 1, k], {n, 2, 15}, {k, n - 1}]

noz(n) = fromdigits(select(sign, digits(n)));
T(n,k) = if (k==1, 1, noz(T(n,k-1) + (k-1)*(n-2) + 1));
matrix(7,7,n,k,T(n+1,k)) \\ Michel Marcus, May 30 2024

A370812 a(1) = 1; for n >= 2, a(n) = noz(a(n-1) + 2*n - 1), where noz(n) = A004719(n).

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 22, 45, 7, 34, 63, 94, 127, 162, 199, 238, 279, 322, 367, 414, 463, 514, 567, 622, 679, 738, 799, 862, 927, 994, 163, 234, 37, 112, 189, 268, 349, 432, 517, 64, 153, 244, 337, 432, 529, 628, 729, 832, 937, 144, 253, 364, 477

Offset: 1

Paolo Xausa, May 24 2024

Zeroless analog of the positive squares.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000290, A004719, A243657, A243658, A373171, A373172.

Row n = 4 of A373169.

noz[n_] := FromDigits[DeleteCases[IntegerDigits[n], 0]];
Block[{n = 1}, NestList[noz[++n*2 - 1 + #] &, 1, 100]]

A373171 a(1) = 1; for n >= 2, a(n) = noz(a(n-1) + 3*n - 2), where noz(n) = A004719(n).

Original entry on oeis.org

1, 5, 12, 22, 35, 51, 7, 29, 54, 82, 113, 147, 184, 224, 267, 313, 362, 414, 469, 527, 588, 652, 719, 789, 862, 938, 117, 199, 284, 372, 463, 557, 654, 754, 857, 963, 172, 284, 399, 517, 638, 762, 889, 119, 252, 388, 527, 669, 814, 962, 1113, 1267, 1424, 1584

Offset: 1

Paolo Xausa, May 28 2024

Zeroless analog of the positive pentagonal numbers.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Row n = 5 of A373169.

Cf. A000326, A004719, A243658, A370812, A373172.

noz[n_] := FromDigits[DeleteCases[IntegerDigits[n], 0]];
Block[{n = 1}, NestList[noz[++n*3 - 2 + #] &, 1, 100]]

noz(n) = fromdigits(select(sign, digits(n))); \\ A004719
lista(nn) = my(va=vector(nn)); for (n=1, nn, va[n] = if (n==1, 1, noz(va[n-1] + 3*n - 2))); va; \\ Michel Marcus, Jun 03 2024

A373172 a(1) = 1; for n >= 2, a(n) = noz(a(n-1) + 4*n - 3), where noz(n) = A004719(n).

Original entry on oeis.org

1, 6, 15, 28, 45, 66, 91, 12, 45, 82, 123, 168, 217, 27, 84, 145, 21, 9, 82, 159, 24, 19, 18, 111, 28, 129, 234, 343, 456, 573, 694, 819, 948, 181, 318, 459, 64, 213, 366, 523, 684, 849, 118, 291, 468, 649, 834, 123, 316, 513, 714, 919, 1128, 1341, 1558, 1779

Offset: 1

Paolo Xausa, May 28 2024

Zeroless analog of the positive hexagonal numbers.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Row n = 6 of A373169.

Cf. A000384, A004719, A243658, A370812, A373171.

noz[n_] := FromDigits[DeleteCases[IntegerDigits[n], 0]];
Block[{n = 1}, NestList[noz[++n*4 - 3 + #] &, 1, 100]]
nxt[{n_,a_}]:={n+1,FromDigits[DeleteCases[IntegerDigits[a+4n+1],0]]}; NestList[nxt,{1,1},60][[;;,2]] (* Harvey P. Dale, Jul 08 2024 *)

noz(n) = fromdigits(select(sign, digits(n))); \\ A004719
lista(nn) = my(va=vector(nn)); for (n=1, nn, va[n] = if (n==1, 1, noz(va[n-1] + 4*n - 3))); va; \\ Michel Marcus, Jun 03 2024

A321480 Zeroless analog of triangular numbers (version 2): a(0) = 0, and for any n > 0, a(n) = noz(1 + noz(2 + ... + noz((n-1) + n))), where noz(n) = A004719(n) omits the zeros from n.

Original entry on oeis.org

0, 1, 3, 6, 1, 15, 3, 28, 9, 18, 19, 39, 6, 28, 15, 12, 1, 9, 99, 37, 39, 177, 64, 69, 39, 19, 72, 99, 37, 12, 69, 64, 87, 12, 289, 27, 54, 82, 39, 42, 19, 6, 57, 37, 27, 54, 82, 12, 69, 64, 69, 12, 64, 27, 27, 82, 12, 87, 289, 69, 39, 289, 72, 99, 64, 57, 24

Offset: 0

Rémy Sigrist, Nov 11 2018

This sequence is a variant of A243658 where the additions are carried in the opposite order; as (i, j) -> noz(i + j) is not associative in general we obtain another sequence.
This sequence is conjectured to be bounded. This could be explained by the fact that the zeros appearing in the last steps of the calculation (when adding small values) erode the number of digits of the intermediate sums.
The distinct values among the first 1000000 terms are: 0, 1, 3, 6, 9, 12, 15, 18, 19, 24, 27, 28, 37, 39, 42, 54, 57, 64, 69, 72, 82, 84, 87, 99, 177, 289.

			For n = 16:
- noz(15 + 16) = noz(31) = 31,
- noz(14 + 31) = noz(45) = 45,
- noz(13 + 45) = noz(58) = 58,
- noz(12 + 58) = noz(70) = 7,
- noz(11 + 7) = noz(18) = 18,
- noz(10 + 18) = noz(28) = 28,
- noz(9 + 28) = noz(37) = 37,
- noz(8 + 37) = noz(45) = 45,
- noz(7 + 45) = noz(52) = 52,
- noz(6 + 52) = noz(58) = 58,
- noz(5 + 58) = noz(63) = 63,
- noz(4 + 63) = noz(67) = 67,
- noz(3 + 67) = noz(70) = 7,
- noz(2 + 7) = noz(9) = 9,
- noz(1 + 9) = noz(10) = 1,
- hence a(16) = 1.

Rémy Sigrist, Table of n, a(n) for n = 0..10000 (corrected by _Paolo Xausa_, Apr 17 2024)

Cf. A000217, A004719, A243658.

noz[n_] := FromDigits[DeleteCases[IntegerDigits[n], 0]];
A321480[n_] := Block[{k = n}, Nest[noz[--k + #] &, n, Max[0, n-1]]];
Array[A321480,100,0] (* Paolo Xausa, Apr 17 2024 *)

a(n, base=10) = { my (t=n); forstep (k=n-1, 1, -1, t = fromdigits(select(sign, digits(t+k, base)), base)); t } \\ corrected by Rémy Sigrist, Apr 17 2024

a(10), a(20), a(30), a(40), a(50) and a(60) corrected by Paolo Xausa, Apr 17 2024

A371911 Zeroless analog of tribonacci numbers.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 17, 31, 57, 15, 13, 85, 113, 211, 49, 373, 633, 155, 1161, 1949, 3265, 6375, 11589, 21229, 39193, 7211, 67633, 11437, 86281, 165351, 26369, 2781, 19451, 4861, 2793, 2715, 1369, 6877, 1961, 127, 8965, 1153, 1245, 11363, 13761, 26369, 51493, 91623, 169485, 31261

Offset: 0

Bryle Morga, Apr 11 2024

At n = 208666297 this sequence enters a cycle that has a period of 300056874 and begins: 2847, 26331, 5851, ... (only the first 3 terms of the cycle are needed to reproduce the entire cycle).

This can be compared with the sequence A243063, which enters a cycle that has a (relatively) small period of 912.

			a(9) = Zr(a(8) + a(7) + a(6)) = Zr(17 + 31 + 57) = Zr(105) = 15.

Cf. A000045, A000213, A004719, A242350, A243657, A243658, A243063.

a[0]=a[1]=a[2]=1; a[n_]:=FromDigits[DeleteCases[IntegerDigits[a[n-1]+a[n-2]+a[n-3]],0]]; Array[a,50,0] (* Stefano Spezia, Apr 12 2024 *)

def a(n):
    a, b, c = 1, 1, 1
    for _ in range(n):
        a, b, c = b, c, int(str(a+b+c).replace('0', ''))
    return a

# faster for initial segment of sequence
from itertools import islice
def agen(): # generator of terms
    a, b, c = 1, 1, 1
    while True: yield a; a, b, c = b, c, int(str(a+b+c).replace("0", ""))
print(list(islice(agen(), 50))) # Michael S. Branicky, Apr 13 2024

a(n) = Zr(a(n-1) + a(n-2) + a(n-3)), where the function Zr(k) removes all zero digits from k.

A374265 Minimized zeroless factorials.

Original entry on oeis.org

1, 1, 2, 6, 24, 12, 72, 54, 432, 3888, 3888, 42768, 47916, 62298, 872172, 13968, 221688, 57996, 143928, 134712, 269154, 563994, 1247868, 286344, 877356, 171864, 513324, 1252728, 3414474, 914616, 41868, 119178, 454716, 127188, 527832, 15642, 91332, 192924, 125892, 29718

Offset: 0

Bryle Morga, Jul 02 2024

a(n) is the smallest f(n) such that f(0) = 1 and for i > 0, f(i) = OpNoz_i(i*f(i-1)),where OpNoz_i is a function that either removes zeros or keeps the value unchanged (the choice is made for each value of i).
Removing zeros at every i gives an upper bound a(n) <= A243657(n); is this a strict inequality for n >= 12?
Is this sequence bounded?

			a(12) = 47916 via the path: 1, 1, 2, 6, 24, 12, 72, 504, 4032, 36288, 362880, 3991680, 47916.

Cf. A000142, A004719, A242350, A243063, A243657, A243658, A356757, A374266.

def a(n):
   reach = {1}
   for i in range(1, n+1):
      newreach = set()
      for m in reach:
         newreach.update([m*i, int(str(m*i).replace('0', ''))])
      reach = newreach
   return min(reach)

cp's OEIS Frontend

A242350 Multiply a(n-1) by 2 and drop all 0's.

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A243657 Zeroless factorials: a(0)=1; thereafter a(n) = noz(n*a(n-1)), where noz(n) = A004719(n) omits the zeros from n.

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A243063 Numbers generated by a Fibonacci-like sequence in which zeros are suppressed.

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A373169 Square array read by ascending antidiagonals: T(n,k) = noz(T(n,k-1) + (k-1)*(n-2) + 1), with T(n,1) = 1, n >= 2, k >= 1, where noz(n) = A004719(n).

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A370812 a(1) = 1; for n >= 2, a(n) = noz(a(n-1) + 2*n - 1), where noz(n) = A004719(n).

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A373171 a(1) = 1; for n >= 2, a(n) = noz(a(n-1) + 3*n - 2), where noz(n) = A004719(n).

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A373172 a(1) = 1; for n >= 2, a(n) = noz(a(n-1) + 4*n - 3), where noz(n) = A004719(n).

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A321480 Zeroless analog of triangular numbers (version 2): a(0) = 0, and for any n > 0, a(n) = noz(1 + noz(2 + ... + noz((n-1) + n))), where noz(n) = A004719(n) omits the zeros from n.

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