A004719 -id:A004719 - cp's OEIS frontend

A243658 a(0)=0; thereafter a(n) = noz(n+a(n-1)), where noz(n) = A004719(n).

0, 1, 3, 6, 1, 6, 12, 19, 27, 36, 46, 57, 69, 82, 96, 111, 127, 144, 162, 181, 21, 42, 64, 87, 111, 136, 162, 189, 217, 246, 276, 37, 69, 12, 46, 81, 117, 154, 192, 231, 271, 312, 354, 397, 441, 486, 532, 579, 627, 676, 726, 777, 829, 882, 936, 991, 147, 24, 82, 141, 21, 82, 144, 27, 91, 156, 222, 289

Offset: 0

N. J. A. Sloane, Jun 11 2014

Zeroless analog of triangular numbers.

N. J. A. Sloane, Table of n, a(n) for n = 0..5000

Cf. A004719, A242350, A243063, A243657.

Row n = 3 of A373169.

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:=[0]; for n from 1 to 50 do t1:=[op(t1),noz(n+t1[n])]; od: t1;

noz[n_] := FromDigits[DeleteCases[IntegerDigits[n], 0]];
Block[{n = 0}, NestList[noz[++n+#] &, 0, 100]] (* Paolo Xausa, Apr 17 2024 *)

from itertools import count, islice
def noz(n): return int(str(n).replace("0", ""))
def agen(): # generator of terms
    yield (an:=0)
    yield from (an:=noz(n+an) for n in count(1))
print(list(islice(agen(), 68))) # Michael S. Branicky, Jul 02 2024

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

A306569 a(n) is the largest value obtained by iterating x -> noz(x * n) starting from 1 (where noz(k) = A004719(k) omits the zeros from k) if any, otherwise a(n) = -1.

Original entry on oeis.org

1, 765257552, 4858337151, 62987487698944, 14281197489647865625, 16756687971893376, 956884714362661, 292452281337511936, 11897243269649199, 1, 824281567746336491, 13552472793415699584, 841944776182612378933, 9434962871842528764976

Offset: 1

Rémy Sigrist, Feb 24 2019

Is every term positive?

			a(2) = max(A242350) = 765257552.

Rémy Sigrist, Table of n, a(n) for n = 1..250
Rémy Sigrist, PARI program for A306569

See A306567 for the additive variant.

Cf. A004719, A242350.

PARI
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See Links section.
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a(10^k) = 1 for any k >= 0.

a(10*n) = a(n) for any n > 0.

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

A306567 a(n) is the largest value obtained by iterating x -> noz(x + n) starting from 0 (where noz(k) = A004719(k) omits the zeros from k).

Original entry on oeis.org

9, 99, 27, 99, 96, 99, 63, 99, 81, 91, 99, 195, 94, 295, 93, 291, 113, 189, 171, 992, 159, 187, 187, 483, 988, 475, 153, 281, 181, 273, 279, 577, 297, 997, 567, 369, 333, 363, 351, 994, 219, 465, 357, 663, 459, 461, 423, 192, 441, 965, 399, 999, 437, 126, 551

Offset: 1

Rémy Sigrist, Feb 24 2019

For any n > 0, a(n) is well defined:
- the set of zeroless numbers (A052382) contains arbitrarily large gaps,
- for example, for any k > 0, the interval I_k = [10^k..(10^(k+1)-1)/9-1] if free of zeroless numbers,
- let i be such that #I_i > n,
- let b_n be defined by b_n(0) = 0, and for any j > 0, b_n(j) = noz(b_n(j-1) + n),
- as b_n starts below 10^i and cannot cross the gap constituted by I_i,
- b_n is bounded (and eventually periodic), QED.

			For n = 1:
- noz(0 + 1) = 1,
- noz(1 + 1) = 2,
- noz(2 + 1) = 3,
  ...
- noz(7 + 1) = 8,
- noz(8 + 1) = 9,
- noz(9 + 1) = noz(10) = 1,
- hence a(1) = 9.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A306567

See A306569 for the multiplicative variant.

Cf. A004719, A052382.

PARI
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\\ See Links section.
```

Empirically, for any k >= 0:
- a(    10^k) =  9 * 10^k +     (10^k-1)/9,
- a(2 * 10^k) = 99 * 10^k + 2 * (10^k-1)/9,
- a(3 * 10^k) = 27 * 10^k + 3 * (10^k-1)/9,
- a(4 * 10^k) = 99 * 10^k + 4 * (10^k-1)/9,
- a(5 * 10^k) = 96 * 10^k + 5 * (10^k-1)/9,
- a(6 * 10^k) = 99 * 10^k + 6 * (10^k-1)/9,
- a(7 * 10^k) = 63 * 10^k + 7 * (10^k-1)/9,
- a(8 * 10^k) = 99 * 10^k + 8 * (10^k-1)/9,
- a(9 * 10^k) = 81 * 10^k + 9 * (10^k-1)/9.

A321475 Zeroless factorials (version 2): a(0) = 1, 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

1, 1, 2, 6, 24, 12, 72, 54, 432, 3888, 3888, 399168, 576, 82728, 879912, 2397168, 337968, 5924736, 8851949568, 143936352, 31644, 92589264, 118459638, 3698784, 1197539136, 2387625984, 954864, 236271168, 3573339984, 238453776, 69587928, 142275168, 33566976

Offset: 0

Rémy Sigrist, Nov 11 2018

This sequence is a variant of A243657 where the multiplications are carried in the opposite order; as (i, j) -> noz(i * j) is not associative in general we obtain another sequence.

Is this sequence bounded?

			For n = 12:
- noz(11 * 12) = noz(132) = 132,
- noz(10 * 132) = noz(1320) = 132,
- noz(9 * 132) = noz(1188) = 1188,
- noz(8 * 1188) = noz(9504) = 954,
- noz(7 * 954) = noz(6678) = 6678,
- noz(6 * 6678) = noz(40068) = 468,
- noz(5 * 468) = noz(2340) = 234,
- noz(4 * 234) = noz(936) = 936,
- noz(3 * 936) = noz(2808) = 288,
- noz(2 * 288) = noz(576) = 576,
- noz(1 * 576) = noz(576) = 576,
- hence a(12) = 576.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A000142, A004719, A243657.

noz[n_] := FromDigits[DeleteCases[IntegerDigits[n], 0]];
A321475[n_] := If[n == 0, 1, Block[{k = n}, Nest[noz[--k * #] &, n, n-1]]];
Array[A321475, 50, 0] (* Paolo Xausa, May 20 2024 *)

a(n, base=10) = my (f=max(1, n)); forstep (k=n-1, 2, -1, f = fromdigits(select(sign, digits(f*k, base)), base)); f

a(10^k) = a(10^k - 1) for any k >= 0.

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

cp's OEIS Frontend

A243658 a(0)=0; thereafter a(n) = noz(n+a(n-1)), where noz(n) = A004719(n).

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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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A306569 a(n) is the largest value obtained by iterating x -> noz(x * n) starting from 1 (where noz(k) = A004719(k) omits the zeros from k) if any, otherwise a(n) = -1.

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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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A306567 a(n) is the largest value obtained by iterating x -> noz(x + n) starting from 0 (where noz(k) = A004719(k) omits the zeros from k).

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