A331440 -id:A331440 - cp's OEIS frontend

A331441 Inverse permutation to A331440.

1, 2, 6, 3, 14, 7, 24, 4, 33, 15, 44, 8, 55, 25, 66, 5, 77, 34, 85, 16, 96, 45, 107, 9, 118, 56, 129, 26, 140, 67, 151, 162, 173, 78, 184, 35, 195, 86, 206, 17, 217, 97, 228, 46, 248, 108, 264, 10, 305, 119, 336, 57, 364, 130, 392, 27, 423, 141, 477, 68, 498

Offset: 1

N. J. A. Sloane, Jan 22 2020

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A331441

Cf. A331440.

PARI
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See Links section.
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More terms from Rémy Sigrist, Jan 22 2020

A331442 Length of n-th chain in A331440.

Original entry on oeis.org

5, 8, 10, 9, 11, 11, 11, 11, 8, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 20, 16, 41, 31, 28, 28, 31, 54, 21, 9, 36, 11, 24, 23, 32, 10, 37, 42, 12, 27, 46, 16, 40, 27, 6, 19, 45, 13, 42, 42, 37, 61, 37, 98, 93, 18, 44, 33, 19, 59, 35, 54, 166, 131

Offset: 1

N. J. A. Sloane, Jan 22 2020

			When starting chain 9, A331440(77) is set to the first free number which is 17, and the chain becomes 17, 34, 68, 136, 272, 544, 1088, 2176 with length 8.

Lars Blomberg, Table of n, a(n) for n = 1..3000
Rémy Sigrist, PARI program for A331442

Cf. A331440, A331441.

PARI
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See Links section.
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Corrected a(9) and more terms from Lars Blomberg, Jan 22 2020

A348433 a(1) = 1; a(n+1) = 2*a(n) if the digit sum of a(n) is already in the sequence, otherwise a(n+1) = digitsum(a(n)).

Original entry on oeis.org

1, 2, 4, 8, 16, 7, 14, 5, 10, 20, 40, 80, 160, 320, 640, 1280, 11, 22, 44, 88, 176, 352, 704, 1408, 13, 26, 52, 104, 208, 416, 832, 1664, 17, 34, 68, 136, 272, 544, 1088, 2176, 4352, 8704, 19, 38, 76, 152, 304, 608, 1216, 2432, 4864, 9728, 19456, 25, 50, 100, 200

Offset: 1

Rodolfo Kurchan, Oct 18 2021

There are no multiples of 3 in this sequence.

Will all other positive integers appear in this sequence?

			The sum of the digits of a(4) = 8 is 8, which is already in the sequence, so a(5) = 2*8 = 16.
The sum of the digits of a(5) = 16 is 7, which is not yet in the sequence, so a(6) = 7.
From _Omar E. Pol_, Oct 19 2021: (Start)
Written as an irregular triangle the sequence begins (see A348408):
   1,  2,  4,   8,  16;
   7, 14;
   5, 10, 20,  40,  80, 160,  320,  640, 1280;
  11, 22, 44,  88, 176, 352,  704, 1408;
  13, 26, 52, 104, 208, 416,  832, 1664;
  17, 34, 68, 136, 272, 544, 1088, 2176, 4352, 8704;
  19, 38, 76, 152, 304, 608, 1216, 2432, 4864, 9728, 19456;
... (End)

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A348433

Cf. A001651, A007953, A008585, A331440 (similar), A348400, A348408, A348483.

seq[len_] := Module[{s = {1}, k, d}, While[Length[s] < len, k = s[[-1]]; If[MemberQ[s, (d = Plus @@ IntegerDigits[k])], AppendTo[s, 2*k], AppendTo[s, d]]]; s]; seq[50] (* Amiram Eldar, Oct 19 2021 *)

lista(nn) = my(s, v=List([1])); for(n=1, nn, if(setsearch(vecsort(v), s=sumdigits(v[n])), listput(v, 2*v[n]), listput(v, s))); v \\ Jinyuan Wang, Oct 21 2021

PARI
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See Links section.
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Definition and examples clarified by N. J. A. Sloane, Oct 24 2021

A331619 a(n) is the smallest positive number k such that the decimal expansion of n*2^k contains the string n.

Original entry on oeis.org

1, 4, 4, 7, 4, 9, 4, 8, 4, 10, 10, 10, 10, 10, 10, 10, 10, 7, 10, 10, 4, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 4, 10, 28, 19, 13, 15, 16, 40, 20, 30, 53, 27, 20, 27, 35, 30, 20, 53, 31, 20, 4, 8, 18, 35, 20, 10, 30, 23, 20

Offset: 0

Rémy Sigrist, Jan 22 2020

			  n   a(n)  n*2^a(n)
  --  ----  --------
   0     1         0
   1     4        16
   2     4        32
   3     7       384
   4     4        64
   5     9      2560
   6     4        96
   7     8      1792
   8     4       128
   9    10      9216
  10    10     10240

Rémy Sigrist, PARI program for A331619

Cf. A030000, A331440.

A331631 Let S = smallest missing positive number, adjoin S, 3S, 9S, 27S, 81S, ... to the sequence until reaching a term that has S as a substring; reset S to the smallest missing positive number, repeat.

Original entry on oeis.org

1, 3, 9, 27, 81, 2, 6, 18, 54, 162, 4, 12, 36, 108, 324, 5, 15, 7, 21, 63, 189, 567, 8, 24, 72, 216, 648, 10, 30, 90, 270, 810, 11, 33, 99, 297, 891, 2673, 8019, 24057, 72171, 216513, 649539, 1948617, 5845851, 17537553, 52612659, 157837977, 473513931, 1420541793, 4261625379, 12784876137, 38354628411, 13, 39

Offset: 1

Eric Angelini and Carole Dubois, Jan 23 2020

This is conjectured to be a permutation of the positive integers (see the Crossrefs section).

			The process begins like this:
Initially S = 1 is the smallest missing number, so we have:
S = 1, 3, 9, 27, 81, stop (because 81 contains S), S = 2, 6, 18, 54, 162, stop, S = 4, 12, 36, 108, 324, stop, S = 5, 15, stop, S = 7, 21, 63, 189, 567, ...

Carole Dubois, Table of n, a(n) for n = 1..14999

Cf. A331440 (where one adjoins 2*S, 4*S, 8*S, 16*S, ... to the sequence).

A341993 a(0)=0. For n > 0, a(n+1) = 2a(n) if the sum of digits of 2a(n) exceeds that of a(n); otherwise, a(n+1) is the smallest unused nonnegative integer.

Original entry on oeis.org

0, 1, 2, 4, 8, 3, 6, 5, 7, 9, 10, 20, 40, 80, 11, 22, 44, 88, 12, 24, 48, 96, 13, 26, 14, 28, 56, 15, 16, 17, 18, 19, 38, 76, 21, 42, 84, 168, 23, 46, 92, 184, 368, 25, 27, 29, 58, 30, 60, 31, 62, 32, 64, 128, 256, 33, 66, 34, 68, 35, 36, 37, 74, 148, 296, 39

Offset: 0

Jamie Robert Creasey, Feb 25 2021

This sequence is a permutation of the nonnegative integers; the inverse permutation begins 0, 1, 2, 5, 3, 7, 6, 8, 4, 9, 10, ...
There exist areas that feature numbers in runs of three or more in arithmetic progression, such as (5, 7, 9) and (15, 16, 17, 18, 19).
Record values are 0, 1, 2, 4, 8, 9, 10, 20, 40, 80, 88, ...

			We start the sequence with 0. Doubling this integer results in 0, but as the sum of digits of 0 is equal to that of 0, we choose the smallest nonnegative integer not yet used, which is 1. We can double 1 three times before the sum of digits of 2*a(n) (i.e., 16) does not exceed that of a(n) (8). Thus the next term after 8 is the next unused nonnegative integer, 3, after which we resume doubling.

Index entries for sequences that are permutations of the nonnegative integers

Cf. A000079 (powers of 2), A331440 (similar principle, except lesser or equal sum of digits replaced by containing the digit S).

cp's OEIS Frontend

A331441 Inverse permutation to A331440.

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A331442 Length of n-th chain in A331440.

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A348433 a(1) = 1; a(n+1) = 2*a(n) if the digit sum of a(n) is already in the sequence, otherwise a(n+1) = digitsum(a(n)).

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A331619 a(n) is the smallest positive number k such that the decimal expansion of n*2^k contains the string n.

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A331631 Let S = smallest missing positive number, adjoin S, 3*S, 9*S, 27*S, 81*S, ... to the sequence until reaching a term that has S as a substring; reset S to the smallest missing positive number, repeat.

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A341993 a(0)=0. For n > 0, a(n+1) = 2*a(n) if the sum of digits of 2*a(n) exceeds that of a(n); otherwise, a(n+1) is the smallest unused nonnegative integer.

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Author

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Crossrefs

A331631 Let S = smallest missing positive number, adjoin S, 3S, 9S, 27S, 81S, ... to the sequence until reaching a term that has S as a substring; reset S to the smallest missing positive number, repeat.

A341993 a(0)=0. For n > 0, a(n+1) = 2a(n) if the sum of digits of 2a(n) exceeds that of a(n); otherwise, a(n+1) is the smallest unused nonnegative integer.