A307092 -id:A307092 - cp's OEIS frontend

A307074 a(n) is the smallest k such that A307092(k) = n.

1, 2, 3, 5, 9, 15, 27, 47, 55, 95, 187, 191, 375, 415, 751, 831, 1503, 1663, 3007, 3327, 6639, 7039, 13279, 14079, 26559, 28159, 53119, 56319, 106239, 112639, 212479, 225279, 424959, 450559, 849919, 901119, 1699839, 1802239, 3399679, 3604479, 6799359, 7208959

Offset: 0

Yancheng Lu, Mar 22 2019

nonn

a(n) is the smallest number k such that exactly n iterations of the mapping x -> x + x^j, where j is a nonnegative integer, are required to reach x=k from x=1 (the j's in each iteration need not be identical).

			n  |a(n)| maps                         | exponents
---+----+------------------------------+------------
1  | 1  | 1                            | []
2  | 2  | 1 -> 2                       | [0]
3  | 3  | 1 -> 2 -> 3                  | [0,0]
4  | 5  | 1 -> 2 -> 4 -> 5             | [0,1,0]
5  | 9  | 1 -> 2 -> 4 -> 8 -> 9        | [0,1,1,0]
6  | 15 | 1 -> 2 -> 6 -> 7 -> 14 -> 15 | [0,2,0,1,0]

Yancheng Lu, Pascal program for sequence
Minecraft Wiki, Execute command

Cf. A307092.

(* To get more terms of the sequence, increase terms and maxx,
   and then set maxi=trunc(lb(maxx)) *)
maxi=16;maxx=65536;terms=10;
a = NestList[
  Function[list,
   DeleteDuplicates[
    Join[list,
     Flatten[Table[If[# + #^i <= maxx, # + #^i, 1], {i, 0, maxi}] & /@
       list]]]], {1}, terms];
b = Prepend[Table[Complement[a[[i + 1]], a[[i]]], {i, Length[a] - 1}],
   First[a]];
Min /@ b

A309997 Number of paths from 2 to n of length A307092(n) - 1 via maps of the form x -> x + x^j, where j is a nonnegative integer.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2

Offset: 2

Peter Kagey, Aug 26 2019

nonn

This sequence counts paths starting from 2 since there are an infinite number of maps from 1 to 2 via 1 -> 1 + 1^j.

Records occur at 2, 12, 226, 372, 744, 1490, 139511, ...

			For n = 520, the a(520) = 3 sequences of A307092(520)-1 = 3 maps are:
2 -> 2 + 2^1 -> 4 + 4^1     -> 8 + 8^3     = 520
2 -> 2 + 2^1 -> 4 + 4^4     -> 260 + 260^1 = 520
2 -> 2 + 2^7 -> 130 + 130^1 -> 260 + 260^1 = 520
With exponents (1,1,3), (1,4,1), and (7,1,1) respectively.

Peter Kagey, Table of n, a(n) for n = 2..10000
Peter Kagey, Count the number of paths to n, Code Golf Stack Exchange.

Cf. A307074, A307092.

A309978 a(n) is the number of positive integers k such that there exists a nonnegative integer m with k + k^m = n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1

Offset: 1

Peter Kagey, Aug 28 2019

nonn

Records occur at 1, 2, 4, 6, 30, ...

Does there exist n such that a(n) >= 5? Do there exist examples besides 30 and 130 such that a(n) = 4? If so in either case, n > A253913(10000) = 87469256.

			For n = 130 the a(130) = 4 positive integers with valid maps are
  129 via 129 + 129^0 = 130,
   65 via  65 +  65^1 = 130,
    5 via   5 +   5^3 = 130, and
    2 via   2 +   2^7 = 130.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A253913, A307074, A307092, A309997.

a(n) = {if (n==1, return (0)); my(d = divisors(n)); 1 + sumdiv(n, d, if ((d>1) && (dMichel Marcus, Oct 16 2019

a(2n+1) = 1 for all n >= 1.

a(2n) >= 2 for all n >= 2.

A328422 Number of paths from 2 to n via maps of the form x -> x + x^j, where j is a nonnegative integer.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 6, 9, 9, 14, 14, 18, 18, 24, 24, 31, 31, 42, 42, 51, 51, 65, 65, 79, 79, 97, 97, 118, 118, 142, 142, 167, 167, 198, 198, 229, 229, 271, 271, 317, 317, 368, 368, 419, 419, 484, 484, 549, 549, 628, 628, 707, 707, 808, 808, 905, 905, 1023

Offset: 2

Peter Kagey, Oct 15 2019

nonn

This sequence is essentially the same as the number of paths from 1 to n. However, starting from 2 removes the ambiguity of how many maps there are from 1 to 2.

a(2n+1) = a(2n) for all n because x + x^j is odd if and only if x is even and j = 0.

			For n = 8 the a(8) = 6 paths are:
2 -> 3 -> 4 -> 5 -> 6 -> 7 -> 8 with j = [0,0,0,0,0,0]
2 -> 3 -> 4 -> 8                with j = [0,0,1]
2 -> 3 -> 6 -> 7 -> 8           with j = [0,1,0,0]
2 -> 4 -> 5 -> 6 -> 7 -> 8      with j = [1,0,0,0,0]
2 -> 4 -> 8                     with j = [1,1]
2 -> 6 -> 7 -> 8                with j = [2,0,0]

Peter Kagey, Table of n, a(n) for n = 2..10000

Cf. A307074, A307092, A309997, A309978.

a(2) = 1, a(n) = Sum_{k=1..A309978(n)} a(A328446(n,k)) for n > 2.

A328446 Table read by rows: the n-th row gives the nonnegative integers k such that n - k is a power of k.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 2, 3, 5, 6, 4, 7, 8, 2, 5, 9, 10, 3, 6, 11, 12, 7, 13, 14, 8, 15, 16, 2, 9, 17, 18, 4, 10, 19, 20, 11, 21, 22, 12, 23, 24, 13, 25, 26, 14, 27, 28, 3, 5, 15, 29, 30, 16, 31, 32, 2, 17, 33, 34, 18, 35, 36, 19, 37, 38, 20, 39, 40, 6, 21, 41, 42, 22

Offset: 1

Peter Kagey, Oct 15 2019

The n-th row has length A309978(n) for n > 1.

			Table begins
   n | n-th row
  ---+----------
   1 | 0
   2 | 1
   3 | 2
   4 | 2, 3
   5 | 4
   6 | 2, 3, 5
   7 | 6
   8 | 4, 7
   9 | 8
  10 | 2, 5, 9
  11 | 10
  12 | 3, 6, 11
  13 | 12
  14 | 7, 13
  15 | 14
  16 | 8, 15
  17 | 16
  18 | 2, 9, 17
For n = 10 the 10th row is 2, 5, 9 because
10 - 2 = 2^3,
10 - 5 = 5^1, and
10 - 9 = 9^0.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A307092, A309978.

row(n) = {if (n==1, return ([0])); my(row = vector(0)); fordiv(n, d, if ((d>1) && (dMichel Marcus, Oct 16 2019

cp's OEIS Frontend

A307074 a(n) is the smallest k such that A307092(k) = n.

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A309997 Number of paths from 2 to n of length A307092(n) - 1 via maps of the form x -> x + x^j, where j is a nonnegative integer.

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A309978 a(n) is the number of positive integers k such that there exists a nonnegative integer m with k + k^m = n.

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PARI

Formula

A328422 Number of paths from 2 to n via maps of the form x -> x + x^j, where j is a nonnegative integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A328446 Table read by rows: the n-th row gives the nonnegative integers k such that n - k is a power of k.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI