A153671 -id:A153671 - cp's OEIS frontend

A153699 Greatest number m such that the fractional part of (10/9)^A153695(m) >= 1-(1/m).

1, 1, 1, 2, 3, 8, 15, 264, 334, 465, 683, 713, 758, 8741, 15912, 18920, 38560, 409895

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(7)=15, since 1-(1/16)=0.9375>fract((10/9)^A153695(7))=fract((10/9)^13)=0.9341...>=1-(1/15).

Cf. A153663, A153671, A153679, A153687, A153695, A154130, A153707, A153715, A153723.

a(n):=floor(1/(1-fract((10/9)^A153695(n)))), where fract(x) = x-floor(x).

A153683 Greatest number m such that the fractional part of (1024/1000)^A153679(m) >= 1-(1/m).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 6, 9, 17, 93, 123, 1061, 1360, 4137, 66910, 571666, 1192010

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(5)=1, since 1-(1/2)=0.5>fract((1024/1000)^A153679(5))=fract((1024/1000)^5)=0.0510...>=1-(1/1).

Cf. A153663, A153671, A153679, A154130, A153691, A153699, A153707, A153715, A153723.

a(n):=floor(1/(1-fract((1024/1000)^A153679(n)))), where fract(x) = x-floor(x).

A153691 Greatest number m such that the fractional part of (11/10)^A153687(m) >= 1-(1/m).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 19, 21, 28, 151, 200, 709, 767, 5727, 15908, 162819, 302991

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(6)=4, since 1-(1/5)=0.8>fract((11/10)^A153687(6))=fract((11/10)^6)=0.771...>=1-(1/4).

Cf. A153663, A153671, A153679, A153683, A154130, A153699, A153707, A153715, A153723.

a(n):=floor(1/(1-fract((11/10)^A153687(n)))), where fract(x) = x-floor(x).

A154132 Minimal exponents m such that the fractional part of (4/3)^m increases monotonically (when starting with m=1).

Original entry on oeis.org

1, 2, 8, 39, 2495, 3895, 4714, 8592

Offset: 1

Hieronymus Fischer, Jan 11 2009

Recursive definition: a(1)=1, a(n) = least number m>a(n-1) such that the fractional part of (4/3)^m is greater than the fractional part of (4/3)^k for all k, 1<=k

The next such number must be greater than 200000.

			a(4)=39, since fract((4/3)^39)= 0.999186..., but fract((4/3)^k)<0.9887... for 1<=k<=38; thus fract((4/3)^39)>fract((4/3)^k) for 1<=k<39 and 39 is the minimal exponent > 8 with this property.

Cf. A153663, A153671, A091560, A153710, A154140, A154148.

Cf. A002379, A064628.

Recursion: a(1):=1, a(k):=min{ m>1 | fract((4/3)^m) > fract((4/3)^a(k-1))}, where fract(x) = x-floor(x).

A373300 Sum of successive integers in a row of length p(n) where p counts integer partitions.

Original entry on oeis.org

1, 5, 15, 45, 105, 264, 555, 1221, 2445, 4935, 9324, 17941, 32522, 59400, 104808, 184569, 315711, 540540, 902335, 1504800, 2462724, 4014513, 6444425, 10316250, 16283707, 25610886, 39841865, 61720659, 94687230, 144731706, 219282679, 330996105, 495901413, 740046425

Offset: 1

Olivier Gérard, May 31 2024

nonn

The length of each row is given by A000041.
As many sequences start like the positive integers, their row sums when disposed in this shape start with the same values.
Here is a sample list by A-number order of the sequences which are sufficiently close to A000027 to have the same row sums for at least 8 terms.
A069782, A088480, A090107, A090108, A090109, A115510, A115511, A130446, A131717, A132086, A153671, A167904, A296879, A296882, A296885, A296888, A296891, A296894, A296897, A296900, A296903, A296906, A303502, A317945, A335280, A361374.

			Let's put the list of integers in a triangle whose rows have length p(n), number of integer partitions of n.
.
    1 |  1
    5 |  2  3
   15 |  4  5  6
   45 |  7  8  9 10 11
  105 | 12 13 14 15 16 17 18
  264 | 19 20 21 22 23 24 25 26 27 28 29
  555 | 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
.
The sequence gives the row sums of this triangle.

Cf. A000027, seen as a triangle with shape A000041.
Cf. A373301, the same principle, but starting from integer zero instead of 1.
Cf. A006003, row sums of the integers but for the linear triangle.

Module[{s = 0},
 Table[s +=
   PartitionsP[n - 1]; (s + PartitionsP[n])*(s + PartitionsP[n] - 1)/2 -
   s*(s - 1)/2, {n, 1, 30}]]

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cp's OEIS Frontend

A153699 Greatest number m such that the fractional part of (10/9)^A153695(m) >= 1-(1/m).

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A153683 Greatest number m such that the fractional part of (1024/1000)^A153679(m) >= 1-(1/m).

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A153691 Greatest number m such that the fractional part of (11/10)^A153687(m) >= 1-(1/m).

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A154132 Minimal exponents m such that the fractional part of (4/3)^m increases monotonically (when starting with m=1).

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A373300 Sum of successive integers in a row of length p(n) where p counts integer partitions.

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Mathematica