Dmitry Kamenetsky - cp's OEIS frontend

A366928 a(n) is the smallest nonnegative k such that A301573(k) = n.

1, 0, 6, 12, 20, 41, 42, 56, 72, 90, 110, 155, 156, 182, 270, 271, 272, 306, 379, 380, 420, 462, 551, 552, 600, 650, 702, 756, 812, 870, 930, 1055, 1056, 1122, 1190, 1260, 1405, 1406, 1482, 1560, 1640, 1805, 1806, 1892, 1980, 2254, 2255, 2256, 2352, 2450, 2550, 2652, 2861, 2862, 2970

Offset: 0

Dmitry Kamenetsky, Oct 28 2023

nonn

If negative values were allowed then one could argue a(n) = 1-n from the name of A301573. - David A. Corneth, Nov 13 2023

			a(3) = 12 as 12 is the smallest positive integer that is 3 away from the closest perfect power (namely 9 = 3^2). - _David A. Corneth_, Nov 12 2023

David A. Corneth, Table of n, a(n) for n = 0..10000
David A. Corneth, PARI program

Cf. A001597, A301573.

ispp(n) = {ispower(n) || n==1}; \\ A001597
f(n) = my(k=0); while(!ispp(n+k) && !ispp(n-k), k++); k; \\ A301573
a(n) = my(k=0); while (f(k) != n, k++); k; \\ Michel Marcus, Oct 29 2023

PARI
```
\\ See PARI link
```

from itertools import count
from sympy import perfect_power
def A366928(n): return next(m for m in count(0) if next(k for k in count(0) if perfect_power(m+k) or perfect_power(m-k) or m-k==1 or m+k==1) == n) # Chai Wah Wu, Nov 12 2023

a(n) > n^2 for n > 1. - David A. Corneth, Nov 13 2023

More terms from Michel Marcus, Oct 29 2023

A364692 Largest number that is the sum of n distinct primes in exactly n ways; 0 if no solution exists.

Original entry on oeis.org

68, 130, 42, 59, 76, 0, 0, 161, 192, 233, 227, 276, 0, 425, 480, 0, 0, 0, 752

Offset: 2

Dmitry Kamenetsky, Aug 03 2023

			68 = 7 + 61 = 31 + 37 and there is no larger number that is the sum of 2 distinct primes in exactly 2 ways, so a(2)=68.
130 = 2 + 19 + 109 = 2 + 31 + 97 = 2 + 61 + 67 and there is no larger number that is the sum of 3 distinct primes in exactly 3 ways, so a(3)=130.

Carlos Rivera, Puzzle 1130. As 68, The Prime Puzzles & Problems Connection.

Cf. A344989 asks for the smallest number with the same properties.

A363274 a(n) is the smallest prime such that a(1)a(2)...*a(n) +/- 1 is prime.

Original entry on oeis.org

2, 2, 2, 2, 2, 3, 2, 2, 2, 11, 3, 3, 2, 2, 5, 2, 2, 3, 3, 5, 2, 2, 5, 2, 2, 7, 3, 3, 2, 2, 23, 47, 2, 2, 2, 29, 2, 53, 13, 5, 17, 29, 5, 3, 3, 3, 5, 17, 2, 3, 3, 79, 31, 11, 7, 23, 2, 7, 3, 7, 2, 13, 5, 7, 5, 7, 23, 17, 23, 13, 31, 29, 7, 67, 47, 7, 2, 23, 13, 23, 23, 31, 73, 13, 23, 3, 17

Offset: 1

Dmitry Kamenetsky, Jul 08 2023

nonn

Cf. A036014, A236465.

a[1] = 2; a[n_] := a[n] = Module[{r = Product[a[k], {k, 1, n - 1}], p = 2}, While[! PrimeQ[r*p - 1] && ! PrimeQ[r*p + 1], p = NextPrime[p]]; p]; Array[a, 100] (* Amiram Eldar, Jul 08 2023 *)

A360803 Numbers whose squares have a digit average of 8 or more.

Original entry on oeis.org

3, 313, 94863, 298327, 987917, 3162083, 9893887, 29983327, 99477133, 99483667, 197483417, 282753937, 314623583, 315432874, 706399164, 773303937, 894303633, 947047833, 948675387, 989938887, 994927133, 994987437, 998398167, 2428989417, 2754991833, 2983284917, 2999833327

Offset: 1

Dmitry Kamenetsky, Feb 21 2023

This sequence is infinite. For example, numbers floor(30*100^k - (5/3)*10^k) beginning with 2 followed by k 9s, followed by 8 and k 3s, have a square whose digit average converges to (but never equals) 8.25. [Corrected and formula added by M. F. Hasler, Apr 11 2023]
Only a few examples are known whose square has a digit average of 8.25 and above: 3^2 = 9, 707106074079263583^2 = 499998999999788997978888999589997889 (digit average 8.25), 94180040294109027313^2 = 8869879989799999999898984986998979999969 (digit average 8.275).
This is the union of A164772 (digit average = 8) and A164841 (digit average > 8). - M. F. Hasler, Apr 11 2023

			94863 is in the sequence, because 94863^2 = 8998988769, which has a digit average of 8.1 >= 8.

Michael S. Branicky, Python program for OEIS A360822 and searching for squares with < given number of digits < 8
Matthieu Dufour and Silvia Heubach, Squares with Large Digit Average, 2020.
Dmitry Kamenetsky, The first 68 terms and their digit average

Cf. A004159, A185679.

Cf. A164772 (digit average = 8), A164841 (digit average > 8).

isok(k) = my(d=digits(k^2)); vecsum(d)/#d >= 8; \\ Michel Marcus, Feb 22 2023

def ok(n): d = list(map(int, str(n**2))); return sum(d) >= 8*len(d)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Feb 22 2023

A360822 Numbers whose squares have at most 2 digits less than 8.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 17, 22, 23, 27, 28, 29, 30, 31, 33, 43, 53, 63, 67, 77, 83, 91, 93, 94, 97, 99, 141, 167, 173, 197, 283, 293, 297, 298, 303, 313, 314, 316, 447, 583, 707, 767, 833, 836, 917, 943, 947, 1378, 2917, 2983, 3033, 5467, 9417, 9433, 29983, 31367, 94863

Offset: 1

Dmitry Kamenetsky, Feb 22 2023

From Michael S. Branicky, Feb 22 2023: (Start)
Conjecture: Sequence has 72 terms, with largest term 940206833.
No terms > 940206833 with less than 17 digits. (End)

			314641 is in the sequence, because 314641^2 = 98998958881 has only two digits that are less than 8.

Michael S. Branicky, Python program

Cf. A360803.

Select[Range[10^5], Count[IntegerDigits[#^2], ?(#1 < 8 &)] < 3 &] (* _Amiram Eldar, Feb 22 2023 *)

isok(k) = #select(x->(x<8), digits(k^2)) <= 2; \\ Michel Marcus, Feb 22 2023

def ok(n): return sum(1 for d in str(n**2) if d < "8") < 3
print([k for k in range(1, 10**5) if ok(k)]) # Michael S. Branicky, Feb 22 2023

# see link for a faster version to find all terms

from itertools import count, islice
def A360822_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:len(s:=str(n**2))<=s.count('8')+s.count('9')+2,count(max(startvalue,1)))
A360822_list = list(islice(A360822_gen(),62)) # Chai Wah Wu, Mar 11 2023

A350155 a(n) is the greatest number of times that a laser can hit a reflector on an n X n grid (see Comments for precise definition).

Original entry on oeis.org

1, 6, 18, 49, 120, 233

Offset: 1

Dmitry Kamenetsky, Dec 17 2021

The problem is the following.
You are given an empty n X n grid. You can place some reflectors into the cells of the grid.
This arrangement consists of thin flat plates, each of which is rotatably mounted about a vertical axis so that two angular positions relative to the grid axes, namely plus and minus 45 degrees, are possible. A particle is shot into this arrangement and when it hits a reflector, it is deflected by 90 degrees without any loss of momentum according to the reflector's current orientation. After a reflector has been hit by the particle, it rotates by 90 degrees and maintains this position until the next hit. Not every grid position has to be assigned a reflector -- grid positions without reflectors are permitted.

			See links for examples.

Benjamin Butin, a(3) = 18
Benjamin Butin, Exponential lower bound for A350155
Dmitry Kamenetsky, Laser and mirrors on a 4x4 grid, Puzzling StackExchange, November 2021.
Dmitry Kamenetsky and Daniel Mathias, Optimal solutions for n <= 6
Dmitry Kamenetsky and Benjamin Butin, Improved patterns.
Topcoder, Marathon Match 132: BouncingBalls.

a(n) >= 3*2^n-6, this value can be obtained with a simple pattern (see links above). - Benjamin Butin, Jan 20 2022

a(3) corrected and a(4)-a(6) confirmed by Benjamin Butin, Jan 20 2022

A346775 Starting from n!+1, the length of the longest sequence of consecutive numbers which all take the same number of steps to reach 1 in the Collatz (or '3x+1') problem.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 5, 1, 7, 1, 1, 3, 1, 3, 1, 1, 1, 30, 1, 30, 3, 7, 1, 3, 3, 7, 1, 1, 7, 15, 3, 1, 1, 3, 15, 26, 15, 1, 1, 1, 1, 7, 7, 26, 7, 1, 7, 3, 1, 1, 3, 1, 7, 3, 7, 1, 1, 26, 15, 7, 30, 1, 1, 1, 1, 3, 15, 3, 1, 1, 31, 648, 26, 26, 30, 90, 1, 1, 3, 15

Offset: 0

Dmitry Kamenetsky, Aug 03 2021

nonn

The largest value known in this sequence is a(219)=78553595.
2^32 < a(238) < 11442739136455298475. - Martin Ehrenstein, Aug 21 2021
Jeremy Sawicki found that a(238) = 107150589645. - Dmitry Kamenetsky, Aug 25 2024

			a(6) = 3, because 6!+1, 6!+2 and 6!+3 all take 46 steps to reach 1, while 6!+4 requires 20 steps to reach 1.

Dmitry Kamenetsky, Table of n, a(n) for n = 0..237
Dmitry Kamenetsky, Smallest m>1 such that the number of Collatz steps needed for 238!+m to reach 1 differs from that of 238!+1., Mathematics StackExchange, Aug 2021.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006577, A070974, A277109.

f[n_] := Length[NestWhileList[If[EvenQ@#, #/2, 3 # + 1] &, n, # != 1 &]] - 1; Table[k = 1; While[f[n! + k] == f[n! + k + 1], k++]; k, {n, 0, 100}] (* Bence Bernáth, Aug 14 2021 *)

a6577(n0)={my(n=n0,k=0);while(n>1,k++;n=if(n%2,3*n+1,n/2));k};
for(n=0,80,my(n0=n!+1,nc=a6577(n0),k=1);while(a6577(n0++)==nc,k++);print1(k,", ")) \\ Hugo Pfoertner, Aug 04 2021

A344719 Minimum number of cells in an n X n grid that need to be painted so that each unpainted cell is orthogonally adjacent to exactly two painted cells.

Original entry on oeis.org

1, 2, 5, 8, 11, 17, 21, 28, 35, 42, 51, 60, 69, 80, 91

Offset: 1

Dmitry Kamenetsky, May 28 2021

Dmitry Kamenetsky, Example solutions for n <= 11.
Puzzling StackExchange, Paint Eleven Squares, 2021.

Cf. A344301.

a(12)-a(15) from Bert Dobbelaere, Jun 13 2021

A344301 Minimum number of cells in an n X n grid that need to be painted so that each unpainted cell is orthogonally adjacent to exactly one painted cell.

Original entry on oeis.org

1, 2, 3, 4, 8, 12, 15, 16, 24, 28, 35, 36, 48, 54, 63, 64, 77, 88, 96, 100, 117, 130, 140, 144

Offset: 1

Dmitry Kamenetsky, May 14 2021

Bert Dobbelaere, Illustration for N=20
Dmitry Kamenetsky, Example solutions for n <= 10
Puzzling StackExchange, Paint Eight Squares, 2021.

a(11)-a(20) from Bert Dobbelaere, Jun 13 2021

a(21)-a(24) from Bert Dobbelaere, Jun 19 2021

A341100 Minimum number of base-2 rectangles needed to tile an n X n square.

Original entry on oeis.org

1, 1, 4, 1, 4, 4, 9, 1, 4, 4, 9, 4, 9, 9, 13, 1, 4, 4, 9, 4, 9, 9, 15, 4, 9, 9, 16, 9, 16, 13, 17, 1, 4, 4, 9, 4, 9, 9, 16, 4, 9, 9, 16, 9, 16, 15, 19, 4, 9, 9, 16, 9, 16, 16, 20, 9, 16, 16, 20, 13, 20, 17, 21

Offset: 1

Dmitry Kamenetsky, Feb 05 2021

nonn

A base-2 rectangle is a rectangle whose dimensions are a power of 2.

			A 5 X 5 square can be covered with 4 such rectangles and this is the minimum, so a(5) = 4. Here is a possible covering:
  1 1 1 1 2
  1 1 1 1 2
  1 1 1 1 2
  1 1 1 1 2
  3 3 3 3 4
n=15 is the smallest n where a(n) < f(n)^2, since a(15) = 13. Here is a possible covering found by Bubbler on Puzzling StackExchange:
  A A A A B B C 1 1 1 1 1 1 1 1
  A A A A B B C 1 1 1 1 1 1 1 1
  A A A A B B C 1 1 1 1 1 1 1 1
  A A A A B B C 1 1 1 1 1 1 1 1
  A A A A B B C 2 2 2 2 2 2 2 2
  A A A A B B C 2 2 2 2 2 2 2 2
  A A A A B B C 3 3 3 3 3 3 3 3
  A A A A B B C 0 X Y Y Z Z Z Z
  7 7 7 7 7 7 7 7 X Y Y Z Z Z Z
  8 8 8 8 8 8 8 8 X Y Y Z Z Z Z
  8 8 8 8 8 8 8 8 X Y Y Z Z Z Z
  9 9 9 9 9 9 9 9 X Y Y Z Z Z Z
  9 9 9 9 9 9 9 9 X Y Y Z Z Z Z
  9 9 9 9 9 9 9 9 X Y Y Z Z Z Z
  9 9 9 9 9 9 9 9 X Y Y Z Z Z Z

ali, Tiling a square with rectangles, Mathematics StackExchange, 2019.
Dmitry Kamenetsky, Covering a 15x15 grid with rectangles, Puzzling StackExchange, 2021.

Cf. A000120.

a(n) <= f(n)^2, where f(n) is the number of 1's in the binary representation of n (A000120).

a(n * 2^k) = a(n) for k >= 0.

cp's OEIS Frontend

User: Dmitry Kamenetsky

A366928 a(n) is the smallest nonnegative k such that A301573(k) = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Python

Formula

Extensions

A364692 Largest number that is the sum of n distinct primes in exactly n ways; 0 if no solution exists.

Views

Author

Keywords

Examples

Links

Crossrefs

A363274 a(n) is the smallest prime such that a(1)*a(2)*...*a(n) +/- 1 is prime.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A360803 Numbers whose squares have a digit average of 8 or more.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

A360822 Numbers whose squares have at most 2 digits less than 8.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Python

Python

A350155 a(n) is the greatest number of times that a laser can hit a reflector on an n X n grid (see Comments for precise definition).

Views

Author

Keywords

Comments

Examples

Links

Formula

Extensions

A346775 Starting from n!+1, the length of the longest sequence of consecutive numbers which all take the same number of steps to reach 1 in the Collatz (or '3x+1') problem.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A344719 Minimum number of cells in an n X n grid that need to be painted so that each unpainted cell is orthogonally adjacent to exactly two painted cells.

Views

Author

Keywords

Links

A363274 a(n) is the smallest prime such that a(1)a(2)...*a(n) +/- 1 is prime.