William Rex Marshall - cp's OEIS frontend

A212308 Numbers with no proper divisor that is not in an arithmetic progression of at least three proper divisors.

1, 6, 12, 15, 18, 24, 30, 36, 45, 48, 54, 60, 66, 72, 75, 84, 90, 91, 96, 108, 120, 132, 135, 144, 150, 162, 168, 180, 192, 198, 216, 225, 240, 252, 264, 270, 276, 288, 300, 306, 312, 324, 330, 336, 360, 375, 384, 396, 405, 420, 432, 435, 450, 480, 486, 504

Offset: 1

William Rex Marshall, Oct 24 2013

nonn

Equivalently, the numbers with exactly one divisor that is not in an arithmetic progression of at least three divisors.

Contains p^j*(2*p-1)^k for j,k>=1 if p and 2*p-1 are primes. - Robert Israel, Apr 13 2020

			36 appears in this sequence because its proper divisors are 1, 2, 3, 4, 6, 9, 12 and 18, each of which appears in at least one of the following arithmetic progressions of at least three proper divisors of 36: {1, 2, 3, 4}, {3, 6, 9, 12}, {6, 12, 18}.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A091010, A091011.

Contains A033845, A129521.

filter:= proc(n) local S,D,tau,a,b;
  S:= numtheory:-divisors(n) minus {n};
  D:= sort(convert(S,list));
  tau:= nops(D);
  for a from 1 to tau-2 do for b from a+1 to tau-1 do
    if member(2*D[b]-D[a],D) then
      S:= S minus {D[a],D[b],2*D[b]-D[a]};
      if S = {} then return true fi;
    fi
  od od;
  false;
end proc:
filter(1):= true:
select(filter, [$1..1000]); # Robert Israel, Apr 13 2020

filterQ[n_] := Module[{S, D, tau, a, b}, S = Most @ Divisors[n]; D = S; tau = Length[D]; For[a = 1, a <= tau - 2, a++, For[b = a + 1, b <= tau - 1, b++, If [MemberQ[D, 2 D[[b]] - D[[a]]], S = S ~Complement~ {D[[a]], D[[b]], 2 D[[b]] - D[[a]]}; If[S == {}, Return[True]]]]]; False];
filterQ[1] = True;
Select[Range[1000], filterQ] (* Jean-François Alcover, Sep 26 2020, after Robert Israel *)

A228653 Numbers m such that if an urn contains m balls, with distinct positive numbers for each of c colors, for some c > 1 there exists a combination of c colors such that it is equally probable for c balls randomly selected from the urn to all be either the same color or distinct colors.

Original entry on oeis.org

4, 8, 9, 11, 12, 16, 25, 29, 36, 46, 48, 49, 55, 57, 64, 66, 81, 88, 89, 97, 99, 100, 120, 121, 144, 145, 147, 156, 162, 164, 165, 166, 169, 176, 196, 217, 225

Offset: 1

William Rex Marshall, Aug 29 2013

William Rex Marshall, Pascal program.

Cf. A228650, A228651, A228652.

Pascal
```
See the link section.
```

A228652 Numbers m such that if an urn contains m balls, with at least one each of c colors, there is no c > 1 for which a combination of c colors exists such that it is equally probable for c balls randomly selected from the urn to all be either the same color or distinct colors.

Original entry on oeis.org

1, 2, 3, 5, 7, 15, 19, 27, 31, 47

Offset: 1

William Rex Marshall, Aug 29 2013

nonn

Is this sequence finite and complete?

William Rex Marshall, Pascal program.

Cf. A228650, A228651, A228653.

Pascal
```
See the link section.
```

A228651 Numbers m such that if an urn contains m balls, with at least one each of c colors, for some c > 1 there exists a combination of c colors such that it is equally probable for c balls randomly selected from the urn to all be either the same color or distinct colors.

Original entry on oeis.org

4, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

Offset: 1

William Rex Marshall, Aug 29 2013

nonn

William Rex Marshall, Pascal program.

Cf. A228650, A228652, A228653.

Pascal
```
See the link section.
```

A228650 Numbers k such that if an urn contains k balls, with at least one each of three colors, there exists a combination of the three colors such that it is equally probable for three balls randomly selected from the urn to all be either the same color or distinct colors.

Original entry on oeis.org

6, 8, 11, 12, 46, 57, 66, 120, 121, 145, 156, 162, 166, 217, 372, 386, 557, 596, 638, 750, 866, 1025, 1038, 1201, 1396, 1857, 2042, 2081, 2146, 2263, 2301, 2452, 2836, 2900, 2926, 2991, 3026, 3053, 3288, 3368, 3963, 3970, 4511, 4656, 5006, 5492, 5890, 5952

Offset: 1

William Rex Marshall, Aug 29 2013

nonn

If the urn contains 596 balls, there exist two inequivalent combinations with the desired property, {86, 246, 264} and {126, 154, 316}.

The analogous sequence for two colors are the square numbers > 1 (A000290 with first two terms truncated).

			46 is a member of the sequence because if the urn contains 6 red, 18 green and 22 blue balls, then there are 6 * 18 * 22 = 2376 selections of three balls with distinct colors, and ((6 * 5 * 4) + (18 * 17 * 16) + (22 * 21 * 20)) / 3! = 2376 selections of three balls all the same color, and 6 + 18 + 22 = 46.

Cf. A228651, A228652, A228653.

program a228650;
var
  p: array[1..6000] of int64;
  b1, b2, b3, k: int64;
  n, s: integer;
begin
  k:=0;
  repeat
    inc(k);
    p[k] := (k * (k - 1) * (k - 2)) div 6;
  until k = 6000;
  n := 0; k := 2;
  repeat
    inc(k); s := 0;
    b1 := 0;
    repeat
      inc(b1);
      b2 := b1 - 1;
        b3 := k - (b1 + b2);
        repeat
        inc(b2); dec(b3);
        if (b3 >= b2) and (b1 * b2 * b3 = p[b1] + p[b2] + p[b3]) then
        begin
          inc(n); inc(s);
          writeln(n,' ',k);
        end;
      until (b3 <= b2) or (s > 0);
    until (3 * b1 >= k) or (s > 0);
  until k = 6000;
end.

A214667 The y-coordinates of prime numbers in an Ulam spiral oriented counterclockwise with first step west.

Original entry on oeis.org

0, -1, -1, 1, 0, -2, -2, 0, 2, -1, -3, -3, 1, 3, 3, 0, -4, -4, -2, 2, 4, 4, 3, -3, -5, -5, -3, 1, 3, 5, 0, -4, -6, -6, -2, 0, 6, 6, 6, 3, -3, -5, -7, -7, -7, -5, 7, 7, 6, 4, 0, -6, -8, -8, -8, -2, 4, 6, 8, 8, 8, 5, -9, -9, -9, -9, -3, 3, 9, 9, 9, 9, 4, -2, -8

Offset: 1

William Rex Marshall, Jul 26 2012

The eight possible orientations of the Ulam spiral can be derived from combining either A214664 or A214666 with either A214665 or A214667 as ordered pairs of coordinates.

This spiral is rotated 180 degrees from the spiral on the March 1964 cover of Scientific American.

William Rex Marshall, Table of n, a(n) for n = 1..10077 (coordinates of primes in a 325 X 325 square)

Cf. A000040, A063826, A214664, A214665, A214666.

A214666 The x-coordinates of prime numbers in an Ulam spiral oriented counterclockwise with first step west.

Original entry on oeis.org

-1, -1, 1, 1, -2, -2, 2, 2, 0, -3, -3, 3, 3, 3, -1, -4, -2, 0, 4, 4, 4, -2, -5, -5, 1, 5, 5, 5, 5, 3, -6, -6, -2, 0, 6, 6, 6, 0, -4, -7, -7, -7, 1, 3, 7, 7, 7, -5, -8, -8, -8, -8, -8, 2, 8, 8, 8, 8, 4, 0, -2, -9, -9, -5, -3, 1, 9, 9, 5, 3, -1, -7, -10, -10, -10

Offset: 1

William Rex Marshall, Jul 26 2012

The eight possible orientations of the Ulam spiral can be derived from combining either A214664 or A214666 with either A214665 or A214667 as ordered pairs of coordinates.

This spiral is rotated 180 degrees from the spiral on the March 1964 cover of Scientific American.

William Rex Marshall, Table of n, a(n) for n = 1..10077 (coordinates of primes in a 325 X 325 square)

Cf. A000040, A063826, A214664, A214665, A214667.

A214665 The y-coordinates of prime numbers in an Ulam spiral oriented counterclockwise with first step east.

Original entry on oeis.org

0, 1, 1, -1, 0, 2, 2, 0, -2, 1, 3, 3, -1, -3, -3, 0, 4, 4, 2, -2, -4, -4, -3, 3, 5, 5, 3, -1, -3, -5, 0, 4, 6, 6, 2, 0, -6, -6, -6, -3, 3, 5, 7, 7, 7, 5, -7, -7, -6, -4, 0, 6, 8, 8, 8, 2, -4, -6, -8, -8, -8, -5, 9, 9, 9, 9, 3, -3, -9, -9, -9, -9, -4, 2, 8, 10

Offset: 1

William Rex Marshall, Jul 26 2012

The eight possible orientations of the Ulam spiral can be derived from combining either A214664 or A214666 with either A214665 or A214667 as ordered pairs of coordinates.

William Rex Marshall, Table of n, a(n) for n = 1..10077 (coordinates of primes in a 325 X 325 square).
Scientific American, Front Matter, Vol. 210, No. 3, March 1964.

Cf. A000040, A063826, A214664, A214666, A214667.

A214664 The x-coordinates of prime numbers in an Ulam spiral oriented counterclockwise with first step east.

Original entry on oeis.org

1, 1, -1, -1, 2, 2, -2, -2, 0, 3, 3, -3, -3, -3, 1, 4, 2, 0, -4, -4, -4, 2, 5, 5, -1, -5, -5, -5, -5, -3, 6, 6, 2, 0, -6, -6, -6, 0, 4, 7, 7, 7, -1, -3, -7, -7, -7, 5, 8, 8, 8, 8, 8, -2, -8, -8, -8, -8, -4, 0, 2, 9, 9, 5, 3, -1, -9, -9, -5, -3, 1, 7, 10, 10

Offset: 1

William Rex Marshall, Jul 26 2012

The eight possible orientations of the Ulam spiral can be derived by combining either A214664 or A214666 with either A214665 or A214667 as ordered pairs of coordinates.

William Rex Marshall, Table of n, a(n) for n = 1..10077 (coordinates of primes in a 325 X 325 square).
Scientific American, Front Matter, Vol. 210, No. 3, March 1964.

Cf. A000040, A063826, A214665, A214666, A214667.

A214261 List of derangements of 1, 2, 3, ..., n for n = 2, 3, 4, ..., in lexicographic order.

Original entry on oeis.org

2, 1, 2, 3, 1, 3, 1, 2, 2, 1, 4, 3, 2, 3, 4, 1, 2, 4, 1, 3, 3, 1, 4, 2, 3, 4, 1, 2, 3, 4, 2, 1, 4, 1, 2, 3, 4, 3, 1, 2, 4, 3, 2, 1, 2, 1, 4, 5, 3, 2, 1, 5, 3, 4, 2, 3, 1, 5, 4, 2, 3, 4, 5, 1, 2, 3, 5, 1, 4, 2, 4, 1, 5, 3, 2, 4, 5, 1, 3, 2, 4, 5, 3, 1, 2, 5, 1

Offset: 2

William Rex Marshall, Jul 08 2012

			The derangements can be written as
21,
231, 312,
2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321, etc.

Cf. A000166, A030298, A065087 (row sums), A086325 (row lengths).

cp's OEIS Frontend

User: William Rex Marshall

A212308 Numbers with no proper divisor that is not in an arithmetic progression of at least three proper divisors.

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Mathematica

A228653 Numbers m such that if an urn contains m balls, with distinct positive numbers for each of c colors, for some c > 1 there exists a combination of c colors such that it is equally probable for c balls randomly selected from the urn to all be either the same color or distinct colors.

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Pascal

A228652 Numbers m such that if an urn contains m balls, with at least one each of c colors, there is no c > 1 for which a combination of c colors exists such that it is equally probable for c balls randomly selected from the urn to all be either the same color or distinct colors.

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Pascal

A228651 Numbers m such that if an urn contains m balls, with at least one each of c colors, for some c > 1 there exists a combination of c colors such that it is equally probable for c balls randomly selected from the urn to all be either the same color or distinct colors.

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Pascal

A228650 Numbers k such that if an urn contains k balls, with at least one each of three colors, there exists a combination of the three colors such that it is equally probable for three balls randomly selected from the urn to all be either the same color or distinct colors.

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Pascal

A214667 The y-coordinates of prime numbers in an Ulam spiral oriented counterclockwise with first step west.

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A214666 The x-coordinates of prime numbers in an Ulam spiral oriented counterclockwise with first step west.

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A214665 The y-coordinates of prime numbers in an Ulam spiral oriented counterclockwise with first step east.

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A214664 The x-coordinates of prime numbers in an Ulam spiral oriented counterclockwise with first step east.

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A214261 List of derangements of 1, 2, 3, ..., n for n = 2, 3, 4, ..., in lexicographic order.

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