Santi Spadaro - cp's OEIS frontend

A093567 Binomial (Binomial (n,2), 3) - Binomial (Binomial (n,3), 2).

0, 1, 14, 75, 265, 735, 1736, 3654, 7050, 12705, 21670, 35321, 55419, 84175, 124320, 179180, 252756, 349809, 475950, 637735, 842765, 1099791, 1418824, 1811250, 2289950, 2869425, 3565926, 4397589, 5384575, 6549215, 7916160, 9512536

Offset: 2

Robert G. Wilson v and Santi Spadaro, Mar 31 2004

nonn

All terms are positive: A093566 >= A054563 ==> C( C(n,2), 3) >= C( C(n,3), 2) ==> n^2*(n^4 + 3n^3 -35n^2 + 69n -38)/144 >= 0 ==> (n - 2)(n - 1)(n^2 + 6n - 19) ==> 0 which it is for all n >= 2.

Solomon W. Golomb, Iterated binomial coefficients, Amer. Math. Monthly, 87 (1980), 719-727.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A054563, A093566.

A093567:=n->binomial(binomial(n, 2), 3) - binomial(binomial(n, 3), 2); seq(A093567(n), n=2..30); # Wesley Ivan Hurt, Feb 02 2014

Table[ Binomial[ Binomial[n, 2], 3] - Binomial[ Binomial[n, 3], 2], {n, 2, 34}]
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,14,75,265,735,1736},40] (* Harvey P. Dale, Jun 12 2016 *)

a(n) = binomial(binomial(n,2), 3) - binomial(binomial(n,3), 2); \\ Michel Marcus, Oct 01 2017

a(n) = A093566(n) - A054563(n).

G.f.: x^3*(-1-7*x+2*x^2+x^3)/(x-1)^7. - R. J. Mathar, Dec 08 2010

A093566 a(n) = n(n-1)(n-2)(n-3)(n^2-3*n-2)/48.

Original entry on oeis.org

0, 0, 0, 0, 1, 20, 120, 455, 1330, 3276, 7140, 14190, 26235, 45760, 76076, 121485, 187460, 280840, 410040, 585276, 818805, 1125180, 1521520, 2027795, 2667126, 3466100, 4455100, 5668650, 7145775, 8930376, 11071620, 13624345, 16649480, 20214480

Offset: 0

Robert G. Wilson v and Santi Spadaro, Mar 31 2004

a(n+1) is the number of chiral pairs of colorings of the faces of a cube (vertices of a regular octahedron) using n or fewer colors. - Robert A. Russell, Sep 28 2020

			For a(3+1) = 1, each of the three colors is applied to a pair of adjacent faces of the cube (vertices of the octahedron). - _Robert A. Russell_, Sep 28 2020

Solomon W. Golomb, Iterated binomial coefficients, Amer. Math. Monthly, 87 (1980), 719-727.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

From Robert A. Russell, Sep 28 2020: (Start)
Cf. A047780 (oriented), A198833 (unoriented), A337898 (achiral) colorings.
a(n+1) = A325006(3,n) (chiral pairs of colorings of orthotope facets or orthoplex vertices).
a(n+1) = A337889(3,n) (chiral pairs of colorings of orthotope faces or orthoplex peaks).
Other polyhedra: A000332 (tetrahedron), A337896 (cube/octahedron).
(End)

Table[ Binomial[ Binomial[n-1, 2], 3], {n,0,32}]
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,0,0,1,20,120},40] (* Harvey P. Dale, Feb 18 2016 *)

a(n)=n*(n-1)*(n-2)*(n-3)*(n^2-3*n-2)/48 \\ Charles R Greathouse IV, Jun 11 2015

[(binomial(binomial(n,2),3)) for n in range(-1, 33)] # Zerinvary Lajos, Nov 30 2009

a(n) = binomial(binomial(n-1, 2), 3).
G.f.: -x^4*(1+13*x+x^2)/(x-1)^7. - R. J. Mathar, Dec 08 2010
a(n+1) = 1*C(n,3) + 16*C(n,4) + 30*C(n,5) + 15*C(n,6), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors. - Robert A. Russell, Sep 28 2020
a(n) = A000217(n-1)*A239352(n-2)/6. - R. J. Mathar, Mar 25 2022

Edited (with a new definition) by N. J. A. Sloane, Jul 02 2008

A064779 Primes such that the sum of their digits and the sum of the reciprocals of their digits is also prime.

Original entry on oeis.org

11, 2441, 4241, 4421, 12163, 12613, 13313, 13331, 16231, 16363, 16633, 21163, 21613, 26113, 31663, 32233, 32323, 32611, 33113, 33223, 33311, 48281, 48821, 61231, 61363, 62131, 62311, 63211, 63361, 88241

Offset: 1

Santi Spadaro, Oct 19 2001

Zero, five, and seven never appear as a digit of any of the terms of this sequence. - Harvey P. Dale, Jul 17 2013

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A034708.

f[ n_ ] := 1/n a[ n_ ] := Apply[ Plus, Map[ f, IntegerDigits[ n ] ] ] b[ n_ ] := Apply[ Plus, IntegerDigits[ n ] ] Select[ Range[ 100000 ], FreeQ[ IntegerDigits[ # ], 0 ] && PrimeQ[ a[ # ] ] && PrimeQ[ b[ # ] ] && PrimeQ[ # ] & ]
sdpQ[n_]:=Module[{idn=IntegerDigits[n]},Min[idn]>0&&And@@PrimeQ[{Total[ idn], Total[ 1/idn]}]]; Select[Prime[Range[10000]],sdpQ] (* Harvey P. Dale, Jul 17 2013 *)

A065438 Complement of A065039.

Original entry on oeis.org

10, 21, 32, 43, 54, 65, 76, 87, 98, 109, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 220, 221, 232, 243, 254, 265, 276, 287, 298, 309, 320, 331, 332, 343, 354, 365, 376, 387, 398, 409, 420, 431, 442, 443, 454, 465, 476, 487, 498, 509, 520, 531, 542

Offset: 1

Santi Spadaro, Nov 17 2001

nonn

A071063 Determinant of n X n matrix defined by m(i,j) = 0 if i+j is a prime, m(i,j) = 1 otherwise.

Original entry on oeis.org

0, 0, -1, 0, 1, 0, -9, -8, 0, 0, 0, 0, 0, 0, 0, -8, 9, 14, -71, -310, 281, 2000, -8004, -9200, 8836, 720, -409, -2710, 67766, 110501, -1117396, -4130160, 381136, 91920, -111376, -36080, 144420, 555581, -311814, -1831958, 1876689, -1648, -3584425, 4768308, 1971637204, 53664688220

Offset: 1

Santi Spadaro, May 26 2002

sign

Let h(i,j) be the matrix defined in A069191, then a(n)=((-1)^n)*Det(h(i,j)-J), where J is the n X n matrix with only 1's as its elements.

Cf. A069191.

a[n_] := Det[Table[If[PrimeQ[i + j], 0, 1], {i, 1, n}, {j, 1, n}]] Table[a[n], {n, 1, 50}]

A071292 Call f(n) the sum of the first n primes then a(n) is the number of squares between f(n) and f(n+1).

Original entry on oeis.org

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

Offset: 1

Santi Spadaro, Jun 11 2002

nonn

a(n)>=1.

Internet Math. Olympiad, February: Problem 1

squareQ[n_] := IntegerQ[Sqrt[n]] f[n_] := Sum[Prime[i], {i, 1, n}] a[n_] := Length[Select[Table[i, {i, a[n], a[n + 1]}], squareQ]]

A064704 Numbers beginning and ending with their multiplicative digital root.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 88, 111, 232, 535, 575, 646, 686, 818, 838, 1111, 2132, 2312, 2472, 2692, 2742, 2962, 5135, 5175, 5315, 5715, 5795, 5975, 6146, 6186, 6226, 6246, 6416, 6426, 6696, 6776, 6816, 6966, 8118, 8138, 8278, 8318, 8728, 11111

Offset: 1

Santi Spadaro, Oct 13 2001

If NA(1)A(2)...A(j)N is MDR strong then also Na(1)a(2)...a(j)N is MDR strong where a(1)a(2)...a(j) is any permutation of A(1)A(2)...A(j) and if any of A(1),A(2),...,A(j) say A(1) is composite then also NB(1)...B(k)A(2)...A(j) is MDR strong where A(1)=B(1)*B(2)*...*B(k), so MDR strong numbers with greater number of digits are likely to be more frequent.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for 10-automatic sequences.

Cf. A031347.

dr[n_]:=NestWhile[Times@@IntegerDigits[#]&,n,#>9&];bemdrQ[n_]:=Module[ {idn=IntegerDigits[n]},First[idn]==Last[idn]==dr[n]]; Select[Range[ 12000], bemdrQ] (* Harvey P. Dale, Oct 21 2011 *)

A065031 In the decimal expansion of n, replace each odd digit with 1 and each even digit with 2.

Original entry on oeis.org

2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 12, 11, 12, 11, 12, 11, 12, 11, 12, 11, 22, 21, 22, 21, 22, 21, 22, 21, 22, 21, 12, 11, 12, 11, 12, 11, 12, 11, 12, 11, 22, 21, 22, 21, 22, 21, 22, 21, 22, 21, 12, 11, 12, 11, 12, 11, 12, 11, 12, 11, 22, 21, 22, 21, 22, 21, 22, 21, 22, 21

Offset: 0

Santi Spadaro, Nov 03 2001

A196563(a(n)) = A196563(n); A196564(a(n)) = A196564(n).

			a(123)=121 because 1 and 3 are odd and 2 is even.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

a065031 n = f n  where
   f x | x < 10    = 2 - x `mod` 2
       | otherwise = 10 * (f x') + 2 - m `mod` 2
       where (x',m) = divMod x 10
-- Reinhard Zumkeller, Feb 22 2012

Table[FromDigits[If[OddQ[#],1,2]&/@IntegerDigits[n]],{n,0,120}] (* Harvey P. Dale, Jun 08 2014 *)

A071906 Sum of digits of 2^n (mod 2).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1

Offset: 0

Santi Spadaro, Jun 13 2002

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A001370.

f[n_] := Mod[Plus @@ IntegerDigits[2^n], 2]; Table[ f@n, {n, 0, 104}] (* Robert G. Wilson v, May 04 2009 *)

a(n) = sumdigits(2^n) % 2; \\ Michel Marcus, Apr 20 2017

A069879 Number of pairs {i,j} with i different from j; 1<=i<=n; 1<= j <=n such that i+j is a prime number.

Original entry on oeis.org

0, 2, 4, 8, 10, 14, 18, 22, 28, 36, 42, 50, 56, 62, 70, 80, 88, 96, 104, 112, 122, 134, 144, 156, 168, 180, 194, 208, 220, 234, 248, 262, 276, 292, 308, 326, 344, 362, 380, 400, 418, 438, 456, 474, 494, 514, 532, 550, 570, 590, 612, 636, 658, 682, 708, 734

Offset: 1

Santi Spadaro, May 04 2002

nonn

Partial sums of 2*A060715(n).

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 0,
      b(n-1)+pi(2*n-1)-pi(n))
    end:
a:= n-> 2*b(n):
seq(a(n), n=1..70);  # Alois P. Heinz, Sep 29 2017

Table[2*Count[Subsets[Range[n],{2}],?(PrimeQ[Total[#]]&)],{n,50}] (* _Harvey P. Dale, Jan 23 2015 *)

a(n) = 2 * A071917(n). - Alois P. Heinz, Sep 29 2017

cp's OEIS Frontend

User: Santi Spadaro

A093567 Binomial (Binomial (n,2), 3) - Binomial (Binomial (n,3), 2).

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A093566 a(n) = n*(n-1)*(n-2)*(n-3)*(n^2-3*n-2)/48.

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A064779 Primes such that the sum of their digits and the sum of the reciprocals of their digits is also prime.

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A065438 Complement of A065039.

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A071063 Determinant of n X n matrix defined by m(i,j) = 0 if i+j is a prime, m(i,j) = 1 otherwise.

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A071292 Call f(n) the sum of the first n primes then a(n) is the number of squares between f(n) and f(n+1).

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A064704 Numbers beginning and ending with their multiplicative digital root.

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Mathematica

A065031 In the decimal expansion of n, replace each odd digit with 1 and each even digit with 2.

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Haskell

Mathematica

A071906 Sum of digits of 2^n (mod 2).

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A093566 a(n) = n(n-1)(n-2)(n-3)(n^2-3*n-2)/48.