cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Andrea Raffetti

Andrea Raffetti's wiki page.

Andrea Raffetti has authored 7 sequences.

A193628 Integer part of A192851 / A191626.

Original entry on oeis.org

1, 4, 4, 3, 14, 21, 60, 69, 72, 101, 99, 98, 119, 95, 110, 93, 106, 112, 111, 92, 138, 144, 141, 120, 119, 132, 122, 109, 106, 98, 89, 106, 102, 102, 108, 91, 97, 95, 96, 102, 106, 122, 121, 121, 127, 142, 141, 139, 141, 132, 135, 134, 140, 141, 138, 137
Offset: 1

Views

Author

Andrea Raffetti, Aug 01 2011

Keywords

Comments

Conjecture: the sequence seems to converge asymptotically to 113.

Links

Crossrefs

Cf. A192851, A191626.

A192933 Triangle read by rows: T(n,k) = Sum_{i <= n, j <= k, (i,j) <> (n,k)} T(i,j), starting with T(1,1) = 1, for n >= 1 and 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 2, 6, 12, 4, 16, 44, 88, 8, 40, 136, 360, 720, 16, 96, 384, 1216, 3152, 6304, 32, 224, 1024, 3712, 11296, 28896, 57792, 64, 512, 2624, 10624, 36416, 108032, 273856, 547712, 128, 1152, 6528, 29056, 109696, 362624, 1056896, 2661504, 5323008, 256, 2560, 15872, 76800, 314880, 1135616, 3659776, 10528768, 26380544, 52761088
Offset: 1

Views

Author

Andrea Raffetti, Jul 13 2011

Keywords

Comments

The outer diagonal is A059435.
The second outer diagonal is A090442.
The third outer diagonal is essentially 2*A068766.
The first column is A011782.
The second column is essentially A057711 (not considering its first two terms).
The second column is essentially A129952 (not considering its first two terms).
The second column is essentially 2*A001792.
The differences between the terms of the second column is essentially 2*A045623.
The third column is essentially 4*A084266.
The cumulative sums of the third column are essentially 4*A176027.
T(n,k) = 0 for n < k. If this overriding constraint is not applied, you get A059576. - Franklin T. Adams-Watters, Jul 24 2011
For n >= 2 and 1 <= k <= n, T(n,k) is the number of bimonotone subdivisions of a 2-row grid with n points on the first row and k points on the second row (with the lower left point of the grid being the origin). A bimonotone subdivision of a convex polygon (the convex hull of the grid) is one where the internal dividing lines have nonnegative (including infinite) slopes. See Robeva and Sun (2020). - Petros Hadjicostas and Michel Marcus, Jul 15 2020

Examples

			Triangle (with rows n >= 1 and columns k = 1..n) begins:
   1;
   1,   2;
   2,   6,   12;
   4,  16,   44,    88;
   8,  40,  136,   360,   720;
  16,  96,  384,  1216,  3152,   6304;
  32, 224, 1024,  3712, 11296,  28896,  57792;
  64, 512, 2624, 10624, 36416, 108032, 273856, 547712;
  ...
Example: T(4,3) = 44 = 1 + 1 + 2 + 2 + 6 + 12 + 4 + 16.
From _Petros Hadjicostas_, Jul 15 2020: (Start)
Consider the following 2-row grid with n = 3 points at the top and k = 2 points at the bottom:
   A  B  C
   *--*--*
   |    /
   |   /
   *--*
   D  E
The sets of the dividing internal lines of the T(3,2) = 6 bimonotone subdivisions of the above 2-row grid are as follows: { }, {DC}, {DB}, {EB}, {DB, DC}, and {DB, EB}. We exclude subdivisions {EA} and {EA, EB} because they have at least one dividing line with a negative slope. (End)

Links

Crossrefs

Cf. A001003, A001792, A006318, A011782, A045623, A057711, A059435, A059576, A068766, A084266, A090442, A129952, A166444, A176027, A336245.

Programs

  • PARI
    lista(nn) = {my(T=matrix(nn, nn)); T[1,1] = 1; for (n=2, nn, for (k=1, n, T[n,k] = sum(i=1, n, sum(j=1, k, if ((i!=n) || (j!=k), T[i,j]))););); vector(nn, k, vector(k, i, T[k, i]));} \\ Michel Marcus, Mar 18 2020

Formula

T(n,1) = 2^(n-2) for n >= 2.
T(n,2) = n*2^(n-2) for n >= 2.
T(n,3) = 2^(n-2)*((n-k+1)^2 + 7*(n-k+1) + 4)/2 = 2^(n-3)*(n^2 + 3*n - 6) for k = 3 and n >= 3.
In general: For 1 <= k <= n with (n,k) <> 1,
T(n,k) = 2^(n-2)*Sum_{i=0..k-1} c(k,i)*(n-k+1)^(k-1-i)/(k-1)! and
T(n,k) = 2^(n-2)*Sum_{j=0..k-1} c(k,k-1-j)*(n-k+1)^j/(k-1)!
with c(k,i) being specific coefficients. Below are the first values for c(k,i):
1;
1, 1;
1, 7, 4;
1, 18, 77, 36;
1, 34, 359, 1238, 528,
1, 55, 1065, 8705, 26654, 10800;
... [Formula corrected by Petros Hadjicostas, Jul 15 2020]
The diagonal of this triangle for c(k,i) divided by (k-1)! (except for the first term) is equal to the Shroeder number sequence A006318(k-1).
From Petros Hadjicostas and Michel Marcus, Jul 15 2020: (Start)
T(n,1) = 2^(n-2) for n >= 2; T(n,k) = 2*(T(n,k-1) + T(n-1,k) - T(n-1,k-1)) for n > k >= 2; T(n,n) = 2*T(n,n-1) for n = k >= 2; and T(n,k) = 0 for 1 <= n < k. [Robeva and Sun (2020)] (They do not specify T(1,1) explicitly since they do not care about subdivisions of a degenerate polygon with only one side.)
T(n,k) = (2^(n-2)/(k-1)!) * P_k(n) = (2^(n-2)/(k-1)!) * Sum_{j=1..k} A336245(k,j)*n^(k-j) for n >= k >= 1 with (n,k) <> (1,1), where P_k(n) is some polynomial with integer coefficients of degree k-1. [Robeva and Sun (2020)]
A336245(k,j) = Sum_{s=0..j-1} c(k,s) * binomial(k-1-s, k-j) * (1-k)^(j-1-s) for 1 <= j <= k, in terms of the above coefficients c(k,i).
So c(k,s) = Sum_{j=1..s+1} A336245(k,j) * binomial(k-j, k-s-1) * (k-1)^(s+1-j) for k >= 1 and 0 <= s <= k-1, obtained by inverting the binomial transform.
Bivariate o.g.f.: x*y*(1 - x)*(1 - 2*y*g(2*x*y))/(1 - 2*x - 2*y + 2*x*y), where g(w) = 2/(1 + w + sqrt(1 - 6*w + w^2)) = g.f. of A001003.
Letting y = 1 in the above joint o.g.f., we get the o.g.f. of the row sums: x*(1-x)*(2*g(2*x) - 1). It can then be easily proved that
Sum_{k=1..n} T(n,k) = 2^n*A001003(n-1) - 2^(n-1)*A001003(n-2) for n >= 3. (End)

Extensions

Offset changed by Andrew Howroyd, Dec 31 2017
Name edited by Petros Hadjicostas, Jul 15 2020

A192851 Integers n such that 6n, 36n, and 216n fall between pairs of twin primes, that is, 6n-1, 6n+1, 36n-1, 36n+1, 216n-1, and 216n+1 are prime.

Original entry on oeis.org

2, 12, 23, 45, 325, 703, 2705, 3598, 4218, 7338, 10698, 13562, 16478, 16665, 20195, 25195, 29678, 32312, 36228, 51882, 79628, 83522, 84513, 84525, 89453, 100028, 106710, 107712, 108868, 114527, 119142, 145590, 147758, 151557, 167155, 173960, 190547, 192588
Offset: 1

Views

Author

Andrea Raffetti, Jul 11 2011

Keywords

Comments

Infinite under Dickson's conjecture. - Charles R Greathouse IV, Jul 24 2011

Examples

			12 is in the list because 12*6=72, 12*36=432, 12*216=2592 are all between a pair of twin primes (71,73 and 431,433 and 2591,2593).

Links

Crossrefs

Subsequence of A191626 and hence A002822.
Cf. A014574.

Programs

  • Mathematica
    Select[Range[1000000], PrimeQ[6 # - 1] && PrimeQ[6 # + 1] && PrimeQ[36 # - 1] && PrimeQ[36 # + 1] && PrimeQ[216 # - 1] && PrimeQ[216 # + 1] &] (* T. D. Noe, Jul 26 2011 *)
Select[Range[193000],AllTrue[{6#-1,6#+1,36#-1,36#+1,216#-1,216#+1}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 21 2020 *)
  • PARI
    is(n)=isprime(6*n-1) && isprime(6*n+1) && isprime(36*n-1) && isprime(36*n+1) && isprime(216*n-1) && isprime(216*n+1) \\ Charles R Greathouse IV, Sep 15 2015

A191626 Integers n such that both 6n and 36n fall between pairs of twin primes, that is, 6n-1, 6n+1, 36n-1, and 36n+1 are prime.

Original entry on oeis.org

2, 3, 5, 12, 23, 32, 45, 52, 58, 72, 107, 137, 138, 175, 182, 270, 278, 287, 325, 562, 577, 578, 597, 703, 747, 753, 872, 980, 1022, 1160, 1325, 1372, 1438, 1477, 1540, 1892, 1950, 2007, 2018, 2313, 2335, 2387, 2597, 2608, 2705, 2742, 2782, 3008
Offset: 1

Views

Author

Andrea Raffetti, Jul 11 2011

Keywords

Comments

Infinite under Dickson's conjecture. [Charles R Greathouse IV, Jul 24 2011]

Examples

			5 is in the list because 5*6=30, 5*36=180 and both fall between a pair of twin primes (29,31 and 179,181).

Links

Crossrefs

Subsequence of A002822.
Cf. A014574.

Programs

  • Mathematica
    Select[Range[3100],And@@PrimeQ[{6#+1,6#-1,36#+1,36#-1}]&] (* Harvey P. Dale, Jul 27 2011 *)

A192720 High-water marks of A062357: record values of prime(n)-n*(prime(n+1)-prime(n)).

Original entry on oeis.org

1, 3, 9, 15, 25, 31, 49, 51, 71, 79, 97, 105, 107, 129, 135, 155, 161, 183, 209, 257, 265, 283, 325, 361, 381, 391, 409, 419, 529, 537, 539, 561, 577, 677, 685, 697, 705, 727, 771, 827, 865, 871, 877, 889, 977, 991
Offset: 1

Views

Author

Andrea Raffetti, Jul 08 2011

Keywords

Comments

The numbers in the sequence are generated in correspondence of twin primes. I.e. a(n)=-A(062357(k)) where k is such as p(k) is the lowest of a pair of twin primes. The sequence seems to have a good linear approximation with something like 11^2*n^2-10*11^4.

Crossrefs

Cf. A000040, A001223, A062357.

A192611 Primes prime(k) such that prime(k)*k+1 is also prime.

Original entry on oeis.org

2, 3, 7, 13, 89, 113, 151, 181, 359, 433, 521, 541, 641, 701, 719, 827, 953, 1033, 1277, 1301, 1439, 1877, 2069, 2111, 2143, 2267, 2357, 2423, 2791, 2801, 2861, 2887, 3191, 3251, 3313, 3557, 3643, 3739, 3797, 3821, 3863, 3931, 3947, 4021, 4447
Offset: 1

Views

Author

Andrea Raffetti, Jul 05 2011

Keywords

Comments

Primes p such that p*pi(p)+1 is prime (see Crossrefs).

Examples

			13 is in the list because, being the 6th prime, 13*6+1=79 is prime.

Links

Crossrefs

Cf. A096064, A086172.
Cf. A000040 The prime numbers.
Cf. A000720 pi(n), the number of primes <= n.

Programs

  • Magma
    [ NthPrime(n): n in [1..650] | IsPrime(NthPrime(n)*n+1) ]; // Bruno Berselli, Jul 05 2011
  • Mathematica
    Select[Prime[Range[700]],PrimeQ[# PrimePi[#]+1]&] (* Harvey P. Dale, May 28 2012 *)
  • PARI
    k=0;forprime(p=2,1e4,if(isprime(k++*p+1),print1(p", "))) \\ Charles R Greathouse IV, Jul 05 2011

A080174 Primes prime(k) such that prime(k)*k falls between twin primes.

Original entry on oeis.org

3, 113, 359, 827, 2069, 2267, 3643, 5179, 7829, 9029, 10223, 10369, 11777, 18169, 21143, 22409, 23173, 23957, 25411, 28051, 28447, 29251, 31991, 32717, 34487, 38561, 43133, 54323, 57097, 61363, 73043, 82493, 86269, 94099
Offset: 1

Views

Author

Andrea Raffetti, Jul 05 2011

Keywords

Comments

Also primes p such that pi(p)*p falls between twin primes (see Crossrefs).

Examples

			113 is in the list because 113 is the 30th prime and 113*30=3390 falls between the twin primes 3389 and 3391.

Links

Crossrefs

The sequence is the intersection of A062291 and A192611.
Cf. A000720 pi(n), the number of primes <= n.
Cf. A001359 Lesser of twin primes; A006512 Greater of twin primes.
Cf. A014574 Average of twin prime pairs.

Programs

  • Magma
    [NthPrime(n): n in [1..10^4] | IsPrime(NthPrime(n)*n-1) and IsPrime(NthPrime(n)*n+1)];  // Bruno Berselli, Jul 06 2011
  • Mathematica
    Prime[Select[Range[10000], PrimeQ[Prime[#]# - 1] && PrimeQ[Prime[#]# + 1] &]] (* Alonso del Arte, Jul 05 2011 *)
  • PARI
    v=List();k=0; forprime(p=2, 1e5, if(isprime(k++*p+1)&&isprime(k*p-1),listput(v,p))); Vec(v)

Extensions

Additional terms from Bruno Berselli, Jul 05 2011

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