A090767 -id:A090767 - cp's OEIS frontend

A047845 a(n) = (m-1)/2, where m is the n-th odd nonprime (A014076(n)).

0, 4, 7, 10, 12, 13, 16, 17, 19, 22, 24, 25, 27, 28, 31, 32, 34, 37, 38, 40, 42, 43, 45, 46, 47, 49, 52, 55, 57, 58, 59, 60, 61, 62, 64, 66, 67, 70, 71, 72, 73, 76, 77, 79, 80, 82, 84, 85, 87, 88, 91, 92, 93, 94, 97, 100, 101, 102, 103, 104, 106, 107, 108, 109, 110, 112, 115

Offset: 1

Enoch Haga

Also (starting with 2nd term) numbers of the form 2xy+x+y for x and y positive integers. This is also the numbers of sticks needed to construct a two-dimensional rectangular lattice of unit squares. See A090767 for the three-dimensional generalization. - John H. Mason, Feb 02 2004
Note that if k is not in this sequence, then 2*k+1 is prime. - Jose Brox (tautocrona(AT)terra.es), Dec 29 2005
Values of k for which A073610(2k+3)=0; values of k for which A061358(2k+3)=0. - Graeme McRae, Jul 18 2006
This sequence also arises in the following way: take the product of initial odd numbers, i.e., the product (2n+1)!/(n!*2^n) and factor it into prime numbers. The result will be of the form 3^f(3)*5^f(5)*7^f(7)*11^f(11)... . Then f(3)/f(5) = 2, f(3)/f(7) = 3, f(3)/f(11) = 5, ... and this sequence forms (for sufficiently large n, of course) the sequence of natural numbers without 4,7,10,12,..., i.e., these numbers are what is lacking in the present sequence. - Andrzej Staruszkiewicz (uszkiewicz(AT)poczta.onet.pl), Nov 10 2007
Also "flag short numbers", i.e., number of dots that can be arranged in successive rows of K, K+1, K, K+1, K, ..., K+1, K (assuming there is a total of L > 1 rows of size K > 0). Adapting Skip Garibaldi's terms, sequence A053726 would be "flag long numbers" because those patterns begin and end with the long lines. If you convert dots to sticks, you get the lattice that John H. Mason mentioned. - Juhani Heino, Oct 11 2014
Numbers k such that (2*k)!/(2*k + 1) is an integer. - Peter Bala, Jan 24 2017
Except for a(1)=0: numbers of the form k == j (mod 2j+1), j >= 1, k > 2j+1. - Bob Selcoe, Nov 07 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Slate, Article about USA flag patterns -- this is where Skip Garibaldi gave definitions.

Complement of A005097.

a047845 = (`div` 2) . a014076  -- Reinhard Zumkeller, Jan 02 2013

[(n-1)/2 : n in [1..350] | (n mod 2) eq 1 and not IsPrime(n)]; // G. C. Greubel, Oct 16 2023

for n from 0 to 120 do
    if irem(factorial(2*n), 2*n+1) = 0 then print(n); end if;
end do:
# Peter Bala, Jan 24 2017

(Select[Range[1, 231, 2], PrimeOmega[#] != 1 &] - 1)/2 (* Jayanta Basu, Aug 11 2013 *)

print1(0,", ");
forcomposite(n=1,250,if(1==n%2,print1((n-1)/2,", "))); \\ Joerg Arndt, Oct 16 2023

from sympy import primepi
def A047845(n):
    if n == 1: return 0
    m, k = n-1, primepi(n) + n - 1 + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n - 1 + (k>>1)
    return m-1>>1 # Chai Wah Wu, Jul 31 2024

[(n-1)/2 for n in (1..350) if n%2==1 and not is_prime(n)] # G. C. Greubel, Oct 16 2023

A193773(a(n)) > 1 for n > 1. - Reinhard Zumkeller, Jan 02 2013

Name edited by Jon E. Schoenfield, Oct 16 2023

A269044 a(n) = 13*n + 7.

Original entry on oeis.org

7, 20, 33, 46, 59, 72, 85, 98, 111, 124, 137, 150, 163, 176, 189, 202, 215, 228, 241, 254, 267, 280, 293, 306, 319, 332, 345, 358, 371, 384, 397, 410, 423, 436, 449, 462, 475, 488, 501, 514, 527, 540, 553, 566, 579, 592, 605, 618, 631, 644, 657, 670, 683, 696, 709, 722, 735

Offset: 0

Bruno Berselli, Feb 18 2016

After 7 (which corresponds to n=0), all terms belong to A090767 because a(n) = 3*n*2*1 + 2*(n*2+2*1+n*1) + (n+2+1).
This sequence is related to A152741 by the recurrence A152741(n+1) = (n+1)*a(n+1) - Sum_{k = 0..n} a(k).
Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 7, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no a(k) is a cube.
The sum of the squares of any two terms of the sequence is also a term of the sequence, that is: a(h)^2 + a(k)^2 = a(h*(13*h+14) + k*(13*k+14) + 7). Therefore: a(h)^2 + a(k)^2 > a(a( h*(h+1) + k*(k+1) )) for h+k > 0.
The primes of the sequence are listed in A140371.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, A description of the recursive method shown in the third comment: website Matem@ticamente (in Italian), 2008.
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A010376, A022271 (partial sums), A088227, A090767, A140371, A152741.
Similar sequences with closed form (2*k-1)*n+k: A001489 (k=0), A000027 (k=1), A016789 (k=2), A016885 (k=3), A017029 (k=4), A017221 (k=5), A017461 (k=6), this sequence (k=7), A164284 (k=8).
Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2), A127547 (q=4), A154609 (q=5), A186113 (q=6), this sequence (q=7), A269100 (q=11).

Magma
```
[13*n+7: n in [0..60]];
```

13 Range[0, 60] + 7 (* or *) Range[7, 800, 13] (* or *) Table[13 n + 7, {n, 0, 60}]
LinearRecurrence[{2, -1}, {7, 20}, 60] (* Vincenzo Librandi, Feb 19 2016 *)

Maxima
```
makelist(13*n+7, n, 0, 60);
```
PARI
```
vector(60, n, n--; 13*n+7)
```
Sage
```
[13*n+7 for n in (0..60)]
```

G.f.: (7 + 6*x)/(1 - x)^2.
a(n) =  A088227(4*n+3).
a(n) = -A186113(-n-1).
Sum_{i=h..h+13*k} a(i) = a(h*(13*k + 1) + k*(169*k + 27)/2).
Sum_{i>=0} 1/a(i)^2 = 0.0257568950542502716970... = polygamma(1, 7/13)/13^2.
E.g.f.: exp(x)*(7 + 13*x). - Stefano Spezia, Aug 02 2021

A277870 Ordered number of unit edges needed to build every 4-orthotope from hypercubes.

Original entry on oeis.org

32, 52, 72, 84, 92, 112, 116, 132, 135, 148, 152, 160, 172, 180, 186, 192, 204, 212, 216, 232, 237, 244, 248, 252, 256, 260, 272, 276, 288, 292, 297, 308, 312, 316, 326, 332, 336, 339, 340, 352, 372, 378, 380, 384, 390, 392, 396, 404, 408, 412, 415, 424, 428

Offset: 1

Eric R. Carter, Nov 02 2016

nonn

Ordered number of edges required to construct every hyperrectangle as a union of unit hypercubes. The sequence gives the n-th smallest such number, and generalizes the two-dimensional A047845 and the three-dimensional A090767 to four dimensions.

Does a(n) ~ n? - Charles R Greathouse IV, Nov 06 2016

			a(1)=32 as this is the number of edges in the unit hypercube.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Wikipedia, Symmetric polynomial

Cf. A047845, A090767.

Edges[x_,y_,z_,w_]:=(4*x*y*z*w)+3*((w*x*z)+(w*y*z)+(w*x*y)+(x*y*z))+2*((w*x)+(w*y)+(w*z)+(x*y)+(x*z)+(y*z))+x+y+z+w;inputs=Tuples[Range[s],4];Union[Table[Edges[inputs[[k]][[1]],inputs[[k]][[2]],inputs[[k]][[3]],inputs[[k]][[4]]],{k,1,Length[inputs]}]]
Accuracy to 170 terms is achieved for s>=5764801, and for the entire list in the limit as s approaches infinity.

list(lim)=my(v=List()); for(w=1,(lim-12)\20, for(x=1, min((lim-8*w-4)\(12*w+8),w), for(y=1,min((lim-5*w*x-3*x-3*w-1)\(7*w*x+5*x+5*w+3),x), forstep(n=((7*w+5)*y+(5*w+3))*x+(5*w + 3)*y+3*w+1, lim, ((4*w+3)*y+3*w+2)*x+(3*w+2)*y+2*w+1, listput(v,n))))); Set(v) \\ Charles R Greathouse IV, Nov 05 2016

These numbers are of the form: 4wxyz + 3(wxz+wyz+wxy+xyz) + 2(wx+wy+wz+xy+xz+yz) + w+x+y+z for any positive integers w, x, y, z.

cp's OEIS Frontend

A047845 a(n) = (m-1)/2, where m is the n-th odd nonprime (A014076(n)).

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A269044 a(n) = 13*n + 7.

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Author

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Programs

Magma

Mathematica

Maxima

PARI

Sage

Formula

A277870 Ordered number of unit edges needed to build every 4-orthotope from hypercubes.

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Author

Keywords

Comments

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Links

Crossrefs

Programs

Mathematica

PARI

Formula