A047845 -id:A047845 - cp's OEIS frontend

A065884 a(n) = A065824(A047845(n+1)).

323, 899, 1763, 5249, 3239, 979801, 5459, 10763, 9179, 9701, 10403, 12319, 5646547, 24569, 19109, 19043, 22499, 50819, 41309, 32639, 46979, 34579, 39059, 125969, 49769, 49949, 154559, 48554797, 114953, 52532203, 56624063, 195499, 75077, 79799, 72899

Offset: 1

Labos Elemer, Nov 27 2001

nonn

By definition (m+1)*phi(a(n)) = m*sigma(a(n)) where m=A065824(n+1).

			A065824(4) = 323, so a(1) = A065824[A047845(1+1)] = 323 A065824(16) = 979801 and a(6) = 979801 = A065824[A047845(1+6)]

Cf. A065824, A047845, A000010, A000203.

from math import prod
from itertools import count
from sympy import factorint, primepi
def A065884(n):
    m, k = n, primepi(n+1) + n + (n+1>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    m = m-1>>1
    for k in count(1):
        f = factorint(k)
        if (m+1)*k*prod((p-1)**2 for p in f)==m*prod(p**(e+2)-p for p,e in f.items()):
            return k # Chai Wah Wu, Aug 12 2024

Name corrected and more terms from Sean A. Irvine, Sep 17 2023

A104085 Coefficient list length of Poincaré-like polynomials made from A047845, indices of odd nonprimes (group dimension equivalent plus one).

Original entry on oeis.org

1, 2, 11, 26, 47, 72, 99, 132, 167, 206, 251, 300, 351, 406, 463, 526, 591, 660, 735, 812, 893, 978, 1065, 1156, 1249, 1344, 1443

Offset: 1

Roger L. Bagula, May 16 2007

There is a possibility that these polynomials also represent groups?

Cf. A047845.

(* Make A047845*) a = Flatten[Table[If[PrimeQ[2*n + 1] == False, n, {}], {n, 0, 50}]]; (* find lengths of polynomials*) Table[Length[CoefficientList[Product[1 + t^(2*a[[n]] + 1), {n, 1, m}], t]], { m, 0, Length[a]}]

P(m)=Product[1 + t^(2*A047845(n) + 1), {n, 1, m}] a(n) =Length[CoefficientList[P(n),x]]

A107615 Coefficient list length of Poincaré-like polynomials made from A047845, indices of 4*n+1 nonprimes as the m(i) exponents.

Original entry on oeis.org

1, 2, 7, 18, 31, 48, 71, 96, 125, 158, 193, 232, 273, 316, 363, 416, 475, 536, 599, 664, 731, 802, 875, 952, 1033, 1116, 1201, 1290, 1383, 1478, 1579

Offset: 1

Roger L. Bagula, May 16 2007

Cf. A045751.

a = Flatten[Table[If[PrimeQ[4*n + 1] == False, n, {}], {n, 0, 50}]]; Table[Length[CoefficientList[Product[1 + t^(2*a[[n]] + 1), {n, 1, m}], t]], { m, 0, Length[a]}]

P(m) = Product[1 + t^(2*A045751(n) + 1), {n, 1, m}] a(n) = Length[CoefficientList[P(n),x]].

A005097 (Odd primes - 1)/2.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156

Offset: 1

N. J. A. Sloane

Or, numbers k such that 2k+1 is prime.
Also numbers not of the form 2xy + x + y. - Jose Brox (tautocrona(AT)terra.es), Dec 29 2005
This sequence arises if you factor the product of a large number of the first odd numbers into the form 3^n(3)5^n(5)7^n(7)11^n(11)... Then n(3)/n(5) = 2, n(3)/n(7) = 3, n(3)/n(11) = 5, ... . - Andrzej Staruszkiewicz (astar(AT)th.if.uj.edu.pl), May 31 2007
Kohen shows: A king invites n couples to sit around a round table with 2n+1 seats. For each couple, the king decides a prescribed distance d between 1 and n which the two spouses have to be seated from each other (distance d means that they are separated by exactly d-1 chairs). We will show that there is a solution for every choice of the distances if and only if 2n+1 is a prime number [i.e., iff n is in A005097], using a theorem known as Combinatorial Nullstellensatz. - Jonathan Vos Post, Jun 14 2010
Starting from 6, positions at which new primes are seen for Goldbach partitions. E.g., 31 is first seen at 34 from 31+3, so position = 1 + (34-6)/2 = 15. - Bill McEachen, Jul 05 2010
Perfect error-correcting Lee codes of word length n over Z: it is conjectured that these always exist when 2n+1 is a prime, as mentioned in Horak. - Jonathan Vos Post, Sep 19 2011
Also solutions to:  A000010(2*n+1) = n * A000005(2*n+1). - Enrique Pérez Herrero, Jun 07 2012
A193773(a(n)) = 1. - Reinhard Zumkeller, Jan 02 2013
I conjecture that the set of pairwise sums of terms of this sequence (A005097) is the set of integers greater than 1, i.e.: 1+1=2, 1+2=3, ..., 5+5=10, ... (This is equivalent to Goldbach's conjecture: every even integer greater than or equal to 6 can be expressed as the sum of two odd primes.) - Lear Young, May 20 2014
See conjecture and comments from Richard R. Forberg, in Links section below, on the relationship of this sequence to rules on values of c that allow both p^q+c and p^q-c to be prime, for an infinite number of primes p. - Richard R. Forberg, Jul 13 2016
The sequence represents the minimum number Ng of gears which are needed to draw a complete graph of order p using a Spirograph(R), where p is an odd prime. The resulting graph consists of Ng hypotrochoids whose respective nodes coincide. If the teethed ring has a circumference p then Ng = (p-1)/2. Examples: A complete graph of order three can be drawn with a Spirograph(R) using a ring with 3n teeth and one gear with n teeth. n is an arbitrary number, only related to the geometry of the gears. A complete graph of order 5 can be drawn using a ring with diameter 5 and 2 gears with diameters 1 and 2 respectively. A complete graph of order 7 can be drawn using a ring with diameter 7 and 3 gears with diameters 1, 2 and 3 respectively. - Bob Andriesse, Mar 31 2017

T. D. Noe, Table of n, a(n) for n = 1..1000
Richard R. Forberg, Comments on A005097
Peter Horak and Bader F. AlBdaiwi, Diameter Perfect Lee Codes, arXiv:1109.3475 [cs.IT], 2011-2012.
Daniel Kohen and Ivan Sadofschi, A New Approach on the Seating Couples Problem, arXiv:1006.2571 [math.CO], 2010.
Dhananjay P. Mehendale, On Hamilton Decompositions, arXiv:0806.0251 [math.GM], 2008.
Eric Weisstein's World of Mathematics, Legendre Symbol

Complement of A047845. Cf. A000040, A006005, A006093.
A130290 is an essentially identical sequence.
Cf. A005384 (subsequence of primes), A266400 (their indices in this sequence).
Numbers n such that 2n+k is prime: this seq(k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).
Cf. also A266409, A294507.

a005097 = (`div` 2) . a065091 -- Reinhard Zumkeller, Jan 02 2013

[n: n in [1..160] |IsPrime(2*n+1)]; // Vincenzo Librandi, Feb 16 2015

with(numtheory): p:=n-> ithprime(n):seq((p(n+1)^2-1)/(2*sigma(p(n+1))), n= 1..64) # Gary Detlefs, May 02 2012

Table[p=Prime[n];(p-1)/2, {n, 2, 22}] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
(Prime[Range[2,70]]-1)/2 (* Harvey P. Dale, Jul 11 2020 *)

forprime(p=3,1e4,print1(p>>1", ")) \\ Charles R Greathouse IV, Jun 16 2011

from sympy import prime
def A005097(n): return prime(n+1)//2 # Chai Wah Wu, Jun 04 2022

a(n) = A006093(n)/2 = A000010(A000040(n+1))/2.
a(n) = (prime(n+1)^2-1)/(2*sigma(prime(n+1))) = (A000040(n+1)^2-1)/(2*A000203(A000040(n+1))). - Gary Detlefs, May 02 2012
a(n) = (A065091(n) - 1) / 2. - Reinhard Zumkeller, Jan 02 2013
a(n) ~ n*log(n)/2. - Ilya Gutkovskiy, Jul 11 2016
a(n) = A294507(n) (mod prime(n+1)). - Jonathan Sondow, Nov 04 2017
a(n) = A130290(n+1). - Chai Wah Wu, Jun 04 2022

A006254 Numbers k such that 2k-1 is prime.

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 10, 12, 15, 16, 19, 21, 22, 24, 27, 30, 31, 34, 36, 37, 40, 42, 45, 49, 51, 52, 54, 55, 57, 64, 66, 69, 70, 75, 76, 79, 82, 84, 87, 90, 91, 96, 97, 99, 100, 106, 112, 114, 115, 117, 120, 121, 126, 129, 132, 135, 136, 139, 141, 142, 147, 154, 156, 157

Offset: 1

Marc LeBrun

a(n) is the inverse of 2 modulo prime(n) for n >= 2. - Jean-François Alcover, May 02 2017
The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004
Positions of prime numbers among odd numbers. - Zak Seidov, Mar 26 2013
Also, the integers remaining after removing every third integer following 2, and, recursively, removing every p-th integer following the next remaining entry (where p runs through the primes, beginning with 5). - Pete Klimek, Feb 10 2014
Also, numbers k such that k^2 = m^2 + p, for some integers m and every prime p > 2. Applicable m values are m = k - 1 (giving p = 2k - 1).  Less obvious is: no solution exists if m equals any value in A047845, which is the complement of (A006254 - 1). - Richard R. Forberg, Apr 26 2014
If you define a different type of multiplication (*) where x (*) y = x * y + (x - 1) * (y - 1), (which has the commutative property) then this is the set of primes that follows. - Jason Atwood, Jun 16 2019

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000040, A067076, A086801.
Equals A005097 + 1. A130291 is an essentially identical sequence.
Cf. A065091.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: this seq(k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

[n: n in [0..1000] | IsPrime(2*n-1)]; // Vincenzo Librandi, Nov 18 2010

Rest@Prime@Range@70/2 + 1/2 (* Robert G. Wilson v, Jun 16 2006 *)
Select[Range[200], PrimeQ[2#-1]&] (* Harvey P. Dale, Apr 06 2014 *)

a(n)=prime(n+1)\2+1 \\ Charles R Greathouse IV, Mar 20 2013

from sympy import prime
def A006254(n): return prime(n+1)+1>>1 # Chai Wah Wu, Aug 02 2024

a(n) = (A000040(n+1) + 1)/2 = A067076(n-1) + 2 = A086801(n-1)/2 + 2.
a(n) = (1 + A065091(n))/2. - Omar E. Pol, Nov 10 2007
a(n) = sqrt((A065091^2 + 2*A065091+1)/4). - Eric Desbiaux, Jun 29 2009
a(n) = A111333(n+1). - Jonathan Sondow, Jan 20 2016

More terms from Erich Friedman

More terms from Omar E. Pol, Nov 10 2007

A153238 Numbers k such that 2*k + 3 is composite.

Original entry on oeis.org

3, 6, 9, 11, 12, 15, 16, 18, 21, 23, 24, 26, 27, 30, 31, 33, 36, 37, 39, 41, 42, 44, 45, 46, 48, 51, 54, 56, 57, 58, 59, 60, 61, 63, 65, 66, 69, 70, 71, 72, 75, 76, 78, 79, 81, 83, 84, 86, 87, 90, 91, 92, 93, 96, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 111, 114

Offset: 1

Vincenzo Librandi, Dec 21 2008

One less than the associated value in A047845. - R. J. Mathar, Jan 05 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Marc Wolf, François Wolf, Representation theorem of composite odd numbers indices, SCIREA Journal of Mathematics (2018) 3.3, 106-117.

Cf. A067076, A047845, A144562.

[ n: n in [0..120] | not IsPrime(2*n+3) ]; // Klaus Brockhaus, Dec 19 2010

lst={};Do[If[ !PrimeQ[2*n+3],AppendTo[lst,n]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 10 2009 *)
Select[Range[120], CompositeQ[2 # + 3] &] (* Michael De Vlieger, Oct 17 2018 *)

for(n=1,115,if(!isprime(2*n+3),print1(n,", "))) \\ Joerg Arndt, Dec 19 2010

a(n) ~ n. - Charles R Greathouse IV, Dec 28 2011

Edited by Ray Chandler, Jan 07 2009

A104275 Numbers k such that 2k-1 is not prime.

Original entry on oeis.org

1, 5, 8, 11, 13, 14, 17, 18, 20, 23, 25, 26, 28, 29, 32, 33, 35, 38, 39, 41, 43, 44, 46, 47, 48, 50, 53, 56, 58, 59, 60, 61, 62, 63, 65, 67, 68, 71, 72, 73, 74, 77, 78, 80, 81, 83, 85, 86, 88, 89, 92, 93, 94, 95, 98, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 113

Offset: 1

Alexandre Wajnberg, Apr 17 2005

Same as A053726 except for the first term of this sequence.
Numbers k such that A064216(k) is not prime. - Antti Karttunen, Apr 17 2015
Union of 1 and terms of the form (u+1)*(v+1) + u*v with 1 <= u <= v. - Ralf Steiner, Nov 17 2021

			a(1) = 1 because 2*1-1=1, not prime.
a(2) = 5 because 2*5-1=9, not prime (2, 3 and 4 give 3, 5 and 7 which are primes).
From _Vincenzo Librandi_, Jan 15 2013: (Start)
As a triangular array (apart from term 1):
   5;
   8,  13;
  11,  18,  25;
  14,  23,  32,  41;
  17,  28,  39,  50,  61;
  20,  33,  46,  59,  72,  85;
  23,  38,  53,  68,  83,  98, 113;
  26,  43,  60,  77,  94, 111, 128, 145;
  29,  48,  67,  86, 105, 124, 143, 162, 181;
  32,  53,  74,  95, 116, 137, 158, 179, 200, 221; etc.
which is obtained by (2*h*k + k + h + 1) with h >= k >= 1. (End)
The above array, which contains the same terms as A053726 but in different order and with some duplicates, has its own entry A144650. - _Antti Karttunen_, Apr 17 2015

Vincenzo Librandi (first 1000 terms) & Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A006254 (complement), A246371 (a subsequence).

Cf. A005097, A008508, A014076, A047845, A053726, A064216, A144650.

[n: n in [1..220]| not IsPrime(2*n-1)]; // Vincenzo Librandi, Jan 28 2011

remove(t -> isprime(2*t-1), [$1..1000]); # Robert Israel, Apr 17 2015

Select[Range[115], !PrimeQ[2#-1] &] (* Robert G. Wilson v, Apr 18 2005 *)

select( {is_A104275(n)=!isprime(n*2-1)}, [1..115]) \\ M. F. Hasler, Aug 02 2022

from sympy import isprime
def ok(n): return not isprime(2*n-1)
print(list(filter(ok, range(1, 114)))) # Michael S. Branicky, May 08 2021

from sympy import primepi
def A104275(n):
    if n <= 2: return ((n-1)<<2)+1
    m, k = n-1, (r:=primepi(n-1)) + n - 1 + (n-1>>1)
    while m != k:
        m, k = k, (r:=primepi(k)) + n - 1 + (k>>1)
    return r+n-1 # Chai Wah Wu, Aug 02 2024

[n for n in (1..250) if not is_prime(2*n-1)] # G. C. Greubel, Oct 17 2023

(define (A104275 n) (if (= 1 n) 1 (A053726 (- n 1)))) ;; More code in A053726. - Antti Karttunen, Apr 17 2015

a(n) = A047845(n-1) + 1.
For n > 1, a(n) = A053726(n-1) = n + A008508(n-1). - Antti Karttunen, Apr 17 2015
a(n) = (A014076(n)+1)/2. - Robert Israel, Apr 17 2015

More terms from Robert G. Wilson v, Apr 18 2005

A046954 Numbers k such that 6*k + 1 is nonprime.

Original entry on oeis.org

0, 4, 8, 9, 14, 15, 19, 20, 22, 24, 28, 29, 31, 34, 36, 39, 41, 42, 43, 44, 48, 49, 50, 53, 54, 57, 59, 60, 64, 65, 67, 69, 71, 74, 75, 78, 79, 80, 82, 84, 85, 86, 88, 89, 92, 93, 94, 97, 98, 99, 104, 106, 108, 109, 111, 113, 114, 116, 117, 119, 120, 124, 127, 129, 130, 132, 133, 134, 136, 139, 140

Offset: 1

Felice Russo

nonn

Equals A171696 U A121763; A121765 U A171696 = A046953; A121763 U A121765 = A067611 where A067611 U A002822 U A171696 = A001477. - Juri-Stepan Gerasimov, Feb 13 2010, Feb 15 2010

These numbers (except 0) can be written as 6xy +-(x+y) for x > 0, y > 0. - Ron R Spencer, Aug 01 2016

			a(2)=8 because 6*8 + 1 = 49, which is composite.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A047845 (2n+1), A045751 (4n+1), A127260 (8n+1).

Cf. A046953, A008588, A016921, subsequence of A067611, complement of A024899.

Filtered([0..250], k-> not IsPrime(6*k+1)) # G. C. Greubel, Feb 21 2019

a046954 n = a046954_list !! (n-1)
a046954_list = map (`div` 6) $ filter ((== 0) . a010051' . (+ 1)) [0,6..]
-- Reinhard Zumkeller, Jul 13 2014

[n: n in [0..250] | not IsPrime(6*n+1)]; // G. C. Greubel, Feb 21 2019

remove(k-> isprime(6*k+1), [$0..140])[]; # Muniru A Asiru, Feb 22 2019

a = Flatten[Table[If[PrimeQ[6*n + 1] == False, n, {}], {n, 0, 50}]] (* Roger L. Bagula, May 17 2007 *)
Select[Range[0, 200], !PrimeQ[6 # + 1] &] (* Vincenzo Librandi, Sep 27 2013 *)

is(n)=!isprime(6*n+1) \\ Charles R Greathouse IV, Aug 01 2016

[n for n in (0..250) if not is_prime(6*n+1)] # G. C. Greubel, Feb 21 2019

Edited by N. J. A. Sloane, Aug 08 2008 at the suggestion of R. J. Mathar
Corrected by Juri-Stepan Gerasimov, Feb 13 2010, Feb 15 2010
Corrected by Vincenzo Librandi, Sep 27 2013

A053726 "Flag numbers": 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 > 1).

Original entry on oeis.org

5, 8, 11, 13, 14, 17, 18, 20, 23, 25, 26, 28, 29, 32, 33, 35, 38, 39, 41, 43, 44, 46, 47, 48, 50, 53, 56, 58, 59, 60, 61, 62, 63, 65, 67, 68, 71, 72, 73, 74, 77, 78, 80, 81, 83, 85, 86, 88, 89, 92, 93, 94, 95, 98, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 113, 116

Offset: 1

Dan Asimov, asimovd(AT)aol.com, Apr 09 2003

Numbers of the form F(K, L) = KL+(K-1)(L-1), K, L > 1, i.e. 2KL - (K+L) + 1, sorted and duplicates removed.
If K=1, L=1 were allowed, this would contain all positive integers.
Positive numbers > 1 but not of the form (odd primes plus one)/2. - Douglas Winston (douglas.winston(AT)srupc.com), Sep 11 2003
In other words, numbers n such that 2n-1, or equally, A064216(n) is a composite number. - Antti Karttunen, Apr 17 2015
Note: the following comment was originally applied in error to the numerically similar A246371. - Allan C. Wechsler, Aug 01 2022
From Matthijs Coster, Dec 22 2014: (Start)
Also area of (over 45 degree) rotated rectangles with sides > 1. The area of such rectangles is 2ab - a - b + 1 = 1/2((2a-1)(2b-1)+1).
Example: Here a = 3 and b = 5. The area = 23.
           *
          ***
         *****
          *****
           *****
            ***
             *
(End)
The smallest integer > k/2 and coprime to k, where k is the n-th odd composite number. - Mike Jones, Jul 22 2024
Numbers k such that A193773(k-1) > 1. - Allan C. Wechsler, Oct 22 2024

Antti Karttunen, Table of n, a(n) for n = 1..10000

Essentially same as A104275, but without the initial one.
A144650 sorted into ascending order, with duplicates removes.
Cf. A006254 (complement, apart from 1, which is in neither sequence).
Cf. also A000720, A008508, A064216, A071904, A199593, A250474.
Differs from its subsequence A246371 for the first time at a(8) = 20, which is missing from A246371.

select( {is_A053726(n)=n>4 && !isprime(n*2-1)}, [1..115]) \\ M. F. Hasler, Aug 02 2022

from sympy import isprime
def ok(n): return n > 1 and not isprime(2*n-1)
print(list(filter(ok, range(1, 117)))) # Michael S. Branicky, May 08 2021

from sympy import primepi
def A053726(n):
    if n == 1: return 5
    m, k = n, (r:=primepi(n)) + n + (n>>1)
    while m != k:
        m, k = k, (r:=primepi(k)) + n + (k>>1)
    return r+n # Chai Wah Wu, Aug 02 2024

;; with Antti Karttunen's IntSeq-library.
(define A053726 (MATCHING-POS 1 1 (lambda (n) (and (> n 1) (not (prime? (+ n n -1)))))))
;; Antti Karttunen, Apr 17 2015

;; with Antti Karttunen's IntSeq-library.
(define (A053726 n) (+ n (A000720 (A071904 n))))
;; Antti Karttunen, Apr 17 2015

a(n) = A008508(n) + n + 1.
From Antti Karttunen, Apr 17 2015: (Start)
a(n) = n + A000720(A071904(n)). [The above formula reduces to this. A000720(k) gives number of primes <= k, and A071904 gives the n-th odd composite number.]
a(n) = A104275(n+1). (End)
a(n) = A116922(A071904(n)). - Mike Jones, Jul 22 2024
a(n) = A047845(n+1)+1. - Amiram Eldar, Jul 30 2024

More terms from Douglas Winston (douglas.winston(AT)srupc.com), Sep 11 2003

A083487 Triangle read by rows: T(n,k) = 2nk + n + k (1 <= k <= n).

Original entry on oeis.org

4, 7, 12, 10, 17, 24, 13, 22, 31, 40, 16, 27, 38, 49, 60, 19, 32, 45, 58, 71, 84, 22, 37, 52, 67, 82, 97, 112, 25, 42, 59, 76, 93, 110, 127, 144, 28, 47, 66, 85, 104, 123, 142, 161, 180, 31, 52, 73, 94, 115, 136, 157, 178, 199, 220, 34, 57, 80, 103, 126, 149, 172, 195, 218, 241, 264

Offset: 1

Artemario Tadeu Medeiros da Silva (artemario(AT)uol.com.br), Jun 09 2003

T(n,k) gives number of edges (of unit length) in a k X n grid.

The values 2*T(n,k)+1 = (2*n+1)*(2*k+1) are nonprime and therefore in A047845.

			Triangle begins:
   4;
   7, 12;
  10, 17, 24;
  13, 22, 31, 40;
  16, 27, 38, 49,  60;
  19, 32, 45, 58,  71,  84;
  22, 37, 52, 67,  82,  97, 112;
  25, 42, 59, 76,  93, 110, 127, 144;
  28, 47, 66, 85, 104, 123, 142, 161, 180;

Vincenzo Librandi, Rows n = 1..100, flattened
Alain Kraus, Cours-Arithmétique et algèbre, 2016-2017, Université de Paris VI. See Exercice 6 p. 13.
OEIS Wiki, Odd composites

Cf. A000217, A016777, A016873, A017017, A017209, A017449, A033954.
Cf. A056220, A082652, A083003, A090288, A182868, A186113, A271625.
See A151890 for another version.

[(2*n*k + n + k): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Jun 01 2014

T[n_,k_]:= 2 n k + n + k; Table[T[n, k], {n, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Jun 01 2014 *)

def T(r, c): return 2*r*c + r + c
a = [T(r, c) for r in range(12) for c in range(1, r+1)]
print(a) # Michael S. Branicky, Sep 07 2022

flatten([[2*n*k +n +k for k in range(1,n+1)] for n in range(1,14)]) # G. C. Greubel, Oct 17 2023

From G. C. Greubel, Oct 17 2023: (Start)
T(n, 1) = A016777(n).
T(n, 2) = A016873(n).
T(n, 3) = A017017(n).
T(n, 4) = A017209(n).
T(n, 5) = A017449(n).
T(n, 6) = A186113(n).
T(n, n-1) = A056220(n).
T(n, n-2) = A090288(n-2).
T(n, n-3) = A271625(n-2).
T(n, n) = 4*A000217(n).
T(2*n, n) = A033954(n).
Sum_{k=1..n} T(n, k) = A162254(n).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A182868((n+1)/2) if n is odd otherwise A182868(n/2) + 1. (End)

Edited by N. J. A. Sloane, Jul 23 2009

Name edited by Michael S. Branicky, Sep 07 2022

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