A193773 -id:A193773 - cp's OEIS frontend

A005097 (Odd primes - 1)/2.

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

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

Original entry on oeis.org

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

A278223 Least number with the same prime signature as the n-th odd number: a(n) = A046523(2n-1).

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 2, 6, 2, 2, 6, 2, 4, 8, 2, 2, 6, 6, 2, 6, 2, 2, 12, 2, 4, 6, 2, 6, 6, 2, 2, 12, 6, 2, 6, 2, 2, 12, 6, 2, 16, 2, 6, 6, 2, 6, 6, 6, 2, 12, 2, 2, 30, 2, 2, 6, 2, 6, 12, 6, 4, 6, 8, 2, 6, 2, 6, 24, 2, 2, 6, 6, 6, 12, 2, 2, 12, 6, 2, 6, 6, 2, 30, 2, 4, 12, 2, 12, 6, 2, 2, 6, 6, 6, 24, 2, 2, 30, 2, 2, 6, 6, 6, 12, 6, 2, 6, 6, 6, 6, 6, 2, 36, 2, 2

Offset: 1

Antti Karttunen, Nov 16 2016

nonn

This sequence works as a filter for sequences related to the prime factorization of odd numbers by matching to any sequence that is obtained as f(2*n - 1), where f(n) is any function that depends only on the prime signature of n (see the index entry for "sequences computed from exponents in ..."). The last line in Crossrefs section lists such sequences that were present in the database as of Nov 11 2016, although some of the matches might be spurious.

Antti Karttunen, Table of n, a(n) for n = 1..32769
Index entries for sequences computed from exponents in factorization of n.

Odd bisection of A046523.
Cf. A064216, A244153, A245611, A278222, A278224, A278531.
Sequences that partition or seem to partition N into same or coarser equivalence classes: A099774, A100007, A193773, A101871, A158280, A158315, A158647, A285716.

a[n_] := Times @@ (Prime[Range[Length[f = FactorInteger[2*n - 1]]]]^Sort[f[[;; , 2]], Greater]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Jul 23 2023 *)

from sympy import factorint
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a(n): return a046523(2*n - 1) # Indranil Ghosh, May 11 2017

from math import prod
from sympy import prime, factorint
def A278223(n): return prod(prime(i+1)**e for i,e in enumerate(sorted(factorint((n<<1)-1).values(),reverse=True))) # Chai Wah Wu, Sep 16 2022

(define (A278223 n) (A046523 (+ n n -1)))
(define (A278223 n) (A046523 (A064216 n)))

a(n) = A046523(2n - 1).
a(n) = A046523(A064216(n)).
From Antti Karttunen, May 31 2017: (Start)
a(n) = A278222(A244153(n)).
a(n) = A278531(A245611(n)).
(End)

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

A125203 Number of ways to write n as 4xy - x - y with 1<=x<=y.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 1, 1, 0, 2, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 2, 0, 0, 2, 1, 0, 1, 0, 0, 1, 2, 0, 1, 0, 0, 2, 0, 1, 1, 0

Offset: 1

Reinhard Zumkeller, Nov 24 2006

nonn

a(A094178(n))=0; a(A124934(n))>0; a(A125217(n))=1; a(A125218(n))>1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A094178, A124934, A125217, A125218.

Cf. A193773.

a125203 n = length [() | x <- [1 .. (n + 1) `div` 3],
                         let (y,m) = divMod (x + n) (4 * x - 1),
                         x <= y, m == 0]
-- Reinhard Zumkeller, Jan 02 2013

a[n_] := Solve[1<=x<=y && n == 4 x y - x - y, {x, y}, Integers] // Length;
Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Oct 12 2021 *)

A305152 Expansion of Sum_{k>0} x^(k^2) / (1 + x^k).

Original entry on oeis.org

1, -1, 1, 0, 1, -2, 1, 0, 2, -2, 1, -1, 1, -2, 2, 1, 1, -3, 1, -1, 2, -2, 1, 0, 2, -2, 2, -1, 1, -4, 1, 1, 2, -2, 2, -1, 1, -2, 2, 0, 1, -4, 1, -1, 3, -2, 1, 1, 2, -3, 2, -1, 1, -4, 2, 0, 2, -2, 1, -2, 1, -2, 3, 2, 2, -4, 1, -1, 2, -4, 1, 0, 1, -2, 3, -1, 2, -4, 1

Offset: 1

Seiichi Manyama, May 26 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A038548, A048272, A193773 (odd bisection), A348608, A228441, A010052, A348952.

{a(n) = polcoeff(sum(k=1, sqrtint(n), x^(k^2)/(1+x^k))+x*O(x^n), n)}

a(n) = sumdiv(n, d, if (d <= sqrtint(n), (-1)^(d + n/d))); \\ Michel Marcus, Nov 03 2021

a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d + n/d). - Ilya Gutkovskiy, Nov 02 2021
a(n) = (A228441(n) + A010052(n))/2. - Ridouane Oudra, Aug 14 2025
a(n) = A010052(n) - A348952(n). - Ridouane Oudra, Aug 20 2025

cp's OEIS Frontend

A005097 (Odd primes - 1)/2.

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

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A278223 Least number with the same prime signature as the n-th odd number: a(n) = A046523(2n-1).

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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).

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A125203 Number of ways to write n as 4*x*y - x - y with 1<=x<=y.

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Comments

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Programs

Haskell

Mathematica

A305152 Expansion of Sum_{k>0} x^(k^2) / (1 + x^k).

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Keywords

Links

Crossrefs

Programs

PARI

PARI

A125203 Number of ways to write n as 4xy - x - y with 1<=x<=y.