A104275 -id:A104275 - cp's OEIS frontend

A172048 a(n) = A104275(n) + A014076(n).

2, 14, 23, 32, 38, 41, 50, 53, 59, 68, 74, 77, 83, 86, 95, 98, 104, 113, 116, 122, 128, 131, 137, 140, 143, 149, 158, 167, 173, 176, 179, 182, 185, 188, 194, 200, 203, 212, 215, 218, 221, 230, 233, 239, 242, 248, 254, 257, 263, 266, 275, 278, 281, 284, 293

Offset: 1

Roger L. Bagula, Jan 24 2010

Alternatively: the sequence of the numbers 3*k-1 for all nonprime 2*k-1, k >= 1.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A104275, A014076.

Flatten[Table[If[PrimeQ[2*n - 1], {}, 3*n - 1], {n, 1, 100}]]

The two equivalent definitions separated by the Assoc. Editors of the OEIS, Feb 02 2010

A246371 Numbers n such that, if 2n-1 = Product_{k >= 1} (p_k)^(c_k) then n > Product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

5, 8, 11, 13, 14, 17, 18, 23, 28, 32, 38, 39, 41, 43, 50, 53, 58, 59, 61, 63, 68, 73, 74, 77, 83, 86, 88, 94, 95, 98, 104, 113, 116, 122, 123, 128, 131, 137, 138, 140, 143, 149, 158, 163, 167, 172, 173, 176, 179, 182, 185, 188, 193, 194, 200, 203, 212, 213, 215, 218, 221, 228, 230, 233, 238, 239, 242, 248, 254, 257

Offset: 1

Antti Karttunen, Aug 24 2014

nonn

Numbers n such that A064216(n) < n.
Numbers n such that A064989(2n-1) < n.
Note: This sequence has remarkable but possibly merely coincidental overlap with A053726. On Dec 22 2014, Matthijs Coster mistakenly attached a comment intended for that sequence to this one. On Apr 17 2015, Antti Karttunen noted the error. I have moved the comment to the correct sequence, and have removed Karttunen's note. - Allan C. Wechsler, Aug 01 2022

Antti Karttunen, Table of n, a(n) for n = 1..10000
Matthijs Coster, Oplossing Zomerprijsvraag, Pythagoras 54/2 (2014) 4-7.

Complement: A246372.
Setwise difference of A246361 and A048674.
Subsequence of A104275 and A053726 (20 is the first term > 1 which is not in this sequence).
Subsequence: A246374 (the primes present in this sequence).
Cf. A000040, A064216, A064989, A246351.

default(primelimit, 2^30);
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A064216(n) = A064989((2*n)-1);
isA246371(n) = (A064216(n) < n);
n = 0; i = 0; while(i < 10000, n++; if(isA246371(n), i++; write("b246371.txt", i, " ", n)));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A246371 (MATCHING-POS 1 1 (lambda (n) (< (A064216 n) n))))

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

A014090 Numbers that are not the sum of a square and a prime.

Original entry on oeis.org

1, 10, 25, 34, 58, 64, 85, 91, 121, 130, 169, 196, 214, 226, 289, 324, 370, 400, 526, 529, 625, 676, 706, 730, 771, 784, 841, 1024, 1089, 1225, 1255, 1351, 1414, 1444, 1521, 1681, 1849, 1906, 1936, 2116, 2209, 2304, 2500, 2809, 2986, 3136, 3364, 3481, 3600

Offset: 1

N. J. A. Sloane, R. K. Guy

Sequence is infinite: if 2n-1 is composite then n^2 is in the sequence. (Proof: If n^2 = x^2 + p with p prime, then p = (n-x)(n+x), so n-x=1 and n+x=p. Hence 2n-1=p is prime, not composite.) - Dean Hickerson, Nov 27 2002
21679 is the last known nonsquare in this sequence. See A020495. - T. D. Noe, Aug 05 2006
A002471(a(n))=0; complement of A014089. - Reinhard Zumkeller, Sep 07 2008
There are no prime numbers in this sequence because at the very least they can be represented as p + 0^2. - Alonso del Arte, May 26 2012
Number of terms <10^k,k=0..8: 1, 8, 27, 75, 223, 719, 2361, 7759, ..., . - Robert G. Wilson v, May 26 2012
So far there are only 21 terms which are not squares and they are the terms of A020495. Those that are squares, their square roots are members of A104275. - Robert G. Wilson v, May 26 2012

			From _Alonso del Arte_, May 26 2012: (Start)
10 is in the sequence because none of 10 - p_i are square (8, 7, 5, 3) and none of 10 - b^2 are prime (10, 9, 6, 1); i goes from 1 to pi(10) or b goes from 0 to floor(sqrt(10)).
11 is not in the sequence because it can be represented as 3^2 + 2 or 0^2 + 11. (End)

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 115 terms from T. D. Noe)

Cf. A020495, A104275.

Cf. A064233 (does not allow 0^2).

t={}; Do[k=0; While[k^2=n, AppendTo[t,n]], {n,25000}]; t (* T. D. Noe, Aug 05 2006 *)
max = 5000; Complement[Range[max], Flatten[Table[Prime[p] + b^2, {p, PrimePi[max]}, {b, 0, Ceiling[Sqrt[max]]}]]] (* Alonso del Arte, May 26 2012 *)
fQ[n_] := Block[{j = Sqrt[n], k}, If[ IntegerQ[j] && !PrimeQ[2j - 1], True, k = Floor[j]; While[k > -1 && !PrimeQ[n - k^2], k--]; If[k == -1, True, False]]]; Select[ Range[3600], fQ] (* Robert G. Wilson v, May 26 2012 *)

A153043 Numbers n > 1 such that 2*n-3 is not a prime.

Original entry on oeis.org

2, 6, 9, 12, 14, 15, 18, 19, 21, 24, 26, 27, 29, 30, 33, 34, 36, 39, 40, 42, 44, 45, 47, 48, 49, 51, 54, 57, 59, 60, 61, 62, 63, 64, 66, 68, 69, 72, 73, 74, 75, 78, 79, 81, 82, 84, 86, 87, 89, 90, 93, 94, 95, 96, 99, 102, 103, 104, 105, 106, 108, 109, 110

Offset: 1

Vincenzo Librandi, Dec 17 2008

One more than the associated value in A104275. - R. J. Mathar, Jan 05 2011
2*A155705(m,n)-3 = (2n+1)*(2m+1) are nonprime: all A155705(.,.) are in this sequence.
The terms after a(1) are the values of 2*h*k + k + h + 2, where h and k are positive integers. - Vincenzo Librandi, Jan 19 2013

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

Cf. A098090, A104275, A153040, A155705.

[n: n in [2..102] | not IsPrime(2*n-3)];  // Bruno Berselli, Mar 05 2011

Select[Range[2,200], !PrimeQ[2*#-3]&] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2012 *)

A104278 Numbers m such that 2m+1 and 2m-1 are not primes.

Original entry on oeis.org

13, 17, 25, 28, 32, 38, 43, 46, 47, 58, 59, 60, 61, 62, 67, 71, 72, 73, 77, 80, 85, 88, 92, 93, 94, 101, 102, 103, 104, 107, 108, 109, 110, 118, 122, 123, 124, 127, 130, 133, 137, 143, 144, 145, 148, 149, 150, 151, 152, 160, 161, 162, 163, 164, 167, 170, 171, 172

Offset: 1

Alexandre Wajnberg, Apr 17 2005

Complement of A147820. - Omar E. Pol, Nov 17 2008

m is in the sequence iff A177961(m)Vladimir Shevelev, May 16 2010

			a(1)=13 is the first number satisfying simultaneously the two rules.

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

Intersection of A047845 and A104275.
Cf. A040040, A147820, A025583.
Cf. A010051, A002808, A099047.

a104278 n = a104278_list !! (n-1)
a104278_list = [m | m <- [1..],
                    a010051' (2 * m - 1) == 0 && a010051' (2 * m + 1) == 0]
-- Reinhard Zumkeller, Aug 04 2015

Select[ Range[300], !PrimeQ[2# + 1] && !PrimeQ[2# - 1] &] (* Robert G. Wilson v, Apr 18 2005 *)
Select[Range[300],NoneTrue[2#+{1,-1},PrimeQ]&] (* The program uses the NoneTrue function from Mathematica version 10 *)  (* Harvey P. Dale, Jul 07 2015 *)

select( {is_A104278(n)=!isprime(2*n-1)&&!isprime(2*n+1)}, [1..222]) \\ M. F. Hasler, Apr 29 2024

a(n) = (A025583-1)/2. - Bill McEachen, Feb 05 2025

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

A144650 Triangle read by rows where T(m,n) = 2m*n + m + n + 1.

Original entry on oeis.org

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, 35, 58, 81, 104, 127, 150, 173, 196, 219, 242, 265

Offset: 1

Vincenzo Librandi, Jan 13 2009

First column: A016789, second column: A016885, third column: A017029, fourth column: A017221, fifth column: A017461. - Vincenzo Librandi, Nov 21 2012

			Triangle begins:
   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.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A001105, A001844, A003307, A007742, A104275, A142463, A160378.

Columns k: A016789 (k=1), A016885 (k=2), A017029 (k=3), A017221 (k=4), A017461 (k=5).

[2*n*k + n + k + 1: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012

T[n_,k_]:= 2 n*k + n + k + 1; Table[T[n, k], {n, 11}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

flatten([[2*n*k+n+k+1 for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 14 2023

Sum_{n=1..m} T(m, n) = m*(2*m+3)*(m+1)/2 = A160378(n+1) (row sums). - R. J. Mathar, Jan 15 2009, Jan 05 2011
From G. C. Greubel, Oct 14 2023: (Start)
T(n, n) = A001844(n).
T(n, n-1) = A001105(n), n >= 2.
T(n, n-2) = A142463(n-1), n >= 3.
T(n, n-3) = (-1)*A147973(n+2), n >= 4.
Sum_{k=1..n} (-1)^k*T(n, k) = (-1)^n*A007742(floor((n+1)/2)).
G.f.: x*y*(5 - 2*x - 2*x*y - 2*x^2*y + x^2*y^2)/((1-x)^2*(1-x*y)^3). (End)

A356360 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+1))))).

Original entry on oeis.org

5, 7, 3, 11, 13, 1, 17, 19, 1, 23, 1, 1, 29, 31, 1, 1, 37, 1, 41, 43, 1, 47, 1, 1, 53, 1, 1, 59, 61, 1, 1, 67, 1, 71, 73, 1, 1, 79, 1, 83, 1, 1, 89, 1, 1, 1, 97, 1, 101, 103, 1, 107, 109, 1, 113, 1, 1, 1, 1, 1, 1, 127, 1, 131, 1, 1, 137, 139, 1, 1, 1, 1, 149, 151, 1, 1, 157, 1, 1, 163, 1, 167

Offset: 3

Mohammed Bouras, Oct 15 2022

nonn

Conjecture: The sequence contains only 1's and the primes.
Similar continued fraction to A356247.
Same as A128059(n), A145737(n-1) and A097302(n-2) for n > 5.
a(n) = 1 positions appear to correspond to A104275(m), m > 2. Conjecture: all odd primes are seen in order after 11. - Bill McEachen, Aug 05 2024

Mohammed Bouras, The Distribution Of Prime Numbers And The Continued Fractions, (2022).

Cf. A051403, A051417, A097302, A128059, A145737, A356247.

Cf. A104275.

from math import gcd, factorial
def A356360(n): return (a:=(n<<1)-1)//gcd(a, a*sum(factorial(k) for k in range(n-2))+n*factorial(n-2)>>1) # Chai Wah Wu, Feb 26 2024

For n >= 3, the formula of the continued fraction is as follows:
(A051403(n-2) + A051403(n-3))/(2n - 1) = 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+1))))).
a(n) = (2n - 1)/gcd(2n - 1, A051403(n-2) + A051403(n-3)).
From the conjecture: Except for n = 5, a(n)= 2n - 1 if 2n-1 is prime, 1 otherwise.

A153041 Numbers n >=10 such that 2*n-19 is not a prime.

Original entry on oeis.org

10, 14, 17, 20, 22, 23, 26, 27, 29, 32, 34, 35, 37, 38, 41, 42, 44, 47, 48, 50, 52, 53, 55, 56, 57, 59, 62, 65, 67, 68, 69, 70, 71, 72, 74, 76, 77, 80, 81, 82, 83, 86, 87, 89, 90, 92, 94, 95, 97, 98, 101, 102, 103, 104

Offset: 1

Vincenzo Librandi, Dec 17 2008

One more than associated values in A153051, two more than A153047. - R. J. Mathar, Jan 05 2011

The terms after a(1) are the values of 2*h*k + k + h + 10, where h and k are positive integers.- Vincenzo Librandi, Jan 19 2013

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

Cf. A097932, A155704.
Numbers n such that 2n-k is not prime: A104275 (k=1), A153043 (k=3), A153040 (k=5), A153039 (k=7), A153044 (k=9), A153045 (k=11), A153049 (k=13), A153047 (k=15), A153051 (k=17), A153041 (k=19).
Similar sequence listed also in A089559, A153144.

[n: n in [10..150] | not IsPrime(2*n - 19)]; // Vincenzo Librandi, Jan 19 2013

Select[Range[10, 200], !PrimeQ[2 # - 19] &] (* Vincenzo Librandi, Jan 19 2013 *)

Edited by N. J. A. Sloane, Jun 22 2010

A153045 Numbers k such that 2*k-11 is not a prime.

Original entry on oeis.org

10, 13, 16, 18, 19, 22, 23, 25, 28, 30, 31, 33, 34, 37, 38, 40, 43, 44, 46, 48, 49, 51, 52, 53, 55, 58, 61, 63, 64, 65, 66, 67, 68, 70, 72, 73, 76, 77, 78, 79, 82, 83, 85, 86, 88, 90, 91, 93, 94, 97, 98, 99, 100, 103, 106, 107, 108, 109, 110, 112, 113, 114

Offset: 1

Vincenzo Librandi, Dec 17 2008

The terms are the values of 2*h*k + k + h + 6, where h and k are positive integers. - Vincenzo Librandi, Jan 19 2013

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

Cf. A097338, A153043, A163652, A104275.

[n: n in [7..120] | not IsPrime(2*n - 11)]; // Vincenzo Librandi, Oct 11 2012

Select[Range[10,200], !PrimeQ[2*#-11]&] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2012 *)

from sympy import isprime
def ok(n): return n > 6 and not isprime(2*n-11)
print(list(filter(ok, range(115)))) # Michael S. Branicky, Oct 13 2021

a(n) = 5+A104275(n+1). [R. J. Mathar, Oct 22 2009]

cp's OEIS Frontend

A172048 a(n) = A104275(n) + A014076(n).

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A246371 Numbers n such that, if 2n-1 = Product_{k >= 1} (p_k)^(c_k) then n > Product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

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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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A014090 Numbers that are not the sum of a square and a prime.

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A153043 Numbers n > 1 such that 2*n-3 is not a prime.

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A104278 Numbers m such that 2m+1 and 2m-1 are not primes.

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A144650 Triangle read by rows where T(m,n) = 2m*n + m + n + 1.

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