A014076 -id:A014076 - 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

A067970 First differences of A014076, the odd nonprimes.

Original entry on oeis.org

8, 6, 6, 4, 2, 6, 2, 4, 6, 4, 2, 4, 2, 6, 2, 4, 6, 2, 4, 4, 2, 4, 2, 2, 4, 6, 6, 4, 2, 2, 2, 2, 2, 4, 4, 2, 6, 2, 2, 2, 6, 2, 4, 2, 4, 4, 2, 4, 2, 6, 2, 2, 2, 6, 6, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 6, 4, 2, 6, 2, 2, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 6, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 6, 4, 2, 2, 2, 2, 2, 4, 2, 4, 2, 2, 2

Offset: 1

Labos Elemer, Feb 04 2002

nonn

In this sequence 8 occurs once, but 2,4,6 may occur several times. No other even number arises. Therefore sequence consists of {8,6,4,2}.
Proof: If x is an odd nonprime, then x+2=next-odd-number is either nonprime[Case1] or it is a prime [Case 2]. In Case 1 the difference is 2. E.g., x=25, x+2=27, the actual difference is d=2.
In Case 2 x+2=p=prime. Distinguish two further subcases. In Case 2a: x+2=p=prime and p+2=x+4=q is also a prime. Then x+2+2+2=x+6 will not be prime because in first difference sequence of prime no d=2 occurs twice except for p+2=3+2=5,5+2=7, i.e., when p is divisible by 3; for 6k+1 and 6k+5 primes it is impossible. Consequently x+6 is not a prime and so the difference between two consecutive odd nonprimes is 6. Example: x=39, x+2=41=smaller twin prime and next odd nonprime x+6=45, d=6
In Case 2b: x+2=p=prime, but x+2+2=x+4 is not a prime, i.e., x+2=p is not a smaller one of a twin-prime pair. Thus x+4 is the next odd nonprime. Thus the difference=4. E.g., x=77, x+2=79, so the next odd nonprime is x+4=81, d=4. No more cases. QED.
Interestingly this sequence picks out the twin primes.
That the first term is special is a reflection of the simple fact that there are no 3 consecutive odd primes except from 3, 5, 7 corresponding to A067970(1) = 8 = 9-1 = (7+2)-(3-2). - Frank Ellermann, Feb 08 2002
There are arbitrarily long runs of 2's, but not of 4's or 6's. - Zak Seidov, Oct 01 2011

Jason Yuen, Table of n, a(n) for n = 1..10000

Cf. A014076, A000230.

Cf. A196274, A196276, A196277.

a067970 n = a067970_list !! (n-1)
a067970_list = zipWith (-) (tail a014076_list) a014076_list
-- Reinhard Zumkeller, Sep 30 2011

a = Select[ Range[300], !PrimeQ[ # ] && !EvenQ[ # ] & ]; Table[ a[[n + 1]] - a[[n]], {n, 1, Length[a] - 1} ]
With[{nn=401},Differences[Complement[Range[1,nn,2],Prime[Range[PrimePi [nn]]]]]]  (* Harvey P. Dale, Feb 05 2012 *)

from sympy import primepi, isprime
def A067970(n):
    if n == 1: return 8
    m, k = n-1, primepi(n) + n - 1 + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n - 1 + (k>>1)
    for d in range(2,7,2):
        if not isprime(m+d):
            return d # Chai Wah Wu, Jul 31 2024

a(n) = A014076(n+1) - A014076(n).

a(n) = 2 * A196274(n); a(A196276(n)) = 2; a(A196277(n)) > 2. - Reinhard Zumkeller, Sep 30 2011

Edited by Robert G. Wilson v, Feb 08 2002

Offset changed to 1 by Jason Yuen, Jan 08 2025

A196274 Half of the gaps A067970 between odd nonprimes A014076.

Original entry on oeis.org

4, 3, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 1, 3, 1, 2, 3, 1, 2, 2, 1, 2, 1, 1, 2, 3, 3, 2, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 2, 1, 3, 1, 1, 1, 3, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 2, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 3, 1, 1, 1, 2

Offset: 1

M. F. Hasler, Sep 30 2011

nonn

a(n) < 4 for n > 1; a(A196276(n)) = 1; a(A196277(n)) > 1. - Reinhard Zumkeller, Sep 30 2011

Lengths of runs of equal terms in A025549. That sequence begins with: 1,1,1,1,3,3,3,45,45,45,..., that is 4 ones, 3 threes, 3 forty-fives, ... - Michel Marcus, Dec 02 2014

			The smallest odd numbers which are not prime are 1, 9, 15, 21, 25, 27,... (sequence A014076).
The gaps between these are: 8, 6, 6, 4, 2,... (sequence A067970), which are of course all even by construction, so it makes sense to divide all of them by 2, which yields this sequence: 4, 3, 3, 2, 1, ...

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

Cf. A142723 for the decimal value of the associated continued fraction.

With[{nn=401},Differences[Complement[Range[1,nn,2],Prime[Range[ PrimePi[ nn]]]]]/2] (* Harvey P. Dale, May 06 2012 *)

L=1;forstep(n=3,299,2,isprime(n)&next;print1((n-L)/2",");L=n)

from sympy import primepi, isprime
def A196274(n):
    if n == 1: return 4
    m, k = n-1, primepi(n) + n - 1 + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n - 1 + (k>>1)
    for d in range(1,4):
        if not isprime(m+(d<<1)):
            return d # Chai Wah Wu, Jul 31 2024

a(n) = (A014076(n+1)-A014076(n))/2 = A067970(n)/2.

More terms from Harvey P. Dale, May 06 2012

A047846 Number of successive odd nonprimes (A014076).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 3, 1, 1, 1, 6, 1, 2, 4, 2, 2, 1, 2, 2, 4, 1, 5, 5, 1, 1, 2, 4, 2, 2, 2, 2, 1, 4, 6, 1, 1, 6, 2, 4, 1, 2, 3, 2, 2, 1, 2, 3, 1, 3, 4, 4, 2, 1, 2, 3, 1, 1, 5, 3, 1, 3, 1, 2, 5, 8, 2, 4, 2, 2, 2, 4, 2, 2, 2, 2, 1, 5

Offset: 1

Enoch Haga

Beginning with a(2), length of n-th run of identical numbers in A160522 [Kyle Stern, Jun 19 2009]

a(n+1) = A196277(n+1) - A196277(n). [Reinhard Zumkeller, Sep 30 2011]

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

Length of n-th run in A014076.

Cf. A160522.

a047846 n = a047846_list !! (n-1)
a047846_list = 1 : zipWith (-) (tail a196277_list) a196277_list
-- Reinhard Zumkeller, Sep 30 2011

Length /@ Split[Select[Range[1, 629, 2], PrimeOmega[#] != 1 &], #2 - #1 <= 2 &] (* Jayanta Basu, Aug 11 2013 *)
Length/@Select[Split[Table[If[PrimeQ[n],0,1],{n,1,701,2}]],#[[1]]==1&] (* Harvey P. Dale, Oct 10 2023 *)

a(51) and a(84) corrected by Reinhard Zumkeller, Sep 30 2011

A062532 Odd nonprimes squared, or A014076(n)^2.

Original entry on oeis.org

1, 81, 225, 441, 625, 729, 1089, 1225, 1521, 2025, 2401, 2601, 3025, 3249, 3969, 4225, 4761, 5625, 5929, 6561, 7225, 7569, 8281, 8649, 9025, 9801, 11025, 12321, 13225, 13689, 14161, 14641, 15129, 15625, 16641, 17689, 18225, 19881, 20449

Offset: 1

Jason Earls, Jul 10 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A014076.

With[{nn=500},Complement[Range[1,nn,2],Prime[Range[PrimePi[nn]]]]^2] (* Harvey P. Dale, Nov 12 2012 *)

je=[]; forstep(n=1,301,2, if(isprime(n), n+1,je=concat(je,n^2))); je

{ n=0; forstep (m=1, 10^9, 2, if(!isprime(m), write("b062532.txt", n++, " ", m^2); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 08 2009

from sympy import primepi
def A062532(n):
    if n == 1: return 1
    m, k = n-1, primepi(n) + n - 1 + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n - 1 + (k>>1)
    return m**2 # Chai Wah Wu, Jul 31 2024

A256253 Number of successive odd nonprimes A014076 and number of successive odd primes A065091, interleaved.

Original entry on oeis.org

1, 3, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 2, 1, 1, 6, 1, 1, 1, 2, 2, 4, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 4, 2, 1, 2, 5, 1, 5, 1, 1, 2, 1, 1, 2, 2, 4, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 4, 1, 6, 1, 1, 2, 1, 1, 6, 1, 2, 1, 4, 2, 1, 1, 2, 1, 3, 1, 2, 1, 2

Offset: 1

Omar E. Pol, Mar 30 2015

nonn

See also A256252 and A256262 which contain similar diagrams.

			Consider an irregular array in which the odd-indexed rows list successive odd nonprimes (A014076) and the even-indexed rows list successive odd primes (A065091), in the sequence of odd numbers (A005408), as shown below:
1;
3, 5, 7;
9;
11, 13;
15;
17; 19;
21,
23;
25, 27;
39, 31;
...
a(n) is the length of the n-th row.
.
Illustration of the first 16 regions of the diagram of the symmetric representation of odd nonprimes (A014076) and of odd primes (A065091):
.            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.           |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |   31
.           |_ _ _ _ _ _ _ _ _ _ _ _ _ _  | |   29
.           | | |_ _ _ _ _ _ _ _ _ _ _  | | |   23
.           | | | |_ _ _ _ _ _ _ _ _  | | | |   19
.           | | | |_ _ _ _ _ _ _ _  | | | | |   17
.           | | | | |_ _ _ _ _ _  | | | | | |   13
.           | | | | |_ _ _ _ _  | | | | | | |   11
.           | | | | | |_ _ _  | | | | | | | |    7
.           | | | | | |_ _  | | | | | | | | |    5
.   A014076 | | | | | |_  | | | | | | | | | |    3
.      1    | | | | | |_|_|_|_| | | | | | | | A065091
.      9    | | | | |_ _ _ _ _|_|_| | | | | |
.     15    | | | |_ _ _ _ _ _ _ _|_|_| | | |
.     21    | | |_ _ _ _ _ _ _ _ _ _ _|_| | |
.     25    | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
.     27    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|_|_|
.
a(n) is also the length of the n-th boundary segment in the zig-zag path of the above diagram, between the two types of numbers, as shown below for n = 1..10:
.
.                       |_ _ _
.                             |_ _
.                                 |_ _
.                                     |_
.                                       |
.                                       |_ _
.
The sequence begins:    1,3,1,2,1,2,1,1,2,2,...
.

Cf. A005408, A014076, A047846, A065091, A175632, A251092, A256252, A256134, A256262.

lista(nn) = {my(nb = 1, isp = 0); forstep (n=3, nn, 2, if (bitxor(isp, ! isprime(n)), nb++, print1(nb, ", "); nb = 1; isp = ! isp););} \\ Michel Marcus, May 25 2015

a(n) = A256252(n-1), n >= 3.

A142723 Decimal expansion of the continued fraction whose terms are half the gaps of the odd nonprimes A014076.

Original entry on oeis.org

4, 3, 0, 3, 2, 3, 9, 3, 0, 1, 9, 8, 3, 1, 2, 2, 5, 4, 7, 5, 0, 7, 2, 5, 6, 5, 3, 7, 1, 2, 9, 4, 6, 1, 0, 1, 1, 0, 0, 5, 8, 7, 4, 9, 8, 2, 5, 6, 1, 5, 9, 3, 3, 2, 7, 6, 9, 9, 6, 6, 3, 7, 1, 8, 1, 0, 8, 6, 7, 0, 5, 5, 2, 1, 6, 2, 6, 3, 9, 5, 7, 8, 9, 0, 1, 9, 6, 0, 0, 2, 4, 3, 7, 4, 8, 7, 1, 5, 5, 8, 7, 3, 6, 9, 2

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 27 2008

Take half of the first difference of odd nonprimes A014076 and treat it as a continued fraction. This sequence gives the decimal expansion of that number. - Charles R Greathouse IV, Feb 03 2011

			4.30323930198312254750725653712946101100587498256159332769966371810867...

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

a = Flatten[Table[If[PrimeQ[2*n + 1], {}, 2*n - 1], {n, 0, 200}]]; b = Table[(a[[n + 1]] - a[[n]])/2, {n, 1, Length[a] - 1}]; FromContinuedFraction[b]; c = N[%, 200]; Table[Floor[Mod[c*10^n, 10]], {n, 0, 201}] (* Bagula and Adamson *)
RealDigits[FromContinuedFraction[Differences[Select[Range[-1, 399, 2], !PrimeQ[# + 2]&]]/2], 10, 201][[1]] (* Charles R Greathouse IV, Feb 03 2011 *)

a=contfracpnqn(D(select(vector(99,n,2*n-1),x->!isprime(x)))/2); a[1,1]/a[2,1]*1.  /* OK for 35 digits. For D(.) see A137822 */ \\ M. F. Hasler, Sep 29 2011

A239114 Exponent of 2 in prime factorization (i.e., 2-adic valuation) of odd nonprimes A014076(n) + 1.

Original entry on oeis.org

1, 1, 4, 1, 1, 2, 1, 2, 3, 1, 1, 2, 3, 1, 6, 1, 1, 2, 1, 1, 1, 3, 2, 1, 5, 2, 1, 4, 2, 1, 3, 1, 2, 1, 1, 1, 3, 1, 4, 1, 2, 1, 2, 5, 1, 1, 1, 2, 4, 1, 3, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 3, 1, 2, 1, 1, 3, 2, 1, 2, 1, 3, 1, 1, 8, 2, 1, 1, 2, 1, 2, 3, 1, 5, 1, 2, 3, 1, 2, 1, 4, 1, 1, 2, 6, 1, 2, 1, 3, 1

Offset: 1

K. G. Stier, Mar 10 2014

nonn

Sequence is counterpart to A023512, i.e., merging these two sequences gives the ruler function A001511.

			a(13) = 3, because the 13th odd nonprime is 55, and the largest power of 2 dividing 55+1 is 3.

K. G. Stier, Table of n, a(n) for n = 1..7739

Cf. A023512, A001511, A014076.

lista(nn) = {forstep(n=1, nn, 2, if (! isprime(n), print1(valuation(n+1, 2), ", ")););} \\ Michel Marcus, Mar 13 2014

a(n) = A001511((A014076(n)+1)/2)

A062533 a(n) = A000010(A014076(n)).

Original entry on oeis.org

1, 6, 8, 12, 20, 18, 20, 24, 24, 24, 42, 32, 40, 36, 36, 48, 44, 40, 60, 54, 64, 56, 72, 60, 72, 60, 48, 72, 88, 72, 96, 110, 80, 100, 84, 108, 72, 92, 120, 112, 84, 96, 120, 104, 132, 80, 156, 108, 120, 116, 120, 144, 160, 108, 96, 132, 168, 160, 132, 180, 140, 168

Offset: 1

Jason Earls, Jul 10 2001

G. C. Greubel, Table of n, a(n) for n = 1..1000

Subset of the totients of the odds (A037225).

Cf. A000010, A014076.

A014076 := Select[Range[1, 350, 2], PrimeOmega[#] != 1 &]; Table[ EulerPhi[A014076[[n]]], {n, 1, 50}] (* G. C. Greubel, Sep 17 2017 *)

je=[]; forstep(n=1,301,2, if(isprime(n), n+1,je=concat(je,eulerphi(n)))); je

from sympy import primepi, totient
def A062533(n):
    if n == 1: return 1
    m, k = n-1, primepi(n) + n - 1 + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n - 1 + (k>>1)
    return totient(m) # Chai Wah Wu, Jul 31 2024

A163631 Partial sums of the odd nonprimes, A014076.

Original entry on oeis.org

1, 10, 25, 46, 71, 98, 131, 166, 205, 250, 299, 350, 405, 462, 525, 590, 659, 734, 811, 892, 977, 1064, 1155, 1248, 1343, 1442, 1547, 1658, 1773, 1890, 2009, 2130, 2253, 2378, 2507, 2640, 2775, 2916, 3059, 3204, 3351, 3504, 3659, 3818, 3979, 4144, 4313

Offset: 1

Juri-Stepan Gerasimov, Aug 02 2009, Oct 10 2009

nonn

All entries besides the first are of the form 2+(k+1)^2-A007504(j), e.g., 10=2+25-17, 25=2+64-41, where the square is the sum of all odd numbers up to 1+2*k, and the 2 and A007504 represent the partial sum over the primes.

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

Cf. A014076.

upto=200; Accumulate[Complement[Range[1,upto,2], Prime[Range[2,PrimePi[upto]]]]] (* Harvey P. Dale, Mar 19 2011 *)

lista(nn) = {my(s = 0); forstep (n=1, nn, 2, if (!isprime(n), s+= n; print1(s, ", ")););} \\ Michel Marcus, Aug 01 2017

a(n) = Sum_{x= 1st odd nonprime..n-th odd nonprime}x.

150 replaced with 250 by R. J. Mathar, Aug 06 2009

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

SageMath

Formula

Extensions

A067970 First differences of A014076, the odd nonprimes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Python

Formula

Extensions

A196274 Half of the gaps A067970 between odd nonprimes A014076.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A047846 Number of successive odd nonprimes (A014076).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A062532 Odd nonprimes squared, or A014076(n)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

A256253 Number of successive odd nonprimes A014076 and number of successive odd primes A065091, interleaved.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A142723 Decimal expansion of the continued fraction whose terms are half the gaps of the odd nonprimes A014076.

Views

Author