A089229 -id:A089229 - cp's OEIS frontend

A162908 Pairs of central divisors of A089229(n).

2, 3, 2, 4, 2, 5, 3, 4, 2, 7, 3, 5, 3, 6, 4, 5, 3, 7, 2, 11, 4, 6, 2, 13, 3, 9, 4, 7, 5, 6, 4, 8, 3, 11, 2, 17, 5, 7, 2, 19, 3, 13, 5, 8, 6, 7, 4, 11, 5, 9, 2, 23, 6, 8, 5, 10, 3, 17, 4, 13, 6, 9, 5, 11, 7, 8, 3, 19, 2, 29, 6, 10, 2, 31, 7, 9, 5, 13, 6, 11, 4, 17, 3, 23, 7, 10, 8, 9, 2, 37, 5, 15, 4, 19

Offset: 1

Juri-Stepan Gerasimov, Jul 17 2009

The central divisors are the two complementary divisors closest to the square

root of a number, as tabulated in A162348.

			a(1)=2 and a(2)=3 since A089229(1)=6 with divisors 1, 2, 3, 4.
a(3)=2 and a(4)=4 since A089229(2)=8 with divisors 1, 2, 4, 8.
a(5)=2 and a(6)=5 since A089229(3)=10 with divisors 1, 2, 5, 10.

Cf. A089229, A162361.

a(2n-1)= A033676(A089229(n)). a(2n)=A033677(A089229(n)). - R. J. Mathar, Jul 19 2009

Missing pairs inserted by R. J. Mathar, Jul 19 2009

A106543 Composite numbers that are not perfect powers.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105, 106

Offset: 1

Alexandre Wajnberg, May 08 2005

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A002808 and A007916.

Cf. A105642, A089229.

perfPQ[n_]:=GCD@@FactorInteger[n][[All,2]]>1; Select[Range[110], CompositeQ[ #] && !perfPQ[#]&] (* Harvey P. Dale, Oct 10 2017 *)

lista(nn)=forcomposite(i=1, nn, if (! ispower(i), print1(i, ", "));); \\ Michel Marcus, Jun 27 2013

is(n)=!isprime(n) && !ispower(n) && n>1 \\ Charles R Greathouse IV, Oct 19 2015

from sympy import primepi, mobius, integer_nthroot
def A106543(n):
    def f(x): return int(n+1+primepi(x)-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 12 2024

def A106543_list(n) : return [k for k in (2..n) if not k.is_prime() and not k.is_perfect_power()]
A106543_list(106) # Terry D. Grant, Jul 17 2016

a(n) = n + O(n/log n). - Charles R Greathouse IV, Oct 03 2011

A245499 Table read by rows: n-th row contains the factors which occur when constructing R. L. Graham's sequence A006255, such that the number of factors and also the product is minimal.

Original entry on oeis.org

1, 2, 3, 6, 3, 6, 8, 4, 5, 8, 10, 6, 8, 12, 7, 8, 14, 8, 10, 12, 15, 9, 10, 12, 15, 18, 11, 18, 22, 12, 15, 20, 13, 18, 26, 14, 15, 18, 20, 21, 15, 18, 20, 24, 16, 17, 18, 34, 18, 24, 27, 19, 32, 38, 20, 24, 30, 21, 27, 28, 22, 24, 33, 23, 32, 46, 24, 27, 32

Offset: 1

Reinhard Zumkeller, Jul 25 2014

A066400(n) = length of n-th row.
A006255(n) = T(n,A066400(n)), last term in n-th row.
A245530(n) = A066401(n)^2 = product of n-th row.
For n > 2: A245508(n) = T(A000040(n),2).
T(n,k) denote b_k in the definition given in A006255.
From David A. Corneth, Oct 22 2016 and Oct 25 2016: (Start)
Frequency of n in this sequence: 1, 1, 2, 1, 1, 3, 1, 5, 1, 3, 1, 4, 1, 2, ... See A277606.
Primes and squares occur once in this sequence except for 3 which occurs twice.
In the first 10000 rows, 9522 occurs most often and appears 60 times. 6498 is a close second with 59 occurrences.
(End)

			.    n |  Row(n)                | A066400(n) | A245530(n) | A066401(n)
. -----+------------------------+------------+------------+-----------
.    1 |  [1]                   |          1 |          1 |          1
.    2 |  [2, 3, 6]             |          3 |         36 |          6
.    3 |  [3, 6, 8]             |          3 |        144 |         12
.    4 |  [4]                   |          1 |          4 |          2
.    5 |  [5, 8, 10]            |          3 |        400 |         20
.    6 |  [6, 8, 12]            |          3 |        576 |         24
.    7 |  [7, 8, 14]            |          3 |        784 |         28
.    8 |  [8, 10, 12, 15]       |          4 |      14400 |        120
.    9 |  [9]                   |          1 |          9 |          3
.   10 |  [10, 12, 15, 18]      |          4 |      32400 |        180
.   11 |  [11, 18, 22]          |          3 |       4356 |         66
.   12 |  [12, 15, 20]          |          3 |       3600 |         60
.   13 |  [13, 18, 26]          |          3 |       6084 |         78
.   14 |  [14, 15, 18, 20, 21]  |          5 |    1587600 |       1260
.   15 |  [15, 18, 20, 24]      |          4 |     129600 |        360
.   16 |  [16]                  |          1 |         16 |          4
.   17 |  [17, 18, 34]          |          3 |      10404 |        102
.   18 |  [18, 24, 27]          |          3 |      11664 |        108
.   19 |  [19, 32, 38]          |          3 |      23104 |        152
.   20 |  [20, 24, 30]          |          3 |      14400 |        120
.   21 |  [21, 27, 28]          |          3 |      15876 |        126
.   22 |  [22, 24, 33]          |          3 |      17424 |        132
.   23 |  [23, 32, 46]          |          3 |      33856 |        184
.   24 |  [24, 27, 32]          |          3 |      20736 |        144
.   25 |  [25]                  |          1 |         25 |          5 .

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., Problem 4.39, pages 147, 616, 533.

Peter Kagey, Table of n, a(n) for n = 1..44055 (Derived from David A. Corneth's unflattened file)
David A. Corneth, First 10000 rows unflattened

Cf. A006255, A006530, A007913, A010051, A010052, A020639, A049084, A066400 (row lengths), A066401, A089229, A151800, A245508, A245530 (row products), A277606.

Table[k = 0; While[Length@ # == 0 &@ Set[f, Select[Rest@ Subsets@ Range@ k, IntegerQ@ Sqrt[n (Times @@ # &[n + #])] &]], k++]; If[IntegerQ@ Sqrt@ n, k = {n}, k = n + Prepend[First@ f, 0]]; k, {n, 22}] (* Michael De Vlieger, Oct 26 2016 *)

Following a suggestion of Peter Kagey, definition clarified by Reinhard Zumkeller, Nov 28 2014. Also removed erroneous program and b-file.

A089237 List of primes and squares.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 36, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 100, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 196, 197, 199, 211, 223, 225, 227

Offset: 1

N. J. A. Sloane, Dec 11 2003

nonn

Starting at a(1) = 1, this is the lexicographically earliest sequence of distinct numbers whose partial products are all exponentially odd numbers (A268335). - Amiram Eldar, Sep 24 2024

Zak Seidov, Table of n, a(n) for n = 1..1000

Complement of A089229.

Cf. A000040, A000290, A010051, A010052, A161187, A161188, A268335.

a089237 n = a089237_list !! (n-1)
a089237_list = merge a000040_list a000290_list where
   merge xs'@(x:xs) ys'@(y:ys) =
         if x < y then x : merge xs ys' else y : merge xs' ys
-- Reinhard Zumkeller, Dec 18 2012

m=100; Sort[Flatten[{Range[0,m]^2, Prime[Range[PrimePi[m^2]]]}]] (* Zak Seidov, Nov 05 2009 *)

is(n)=isprime(n) || issquare(n) \\ Charles R Greathouse IV, Oct 14 2016

{A89237=List([0..5]); A089237(n)=while(#A89237A340389. - M. F. Hasler, Jul 24 2021, edited Sep 01 2021

from math import isqrt
from sympy import primepi
def A089237(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n-1+x-primepi(x)-isqrt(x))
    return bisection(f,n-1,n-1) # Chai Wah Wu, Oct 12 2024

a(A161187(n)+1) = A000290(n); a(A161188(n)+1) = A000040(n). - Reinhard Zumkeller, Dec 18 2012
A010051(a(n)) + A010052(a(n)) = 1. - Reinhard Zumkeller, Jul 07 2014
a(n) ~ n log n. - Charles R Greathouse IV, Oct 14 2016

A202137 Numbers k such that 24k + 1 is neither square nor prime.

Original entry on oeis.org

6, 9, 11, 16, 20, 21, 23, 27, 29, 30, 31, 33, 34, 36, 37, 38, 41, 44, 45, 46, 49, 53, 56, 58, 59, 60, 61, 63, 64, 65, 66, 68, 71, 72, 76, 79, 80, 81, 82, 85, 86, 91, 93, 94, 96, 97, 98, 101, 102, 104, 106, 107, 110, 111, 114, 115, 116, 120, 121, 122, 124

Offset: 1

Zak Seidov, Dec 15 2011

nonn

Conjecture: sequence contains arbitrarily long runs of consecutive integers.
First runs with lengths 1..4 are 6; 20, 21; 29, 30, 31; 58, 59, 60, 61.
Records in run lengths are 1, 2, 3, 4, 5, 6, 7, 9, 13, 17, 20, 23, 32, 33, 36, 40, 41, 43, 48, 49, 52, 69, 77, 89, 97, 99, 108, 126, 135, 148, 149
with corresponding first terms of runs: 6, 20, 29, 58, 148, 163, 378, 449, 936, 1675, 5740, 7075, 15915, 35545, 112303, 229944, 469454, 628921, 775480, 902518, 1003826, 1208039, 12542948, 29223210, 33015691, 224430268, 260333109, 530363391, 3713119689, 7962252405, 9312173798.
Conjecture is easy to prove using the Chinese Remainder Theorem and the fact that the gaps between squares grow. - Robert Israel, Jan 25 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A089237 (list of primes and squares), A089229 (neither primes nor squares).

[n: n in [1..200] | not IsSquare(24*n+1) and not IsPrime(24*n+1)]; // Vincenzo Librandi, Jan 26 2018

filter:= n -> not issqr(24*n+1) and not isprime(24*n+1):
select(filter, [$1..200]); # Robert Israel, Jan 25 2018

Select[Range[150],!PrimeQ[24#+1]&&!IntegerQ[Sqrt[24#+1]]&] (* Harvey P. Dale, Dec 01 2015 *)

for(n=1,200,m=24*n+1;if(isprime(m)+issquare(m),,print1(n",")))

A228363 Sorted entries of the multiplication table a*b, with a>1, b>1.

Original entry on oeis.org

4, 6, 6, 8, 8, 9, 10, 10, 12, 12, 12, 12, 14, 14, 15, 15, 16, 16, 16, 18, 18, 18, 18, 20, 20, 20, 20, 21, 21, 22, 22, 24, 24, 24, 24, 24, 24, 25, 26, 26, 27, 27, 28, 28, 28, 28, 30, 30, 30, 30, 30, 30, 32, 32, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 36, 36, 36, 36, 36, 38, 38, 39, 39, 40, 40, 40, 40

Offset: 1

Ralf Stephan, Aug 21 2013

nonn

Consists of the composite numbers (A002808), with the composite nonsquares (A089229) occurring more than once, i.e., the n-th composite appears A144925(n) times.

A complement of the primes (A000040).

Cf. A061017 (sorted entries of standard multiplication table)

L=listcreate();for(k=2,250,for(l=2,250,listput(L,k*l)));v=vecsort(Vec(L));for(n=1,100,print1(v[n],","))

A274172 Nonsquare composites with all prime factors larger than 5.

Original entry on oeis.org

77, 91, 119, 133, 143, 161, 187, 203, 209, 217, 221, 247, 253, 259, 287, 299, 301, 319, 323, 329, 341, 343, 371, 377, 391, 403, 407, 413, 427, 437, 451, 469, 473, 481, 493, 497, 511, 517, 527, 533, 539, 551, 553, 559, 581, 583, 589

Offset: 1

Dimitris Valianatos, Jun 12 2016

nonn

Nonsquare composites not divisible by 2,3,5.

			377 = 13*29 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A007775 and A089229. - Felix Fröhlich, Jun 12 2016

filter:= n -> igcd(n,30)=1 and not issqr(n) and not isprime(n):
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, May 30 2021

{ for(n=1,600, if(!(isprime(n) || n%2==0 || n%3==0 || n%5==0 || issquare(n)), print1(n", ")))}

is(n) = my(f=factor(n)); if(!issquare(n),f[1,1]>5 && matsize(f)[1]>1,0) \\ David A. Corneth, Jun 12 2016

A162915 Absolute values of the first differences of A162908.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 1, 2, 5, 4, 2, 2, 3, 2, 1, 2, 4, 5, 9, 7, 2, 4, 11, 10, 6, 5, 3, 2, 1, 2, 4, 5, 8, 9, 15, 12, 2, 5, 17, 16, 10, 8, 3, 2, 1, 3, 7, 6, 4, 7, 21, 17, 2, 3, 5, 7, 14, 13, 9, 7, 3, 4, 6, 4, 1, 5, 16, 17, 27, 23, 4, 8, 29, 24, 2, 4, 8, 7, 5, 7, 13, 14, 20, 16, 3, 2, 1, 7, 35, 32, 10, 11, 15

Offset: 1

Juri-Stepan Gerasimov, Jul 17 2009

Cf. A089229, A162361, A162908.

Corrected as a followup to correcting A162908. - R. J. Mathar, Jul 19 2009

A340449 Nonsquare composites n whose smallest prime factor is greater than or equal to n^(2/5).

Original entry on oeis.org

15, 35, 55, 77, 91, 119, 143, 187, 209, 221, 247, 253, 299, 319, 323, 341, 377, 391, 403, 437, 481, 493, 527, 533, 551, 559, 589, 629, 667, 697, 703, 713, 731, 779, 799, 817, 851, 893, 899, 901, 943, 989, 1003, 1007, 1037, 1073, 1081, 1121, 1139, 1147, 1159, 1189

Offset: 1

Phil Scovis, Jan 07 2021

nonn

Referred to as "freak-show composites" by Doug Massey (see link), who noted that they are exceptionally difficult to factor by hand using trial division.
4181, 4183, 4187, and 4189 are the first four consecutive terms that differ only in the last decimal digit.  The next such set of four consecutive terms begins at 806621.
Ankit Bisain says that for sufficiently large x, there are fewer values in this sequence less than x than there are primes less than x.

			The smallest prime factor of 4189 is 59, which is greater than 4189^(2/5).

Quora, What's the definition of a freak-show composite number?

Superset of A006094, except for initial 6 in that sequence.

Subsequence of A089229.

q:= n-> not (isprime(n) or issqr(n) or min(numtheory[factorset](n))^5Alois P. Heinz, Jan 07 2021

nscQ[n_]:=CompositeQ[n]&&!IntegerQ[Sqrt[n]]&&FactorInteger[n][[1,1]]>= Surd[n^2,5]; Select[Range[1200],nscQ] (* Harvey P. Dale, Jul 25 2021 *)

isok(n) = (n>1) && !isprime(n) && !issquare(n) && (factor(n)[1,1]^5 >= n^2); \\ Michel Marcus, Jan 07 2021

cp's OEIS Frontend

A162908 Pairs of central divisors of A089229(n).

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A106543 Composite numbers that are not perfect powers.

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A245499 Table read by rows: n-th row contains the factors which occur when constructing R. L. Graham's sequence A006255, such that the number of factors and also the product is minimal.

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A089237 List of primes and squares.

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A202137 Numbers k such that 24k + 1 is neither square nor prime.

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A228363 Sorted entries of the multiplication table a*b, with a>1, b>1.

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A274172 Nonsquare composites with all prime factors larger than 5.

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