A144925 -id:A144925 - cp's OEIS frontend

A167131 Numbers k such that A002808(k) - A144925(k) is prime.

1, 8, 9, 12, 21, 24, 26, 30, 38, 44, 45, 49, 53, 61, 66, 81, 84, 86, 97, 100, 106, 109, 116, 121, 131, 140, 154, 165, 183, 189, 191, 198, 203, 205, 208, 216, 232, 245, 252, 257, 270, 283, 290, 305, 308, 310, 313, 323, 325, 330, 340, 342, 358, 363, 367, 377, 388

Offset: 1

Juri-Stepan Gerasimov, Oct 28 2009

nonn

Cf. A000040, A002808, A144925.

A070824 := proc(n) if n <=3 then 0; else numtheory[tau](n)-2 ; end if; end proc: A144925 := proc(n) A070824(A002808(n)) ; end proc: isA167131 := proc(n) isprime( A002808(n)-A144925(n)) ; end proc: for n from 1 to 1000 do if isA167131(n) then printf("%d,",n) ; end if; od: # R. J. Mathar, Nov 02 2009

Edited (but not checked) by N. J. A. Sloane, Nov 01 2009

105 replaced with 106 and sequence extended by R. J. Mathar, Nov 02 2009

A163870 Triangle read by rows: row n lists the nontrivial divisors of the n-th composite.

Original entry on oeis.org

2, 2, 3, 2, 4, 3, 2, 5, 2, 3, 4, 6, 2, 7, 3, 5, 2, 4, 8, 2, 3, 6, 9, 2, 4, 5, 10, 3, 7, 2, 11, 2, 3, 4, 6, 8, 12, 5, 2, 13, 3, 9, 2, 4, 7, 14, 2, 3, 5, 6, 10, 15, 2, 4, 8, 16, 3, 11, 2, 17, 5, 7, 2, 3, 4, 6, 9, 12, 18, 2, 19, 3, 13, 2, 4, 5, 8, 10, 20, 2, 3, 6, 7, 14, 21, 2, 4, 11, 22, 3, 5, 9, 15, 2, 23

Offset: 1

Juri-Stepan Gerasimov, Aug 06 2009

Row n contains row A002808(n) of table A027750.

T(n,k) = A027751(A002808(n),k+1), k = 1..A144925(n). - Reinhard Zumkeller, Mar 29 2014

			The table starts in row n=1 (with the composite 4) as
  2;
  2,3;
  2,4;
  3;
  2,5;
  2,3,4,6;
  2,7;
  3,5;
  2,4,8;
  2,3,6,9;
  2,4,5,10.

Reinhard Zumkeller, Rows n = 1..1000 of table, flattened

Cf. A002808, A027750.

Cf. A144925 (row lengths), A062825 (row sums), A056608 (left edge), A160180 (right edge).

a163870 n k = a163870_tabf !! (n-1) !! (k-1)
a163870_row n = a163870_tabf !! (n-1)
a163870_tabf = filter (not . null) $ map tail a027751_tabf
-- Reinhard Zumkeller, Mar 29 2014

Divisors[Select[Range[50], CompositeQ]][[All, 2 ;; -2]] (* Paolo Xausa, Dec 26 2024 *)

from itertools import islice
def g():
    n, j = 1, 2
    while True:
        n = (n << 1) | 1
        p = 1
        for k in range(2, (j >> 1) + 1):
            p = (p << 1) | 1
            if n % p == 0: yield k
        j+=1
print(list(islice(g(),95))) # Darío Clavijo, Dec 16 2024

Entries checked by R. J. Mathar, Sep 22 2009

A160180 Largest proper divisor of the n-th composite number.

Original entry on oeis.org

2, 3, 4, 3, 5, 6, 7, 5, 8, 9, 10, 7, 11, 12, 5, 13, 9, 14, 15, 16, 11, 17, 7, 18, 19, 13, 20, 21, 22, 15, 23, 24, 7, 25, 17, 26, 27, 11, 28, 19, 29, 30, 31, 21, 32, 13, 33, 34, 23, 35, 36, 37, 25, 38, 11, 39, 40, 27, 41, 42, 17, 43, 29, 44, 45, 13, 46, 31, 47, 19, 48, 49, 33, 50, 51, 52

Offset: 1

Kyle Stern, May 03 2009, May 04 2009

Old name: The n-th positive composite number divided by its lowest nontrivial factor.

			a(1) = 4/2 = 2, a(2) = 6/2 = 3, a(3) = 8/2 = 4, a(4) = 9/3 = 3, a(5) = 10/2 = 5.

K. Stern, Table of n, a(n) for n = 1..9999

Cf. A002808, A032742, A056608, A163870, A144925.

a160180 = a032742 . a002808  -- Reinhard Zumkeller, Mar 29 2014

function [a] = A160180(k) j = 0; n = 1; while j < k if isprime(n) == 1 skip elseif isprime(n) == 0 j = j + 1; factors = factor(n); lowfactor = factors(1,1); a(j,1) = n/lowfactor; end n = n + 1; end - Kyle Stern, May 04 2009

f[n_] := Block[{k = n + PrimePi@ n + 1}, While[k != n + PrimePi@ k + 1, k++ ]; k/FactorInteger[k][[1, 1]]]; Array[f, 75] (* Robert G. Wilson v, May 11 2012 *)
Divisors[#][[-2]]&/@Select[Range[200],CompositeQ] (* Harvey P. Dale, Dec 06 2021 *)
(# / FactorInteger[#][[1, 1]])& /@ Select[Range[300], CompositeQ] (* Amiram Eldar, Jun 18 2022 *)

a(n) = A032742(A002808(n)) = A002808(n) / A056608(n) = A163870(n,A144925(n)). - Reinhard Zumkeller, Mar 29 2014

Indices of b-file corrected, more terms added using b-file. - N. J. A. Sloane, Aug 31 2009
New name from Reinhard Zumkeller, Mar 29 2014
Incorrect formula removed by Ridouane Oudra, Oct 15 2021

A081619 Numbers whose divisors can be arranged as equilateral triangle.

Original entry on oeis.org

1, 4, 9, 12, 18, 20, 25, 28, 32, 44, 45, 48, 49, 50, 52, 63, 68, 75, 76, 80, 92, 98, 99, 112, 116, 117, 121, 124, 144, 147, 148, 153, 162, 164, 169, 171, 172, 175, 176, 188, 207, 208, 212, 236, 242, 243, 244, 245, 261, 268, 272, 275, 279, 284, 289, 292, 304, 316

Offset: 1

Reinhard Zumkeller, Mar 24 2003

nonn

A000005(a(n))=A000217(m) for some m; A000005(A081620(n))=n*(n+1)/2.
Unit together with natural numbers n with number of nontrivial divisors equal to a perfect power. - Juri-Stepan Gerasimov, Oct 30 2009
This is wrong, e.g. a(29)=144, A000005(144)=15 and A075802(15-2)=0, see also example. - Reinhard Zumkeller, Jul 12 2013

			n = 48, A000005(48) = 10, A010054(10) = 1 or A000217(4) = 10:
.            1
.          2   3
.        4   6   8
.     12  16  24  48     therefore 48 is a term: a(12) = 48;
n = 144, A000005(144) = 15, A010054(15) = 1 or A000217(5) = 15:
.             1
.           2   3
.         4   6   8
.       9  12  16  18
.    24  36  48  72  144    therefore 48 is a term: a(29) = 144.

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

Cf. A000027, A001694, A081619, A144925.

a081619 n = a081619_list !! (n-1)
a081619_list = filter ((== 1) . a010054 . a000005) [1..]
-- Reinhard Zumkeller, Jul 12 2013

is(n)=ispolygonal(numdiv(n),3) \\ Charles R Greathouse IV, Oct 16 2015

A010054(A000005(n)) = 1. - Reinhard Zumkeller, Jul 12 2013

Example revised and extended by Reinhard Zumkeller, Jul 12 2013

A259981 Let b be the n-th composite number, A002808(n); a(n) is number of base-b digits x,y,z such that (xb+y)/(zb+x)=y/z.

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 2, 6, 7, 4, 4, 10, 6, 6, 6, 4, 6, 10, 6, 4, 8, 6, 6, 21, 2, 6, 18, 6, 4, 18, 10, 8, 10, 10, 12, 12, 6, 16, 22, 14, 6, 10, 2, 12, 21, 12, 20, 4, 10, 22, 10, 2, 12, 20, 14, 24, 8, 24, 8, 10, 28, 6, 6, 18, 10, 28, 16, 10, 6, 6, 30, 4, 24, 37, 6, 6, 46, 14, 10, 6, 18, 24, 6, 18

Offset: 1

N. J. A. Sloane, Jul 12 2015, following a suggestion from R. P. Boas, May 19 1974

nonn

R. P. Boas writes (slightly edited): The problem originated in the rather silly observation that 64/16 = 4/1 ("cancel" the 6's). I once asked what happens in base b, i.e., when is (xb+y)/(zb+x) = y/z? There are no nontrivial instances of the cancellation phenomenon when b is prime, so we restrict b to A002808; the sequence gives the number of instances of the phenomenon for each composite b. When b-1 is prime the only instances have x=b-1 and the number of them is the number of proper divisors of b (see A144925). A259983 is the subsequence of this sequence corresponding to bases b in A005381.

Chai Wah Wu, Table of n, a(n) for n = 1..1000
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
Eric Weisstein's World of Mathematics, Anomalous Cancellation

Cf. A002808, A005381, A144925, A259983.

Table[Count[Flatten[Table[(x b + y) z == y (z b + x), {x, b}, {y, b}, {z, y - 1}], 2], True], {b, Select[Range[115], CompositeQ]}] (* Eric W. Weisstein, Oct 16 2015 *)

from sympy import primepi
def A002808(n):
    m = n
    while m != primepi(m) + 1 + n:
        m += 1
    return m
def A259981(n):
    b, c = A002808(n), 0
    for x in range(1,b):
        for y in range(1,b):
            if x != y:
                w = b*(x-y)
                for z in range(1,b):
                    if x != z:
                        if z*w == y*(x-z):
                            c += 1
    return c # Chai Wah Wu, Jul 15 2015

Typo in a(61) corrected by Chai Wah Wu, Jul 15 2015

A163838 a(n) = (n-th composite) * (number of nontrivial divisors of n-th composite).

Original entry on oeis.org

4, 12, 16, 9, 20, 48, 28, 30, 48, 72, 80, 42, 44, 144, 25, 52, 54, 112, 180, 128, 66, 68, 70, 252, 76, 78, 240, 252, 176, 180, 92, 384, 49, 200, 102, 208, 324, 110, 336, 114, 116, 600, 124, 252, 320, 130, 396, 272, 138, 420, 720, 148, 300, 304, 154, 468, 640, 243

Offset: 1

Juri-Stepan Gerasimov, Aug 05 2009, Aug 06 2009

nonn

The trivial divisors of the n-th composite are 1 and the n-th composite.

			a(1) =  4 (= 4*1);
a(2) = 12 (= 6*2);
a(3) = 16 (= 8*2);
a(4) =  9 (= 9*1);
a(5) = 20 (= 10*2);
a(6) = 48 (= 12*4).

Cf. A002808, A070824, A144925.

A002808 := proc(n) option remember; local a; if n = 1 then 4; else for a from procname(n-1)+1 do if not isprime(a) then return a; end if; end do: end if; end proc: A070824 := proc(n) numtheory[tau](n)-2 ; end: A144925 := proc(n) A070824(A002808(n)) ; end: A163838 := proc(n) A144925(n)*A002808(n) ; end: seq(A163838(n),n=1..80) ; # R. J. Mathar, Oct 10 2009

# (DivisorSigma[0,#]-2)&/@Select[Range[100],CompositeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 23 2021 *)

a(n) = A002808(n)*A144925(n).

56 replaced with 112, 310 replaced with 320, and 468 inserted by R. J. Mathar, Oct 10 2009

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],","))

A228372 Number of nontrivial divisors in the first n composites.

Original entry on oeis.org

1, 3, 5, 6, 8, 12, 14, 16, 19, 23, 27, 29, 31, 37, 38, 40, 42, 46, 52, 56, 58, 60, 62, 69, 71, 73, 79, 85, 89, 93, 95, 103, 104, 108, 110, 114, 120, 122, 128, 130, 132, 142, 144, 148, 153, 155, 161, 165, 167, 173, 183, 185, 189, 193, 195, 201, 209, 212, 214, 224, 226

Offset: 1

Ralf Stephan, Aug 21 2013

nonn

Also, positions where values change in A228363.

Partial sums of A144925.

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Accumulate[DivisorSigma[0,#]-2&/@Select[Range[100],CompositeQ]] (* Harvey P. Dale, Nov 10 2017 *)

L=listcreate();for(n=1,1000,if(!isprime(n),listput(L,n)));co=Vec(L);s=0;for(n=2,100,s=s+numdiv(co[n])-2;print1(s,","))

cp's OEIS Frontend

A167131 Numbers k such that A002808(k) - A144925(k) is prime.

Views

Author

Keywords

Crossrefs

Programs

Maple

Extensions

A163870 Triangle read by rows: row n lists the nontrivial divisors of the n-th composite.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Python

Extensions

A160180 Largest proper divisor of the n-th composite number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

MATLAB

Mathematica

Formula

Extensions

A081619 Numbers whose divisors can be arranged as equilateral triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

PARI

Formula

Extensions

A259981 Let b be the n-th composite number, A002808(n); a(n) is number of base-b digits x,y,z such that (xb+y)/(zb+x)=y/z.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A163838 a(n) = (n-th composite) * (number of nontrivial divisors of n-th composite).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI