A001749 -id:A001749 - cp's OEIS frontend

A239929 Numbers n with the property that the symmetric representation of sigma(n) has two parts.

3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 67, 68, 71, 73, 74, 76, 78, 79, 82, 83, 86, 89, 92, 94, 97, 101, 102, 103, 106, 107, 109, 113, 114, 116, 118, 122, 124, 127, 131, 134, 136, 137, 138

Offset: 1

Omar E. Pol, Apr 06 2014

nonn

All odd primes are in the sequence because the parts of the symmetric representation of sigma(prime(i)) are [m, m], where m = (1 + prime(i))/2, for i >= 2.
There are no odd composite numbers in this sequence.
First differs from A173708 at a(13).
Since sigma(p*q) >= 1 + p + q + p*q for odd p and q, the symmetric representation of sigma(p*q) has more parts than the two extremal ones of size (p*q + 1)/2; therefore, the above comments are true. - Hartmut F. W. Hoft, Jul 16 2014
From Hartmut F. W. Hoft, Sep 16 2015: (Start)
The following two statements are equivalent:
  (1) The symmetric representation of sigma(n) has two parts, and
  (2) n = q * p where q is in A174973, p is prime, and 2 * q < p.
For a proof see the link and also the link in A071561.
This characterization allows for much faster computation of numbers in the sequence - function a239929F[] in the Mathematica section - than computations based on Dyck paths. The function a239929Stalk[] gives rise to the associated irregular triangle whose columns are indexed by A174973 and whose rows are indexed by A065091, the odd primes. (End)
From Hartmut F. W. Hoft, Dec 06 2016: (Start)
For the respective columns of the irregular triangle with fixed m: k = 2^m * p, m >= 1, 2^(m+1) < p and p prime:
(a) each number k is representable as the sum of 2^(m+1) but no fewer consecutive positive integers [since 2^(m+1) < p].
(b) each number k has 2^m as largest divisor <= sqrt(k) [since 2^m < sqrt(k) < p].
(c) each number k is of the form 2^m * p with p prime [by definition].
m = 1: (a) A100484 even semiprimes (except 4 and 6)
       (b) A161344 (except 4, 6 and 8)
       (c) A001747 (except 2, 4 and 6)
m = 2: (a) A270298
       (b) A161424 (except 16, 20, 24, 28 and 32)
       (c) A001749 (except 8, 12, 20 and 28)
m = 3: (a) A270301
       (b) A162528 (except 64, 72, 80, 88, 96, 104, 112 and 128)
       (c) sequence not in OEIS
b(i,j) = A174973(j) * {1,5) mod 6 * A174973(j), for all i,j >= 1; see A091999 for j=2. (End)

			From _Hartmut F. W. Hoft_, Sep 16 2015: (Start)
a(23) = 52 = 2^2 * 13 = q * p with q = 4 in A174973 and 8 < 13 = p.
a(59) = 136 = 2^3 * 17 = q * p with q = 8 in A174973 and 16 < 17 = p.
The first six columns of the irregular triangle through prime 37:
   1    2    4    6    8   12 ...
  -------------------------------
   3
   5   10
   7   14
  11   22   44
  13   26   52   78
  17   34   68  102  136
  19   38   76  114  152
  23   46   92  138  184
  29   58  116  174  232  348
  31   62  124  186  248  372
  37   74  148  222  296  444
  ...
(End)

Hartmut F. W. Hoft, Proof of Characterization Theorem

Column 2 of A240062.
Cf. A000203, A006254, A065091, A071561, A174973, A196020, A236104, A235791, A237591, A237593, A237270, A237271, A238443, A239660, A239663, A239665, A239931-A239934, A244050, A245062, A262626.
Cf. A000040, A001747, A001749, A091999, A100484, A161344, A161424, A162528, A270298, A270301. - Hartmut F. W. Hoft, Dec 06 2016

isA174973 := proc(n)
    option remember;
    local k,dvs;
    dvs := sort(convert(numtheory[divisors](n),list)) ;
    for k from 2 to nops(dvs) do
        if op(k,dvs) > 2*op(k-1,dvs) then
            return false;
        end if;
    end do:
    true ;
end proc:
A174973 := proc(n)
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA174973(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
isA239929 := proc(n)
    local i,p,j,a73;
    for i from 1 do
        p := ithprime(i+1) ;
        if p > n then
            return false;
        end if;
        for j from 1 do
            a73 := A174973(j) ;
            if a73 > n then
                break;
            end if;
            if p > 2*a73 and n = p*a73 then
                return true;
            end if;
        end do:
    end do:
end proc:
for n from 1 to 200 do
    if isA239929(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Oct 04 2018

(* sequence of numbers k for m <= k <= n having exactly two parts *)
(* Function a237270[] is defined in A237270 *)
a239929[m_, n_]:=Select[Range[m, n], Length[a237270[#]]==2&]
a239929[1, 260] (* data *)
(* Hartmut F. W. Hoft, Jul 07 2014 *)
(* test for membership in A174973 *)
a174973Q[n_]:=Module[{d=Divisors[n]}, Select[Rest[d] - 2 Most[d], #>0&]=={}]
a174973[n_]:=Select[Range[n], a174973Q]
(* compute numbers satisfying the condition *)
a239929Stalk[start_, bound_]:=Module[{p=NextPrime[2 start], list={}}, While[start p<=bound, AppendTo[list, start p]; p=NextPrime[p]]; list]
a239929F[n_]:=Sort[Flatten[Map[a239929Stalk[#, n]&, a174973[n]]]]
a239929F[138] (* data *)(* Hartmut F. W. Hoft, Sep 16 2015 *)

Entries b(i, j) in the irregular triangle with rows indexed by i>=1 and columns indexed by j>=1 (alternate indexing of the example):

b(i,j) = A000040(i+1) * A174973(j) where A000040(i+1) > 2 * A174973(j). - Hartmut F. W. Hoft, Dec 06 2016

Extended beyond a(56) by Michel Marcus, Apr 07 2014

A246955 Numbers j for which the symmetric representation of sigma(j) has two parts, each of width one.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 67, 68, 71, 73, 74, 76, 79, 82, 83, 86, 89, 92, 94, 97, 101, 103, 106, 107, 109, 113, 116, 118, 122, 124, 127, 131, 134, 136, 137, 139, 142, 146, 148, 149, 151, 152, 157, 158, 163, 164, 166, 167, 172, 173, 178, 179, 181, 184, 188, 191, 193, 194, 197, 199

Offset: 1

Hartmut F. W. Hoft, Sep 08 2014

nonn

The sequence is the intersection of A239929 (sigma(j) has two parts) and of A241008 (sigma(j) has an even number of parts of width one).
The numbers in the sequence are precisely those defined by the formula for the triangle, see the link. The symmetric representation of sigma(j) has two parts, each part having width one, precisely when j = 2^(k - 1) * p where 2^k <= row(j) < p, p is prime and row(j) = floor((sqrt(8*j + 1) - 1)/2). Therefore, the sequence can be written naturally as a triangle as shown in the Example section.
The symmetric representation of sigma(j) = 2*j - 2 consists of two regions of width 1 that meet on the diagonal precisely when j = 2^(2^m - 1)*(2^(2^m) + 1) where 2^(2^m) + 1 is a Fermat prime (see A019434). This subsequence of numbers j is 3, 10, 136, 32896, 2147516416, ...[?]... (A191363).
The k-th column of the triangle starts in the row whose initial entry is the first prime larger than 2^(k+1) (that sequence of primes is A014210, except for 2).
Observation: at least the first 82 terms coincide with the numbers j with no middle divisors whose largest divisor <= sqrt(j) is a power of 2, or in other words, coincide with the intersection of A071561 and A365406. - Omar E. Pol, Oct 11 2023

			We show portions of the first eight columns, 0 <= k <= 7, of the triangle.
0    1    2     3     4     5     6     7
3
5    10
7    14
11   22   44
13   26   52
17   34   68    136
19   38   76    152
23   46   92    184
29   58   116   232
31   62   124   248
37   74   148   296   592
41   82   164   328   656
43   86   172   344   688
47   94   188   376   752
53   106  212   424   848
59   118  236   472   944
61   122  244   488   976
67   134  268   536   1072  2144
71   142  284   568   1136  2272
.    .    .     .     .     .
.    .    .     .     .     .
127  254  508   1016  2032  4064
131  262  524   1048  2096  4192  8384
137  274  548   1096  2192  4384  8768
.    .    .     .     .     .     .
.    .    .     .     .     .     .
251  502  1004  2008  4016  8032  16064
257  514  1028  2056  4112  8224  16448  32896
263  526  1052  2104  4208  8416  16832  33664
Since 2^(2^4) + 1 = 65537 is the 6543rd prime, column k = 15 starts with 2^15*(2^(2^16) + 1) = 2147516416 in row 6542 with 65537 in column k = 0.
For an image of the symmetric representations of sigma(m) for all values m <= 137 in the triangle see the link.
The first column is the sequence of odd primes, see A065091.
The second column is the sequence of twice the primes starting with 10, see A001747.
The third column is the sequence of four times the primes starting with 44, see A001749.
For related references also see A033676 (largest divisor of n less than or equal to sqrt(n)).

Hartmut F. W. Hoft, Equivalence proof of sequence and triangle
Hartmut F. W. Hoft, Image for sigma(n) values

Cf. A000203, A033676, A071561, A163280, A237270, A237271, A237593, A241008, A241010, A247687, A250068, A250070, A250071, A365406.

(* functions path[] and a237270[ ] are defined in A237270 *)
atmostOneDiagonalsQ[n_]:=SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], - 1] - path[n - 1], 1]]]
(* data *)
Select[Range[200], Length[a237270[#]]==2 && atmostOneDiagonalsQ[#]&]
(* function for computing triangle in the Example section through row 55 *)
TableForm[Table[2^k Prime[n], {n, 2, 56}, {k, 0, Floor[Log[2, Prime[n]] - 1]}], TableDepth->2]

Formula for the triangle of numbers associated with the sequence:

P(n, k) = 2^k * prime(n) where n >= 2, 0 <= k <= floor(log_2(prime(n)) - 1).

A138636 a(n) = 6 * prime(n).

Original entry on oeis.org

12, 18, 30, 42, 66, 78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362, 1374

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 14 2008

Column 5 of A272214. - Omar E. Pol, Apr 29 2016

			2*6=12, 3*6=18, ...

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

Cf. A000040, A001748, A001749, A001750, A100484.

[6*p: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

Mathematica
```
6*Prime[Range[100]]
```

vector(50, n, 6*prime(n)) \\ G. C. Greubel, Feb 02 2019

[6*nth_prime(n) for n in (1..50)] # G. C. Greubel, Feb 02 2019

A264828 Nonprimes that are not twice a prime.

Original entry on oeis.org

1, 8, 9, 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 81, 84, 85, 87, 88, 90, 91, 92, 93, 95, 96, 98, 99, 100, 102, 104

Offset: 1

Giovanni Teofilatto, Nov 26 2015

Except for the initial 1, if n is in the sequence, so is k*n for all k > 1. So the odd semiprimes (A046315) and numbers of the form 4*p (A001749) where p is prime are core subsequences which give the initial terms of arithmetic progressions in this sequence. - Altug Alkan, Nov 29 2015

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [Terms 1 to 10000 from _Robert Israel_.]

Cf. A009188, A018252, A100484.

Primes, Nonprimes:= selectremove(isprime, {$1..1000}):
sort(convert(Nonprimes minus map(`*`,Primes,2),list)); # Robert Israel, Nov 30 2015

Select[Range@ 104, And[! PrimeQ@ #, Or[PrimeOmega@ # != 2, OddQ@ #]] &] (* Michael De Vlieger, Nov 27 2015 *)
Select[Range@110, Nor[PrimeQ[#], PrimeQ[#/2]] &] (* Vincenzo Librandi, Jan 22 2016 *)

print1(1, ", "); forcomposite(n=1, 1e3, if(n % 2 == 1 || !isprime(n/2), print1(n, ", "))) \\ Altug Alkan, Dec 01 2015

from itertools import count, islice
from sympy import isprime
def A264828_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not (isprime(n) or (n&1^1 and isprime(n>>1))),count(max(startvalue,1)))
A264828_list = list(islice(A264828_gen(),20)) # Chai Wah Wu, Mar 26 2024

from sympy import primepi
def A264828(n):
    def f(x): return int(n+primepi(x)+primepi(x>>1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 17 2024

a(n) = A009188(n-2) for n>=3. - Alois P. Heinz, Oct 17 2024

A377872 Numbers k for which A276085(k) is a multiple of 27, where A276085 is fully additive with a(p) = p#/p.

Original entry on oeis.org

1, 55, 95, 115, 155, 174, 187, 203, 232, 265, 282, 297, 323, 325, 329, 335, 376, 391, 396, 438, 462, 474, 511, 513, 515, 527, 528, 539, 553, 584, 606, 616, 621, 632, 649, 654, 678, 684, 704, 707, 745, 763, 791, 798, 808, 828, 837, 872, 901, 904, 906, 912, 913, 931, 966, 978, 1002, 1057, 1064, 1073, 1074, 1075, 1104, 1105

Offset: 1

Antti Karttunen, Nov 10 2024

nonn

A multiplicative semigroup; if m and n are in the sequence then so is m*n.
From Antti Karttunen, Nov 17 2024: (Start)
Question: What is the asymptotic density of this sequence? There are 1, 3, 56, 484, 4899, 50034, 508254 terms <= 10^k, for k=1..7. See also questions in A377869 and in A377878.
If 3*x is a term, then 4*x is also a term, and vice versa.
Contains no even semiprimes (A100484), semiprimes of the form 3*prime (A001748), nor terms of the form 4*prime (A001749).
(End)

Antti Karttunen, Table of n, a(n) for n = 1..12000
Index entries for sequences related to primorial numbers

Cf. A276085, A377869, A377876, A377878.
Subsequence of A339746, and of A377873.
Cf. also A369007, A377875.

isA377872(n) = { my(m=27, f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= Mod(prime(i),m); i++); s += f[k, 2]*pr); (0==lift(s)); };

{k such that Sum e*A377876(A000720(p)-1) == 0 (mod 27), when k = Product(p^e)}.

A340786 Number of factorizations of 4n into an even number of even factors > 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 6, 1, 3, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 4, 1, 7, 2, 2, 2, 7, 1, 2, 2, 6, 1, 4, 1, 4, 3, 2, 1, 10, 2, 3, 2, 4, 1, 4, 2, 6, 2, 2, 1, 8, 1, 2, 3, 12, 2, 4, 1, 4, 2, 4, 1, 10, 1, 2, 3, 4, 2, 4, 1, 10, 3, 2, 1, 8, 2, 2

Offset: 1

Gus Wiseman, Jan 31 2021

nonn

			The a(n) factorizations for n = 6, 12, 24, 36, 60, 80, 500:
  4*6   6*8      2*48      2*72      4*60      4*80          40*50
  2*12  2*24     4*24      4*36      6*40      8*40          4*500
        4*12     6*16      6*24      8*30      10*32         8*250
        2*2*2*6  8*12      8*18      10*24     16*20         10*200
                 2*2*4*6   12*12     12*20     2*160         20*100
                 2*2*2*12  2*2*6*6   2*120     2*2*2*40      2*1000
                           2*2*2*18  2*2*2*30  2*2*4*20      2*2*10*50
                                     2*2*6*10  2*2*8*10      2*2*2*250
                                               2*4*4*10      2*10*10*10
                                               2*2*2*2*2*10

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

Note: A-numbers of Heinz-number sequences are in parentheses below.
Positions of ones are 1 and A000040, or A008578.
A version for partitions is A027187 (A028260).
Allowing odd length gives A108501 (bisection of A340785).
Allowing odd factors gives A339846.
An odd version is A340102.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors.
A340653 counts balanced factorizations.
A340831/A340832 count factorizations with odd maximum/minimum.
- Even -
A027187 counts partitions of even maximum (A244990).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts partitions of even length and sum (A340784).
A340601 counts partitions of even rank (A340602).
Cf. A001147, A001222, A001749, A035363, A050320, A066207, A066208, A160786, A174725, A320655, A320656, A339890, A340851.

g:= proc(n, m, p)
 option remember;
 local F,r,x,i;
 # number of factorizations of n into even factors > m with number of factors == p (mod 2)
 if n = 1 then if p = 0 then return 1 else return 0 fi fi;
 if m > n  or n::odd then return 0 fi;
 F:= sort(convert(select(t -> t > m and t::even, numtheory:-divisors(n)),list));
 r:= 0;
 for x in F do
   for i from 1 while n mod x^i = 0 do
     r:= r + procname(n/x^i, x, (p+i) mod 2)
 od od;
 r
end proc:
f:= n -> g(4*n, 1, 0):
map(f, [$1..100]); # Robert Israel, Mar 16 2023

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[4n],EvenQ[Length[#]]&&Select[#,OddQ]=={}&]],{n,100}]

A340786aux(n, m=n, p=0) = if(1==n, (0==p), my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&!(d%2), s += A340786aux(n/d, d, 1-p))); (s));
A340786(n) = A340786aux(4*n); \\ Antti Karttunen, Apr 14 2022

A171565 Number of partitions of n into odd divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 1, 5, 3, 2, 5, 2, 3, 14, 1, 2, 12, 2, 5, 18, 3, 2, 9, 7, 3, 23, 5, 2, 54, 2, 1, 26, 3, 26, 35, 2, 3, 30, 9, 2, 72, 2, 5, 286, 3, 2, 17, 9, 18, 38, 5, 2, 93, 38, 9, 42, 3, 2, 275, 2, 3, 493, 1, 44, 108, 2, 5, 50, 110, 2, 117, 2, 3, 698, 5, 50, 126, 2, 17, 239, 3, 2, 375, 56

Offset: 0

Reinhard Zumkeller, Dec 11 2009

nonn

a(2*n+1) = A018818(2*n+1), a(A005408(n))=A018818(A005408(n));
a(2^k) = 1, a(A000079(n))=1;
for odd primes p: a(p*2^k) = 2^k + 1,
especially for n>1: a(A000040(n))=2, a(A100484(n))=3, a(A001749(n))=5.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A038550, A065091.

with(numtheory):
a:= proc(n) option remember; local b, l; l, b:= sort(
      [select(x-> is(x:: odd), divisors(n))[]]),
      proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
        b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
      end; b(n, nops(l))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 30 2017

a[0] = 1; a[n_] := a[n] = Module[{b, l}, l = Select[Divisors[n], OddQ]; b[m_, i_] := b[m, i] = If[m == 0, 1, If[i < 1, 0, b[m, i-1] + If[l[[i]] > m, 0, b[m - l[[i]], i]]]]; b[n, Length[l]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 11 2017, after Alois P. Heinz *)

a(n) = f(n,n,1) with f(n,m,k) = if k<=m then f(n,m,k+2)+f(n,m-k,k)*0^(n mod k) else 0^m.

A335738 Factorize each integer m >= 2 as the product of powers of nonunit squarefree numbers with distinct exponents that are powers of 2. The sequence lists m such that the factor with the largest exponent is a power of 2.

Original entry on oeis.org

2, 4, 8, 12, 16, 20, 24, 28, 32, 40, 44, 48, 52, 56, 60, 64, 68, 76, 80, 84, 88, 92, 96, 104, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 184, 188, 192, 204, 208, 212, 220, 224, 228, 232, 236, 240, 244, 248, 256, 260, 264, 268, 272

Offset: 1

Peter Munn, Jun 20 2020

nonn

2 is the only term not divisible by 4. All powers of 2 are present. Every term divisible by an odd square is divisible by 16, the first such being 144.
The defined factorization is unique. Every positive number is a product of at most one squarefree number (A005117), at most one square of a squarefree number (A062503), at most one 4th power of a squarefree number (A113849), at most one 8th power of a squarefree number, and so on.
Iteratively map m using A000188, until 1 is reached, as A000188^k(m), for some k >= 1. m is in the sequence if and only if the preceding number, A000188^(k-1)(m), is 2. k can be shown to be A299090(m).
Closed under squaring, but not closed under multiplication: 12 = 3^1 * 2^2 and 432 = 3^1 * 3^2 * 2^4 are in the sequence, but 12 * 432 = 5184 = 3^4 * 2^6 = 2^2 * 6^4 is not.
The asymptotic density of this sequence is Sum_{k>=0} (d(2^(k+1)) - d(2^k))/2^(2^(k+1)-1) = 0.21363357193921052068..., where d(k) = 2^(k-1)/((2^k-1)*zeta(k)) is the asymptotic density of odd k-free numbers for k >= 2, and d(1) = 0. - Amiram Eldar, Feb 10 2024

			6 is a squarefree number, so its factorization for the definition (into powers of nonunit squarefree numbers with distinct exponents that are powers of 2) is the trivial "6^1". 6^1 is therefore the factor with the largest exponent, and is not a power of 2, so 6 is not in the sequence.
48 factorizes for the definition as 3^1 * 2^4. The factor with the largest exponent is 2^4, which is a power of 2, so 48 is in the sequence.
10^100 (a googol) factorizes in this way as 10^4 * 10^32 * 10^64. The factor with the largest exponent, 10^64, is a power of 10, not a power of 2, so 10^100 is not in the sequence.

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

Complement within A020725 of A335740.
A000188, A299090 are used in a formula defining this sequence.
Powers of squarefree numbers: A005117(1), A144338(1), A062503(2), A113849(4).
Subsequences: A000079\{1}, A001749, A181818\{1}, A273798.
Numbers in the even bisection of A336322.
Row m of A352780 essentially gives the defined factorization of m.

f[p_, e_] := p^Floor[e/2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[2, 300], FixedPointList[s, #] [[-3]] == 2 &] (* Amiram Eldar, Nov 27 2020 *)

is(n) = {my(e = valuation(n, 2), o = n >> e); if(e == 0, 0, if(o == 1, n > 1, floor(logint(e, 2)) > floor(logint(vecmax(factor(o)[,2]), 2))));} \\ Amiram Eldar, Feb 10 2024

{a(n)} = {m : m >= 2 and A000188^(k-1)(m) = 2, where k = A299090(m)}.

{a(n)} = {m : m >= 2 and A352780(m,e) = 2^(2^e), where e = A299090(m)-1}. - Peter Munn, Jun 24 2022

A053661 For n > 1: if n is present, 2n is not.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 12, 13, 15, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 57, 59, 60, 61, 63, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 89, 91, 92, 93, 95, 97, 99, 100, 101, 103, 105

Offset: 1

Jeevan Chana Rai (Karanjit.Rai(AT)btinternet.com), Feb 16 2000

The Name line gives a property of the sequence, not a definition. The sequence can be defined simultaneously with b(n) := A171945(n) via a(n) = mex{a(i), b(i) : 0 <= i < n} (n >= 0}, b(n)=2a(n). The two sequences are complementary, hence A053661 is identical to A171944 (except for the first terms). Furthmore, A053661 is the same as A003159 except for the replacement of vile by dopey powers of 2. - Aviezri S. Fraenkel, Apr 28 2011
For n >= 2, either n = 2^k where k is odd or n = 2^k*m where m > 1 is odd and k is even (found by Kirk Bresniker and Stan Wagon). [Robert Israel, Oct 10 2010]
Subsequence of A175880; A000040, A001749, A002001, A002042, A002063, A002089, A003947, A004171 and A081294 are subsequences.

Robert Israel, Table of n, a(n) for n = 1..10000
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.

Essentially identical to A171944 and the complement of A171945.

a053661 n = a053661_list !! (n-1)
a053661_list = filter (> 0) a175880_list -- Reinhard Zumkeller, Feb 09 2011

N:= 1000: # to get all terms <= N
sort([1,seq(2^(2*i+1),i=0..(ilog2(N)-1)/2), seq(seq(2^(2*i)*(2*j+1),j=1..(N/2^(2*i)-1)/2),i=0..ilog2(N)/2)]); # Robert Israel, Jul 24 2015

Clear[T]; nn = 105; T[n_, k_] := T[n, k] = If[n < 1 || k < 1, 0, If[n == 1 || k == 1, 1, If[k > n, T[k, n], If[n > k, T[k, Mod[n, k, 1]], -Product[T[n, i], {i, n - 1}]]]]]; DeleteCases[Table[If[T[n, n] == -1, n, ""], {n, 1, nn}], ""] (* Mats Granvik, Aug 25 2012 *)

More terms from James Sellers, Feb 22 2000

A272214 Square array read by antidiagonals upwards in which T(n,k) is the product of the n-th prime and the sum of the divisors of k, n >= 1, k >= 1.

Original entry on oeis.org

2, 3, 6, 5, 9, 8, 7, 15, 12, 14, 11, 21, 20, 21, 12, 13, 33, 28, 35, 18, 24, 17, 39, 44, 49, 30, 36, 16, 19, 51, 52, 77, 42, 60, 24, 30, 23, 57, 68, 91, 66, 84, 40, 45, 26, 29, 69, 76, 119, 78, 132, 56, 75, 39, 36, 31, 87, 92, 133, 102, 156, 88, 105, 65, 54, 24, 37, 93, 116, 161, 114, 204, 104, 165, 91, 90, 36, 56

Offset: 1

Omar E. Pol, Apr 28 2016

From Omar E. Pol, Dec 21 2021: (Start)
Also triangle read by rows: T(n,j) = A000040(n-j+1)*A000203(j), 1 <= j <= n.
For a visualization of T(n,j) first consider a tower (a polycube) in which the terraces are the symmetric representation of sigma(j), for j = 1 to n, starting from the top, and the heights of the terraces are the first n prime numbers respectively starting from the base. Then T(n,j) can be represented with a set of A237271(j) right prisms of height A000040(n-j+1) since T(n,j) is also the total number of cubes that are exactly below the parts of the symmetric representation of sigma(j) in the tower.
The sum of the n-th row of triangle is A086718(n) equaling the volume of the tower whose largest side of the base is n and its total height is the n-th prime.
The tower is an member of the family of the stepped pyramids described in A245092 and of the towers described in A221529. That is an infinite family of symmetric polycubes whose volumes represent the convolution of A000203 with any other integer sequence. (End)

			The corner of the square array begins:
   2,  6,   8,  14,  12,  24,  16,  30,  26,  36, ...
   3,  9,  12,  21,  18,  36,  24,  45,  39,  54, ...
   5, 15,  20,  35,  30,  60,  40,  75,  65,  90, ...
   7, 21,  28,  49,  42,  84,  56, 105,  91, 126, ...
  11, 33,  44,  77,  66, 132,  88, 165, 143, 198, ...
  13, 39,  52,  91,  78, 156, 104, 195, 169, 234, ...
  17, 51,  68, 119, 102, 204, 136, 255, 221, 306, ...
  19, 57,  76, 133, 114, 228, 152, 285, 247, 342, ...
  23, 69,  92, 161, 138, 276, 184, 345, 299, 414, ...
  29, 87, 116, 203, 174, 348, 232, 435, 377, 522, ...
  ...
From _Omar E. Pol_, Dec 21 2021: (Start)
Written as a triangle the sequence begins:
   2;
   3,  6;
   5,  9,  8;
   7, 15, 12,  14;
  11, 21, 20,  21,  12;
  13, 33, 28,  35,  18,  24;
  17, 39, 44,  49,  30,  36, 16;
  19, 51, 52,  77,  42,  60, 24,  30;
  23, 57, 68,  91,  66,  84, 40,  45, 26;
  29, 69, 76, 119,  78, 132, 56,  75, 39, 36;
  31, 87, 92, 133, 102, 156, 88, 105, 65, 54, 24;
...
Row sums give A086718. (End)

Ivan Neretin, Table of n, a(n) for n = 1..8128

Rows 1-4 of the square array: A074400, A272027, A274535, A319527.
Columns 1-5 of the square array: A000040, A001748, A001749, A138636, A272470.
Main diagonal of the square array gives A272211.
Cf. A086718 (antidiagonal sums of the square array, row sums of the triangle).
Cf. A000203, A221529, A237270, A237271, A237593, A245092, A272173, A272400, A274824.

Table[Prime[#] DivisorSigma[1, k] &@(n - k + 1), {n, 12}, {k, n}] // Flatten (* Michael De Vlieger, Apr 28 2016 *)

T(n,k) = prime(n)*sigma(k) = A000040(n)*A000203(k), n >= 1, k >= 1.

T(n,k) = A272400(n+1,k).

cp's OEIS Frontend

A239929 Numbers n with the property that the symmetric representation of sigma(n) has two parts.

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A246955 Numbers j for which the symmetric representation of sigma(j) has two parts, each of width one.

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A138636 a(n) = 6 * prime(n).

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A264828 Nonprimes that are not twice a prime.

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A377872 Numbers k for which A276085(k) is a multiple of 27, where A276085 is fully additive with a(p) = p#/p.

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A340786 Number of factorizations of 4n into an even number of even factors > 1.

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A171565 Number of partitions of n into odd divisors of n.

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