A139257 -id:A139257 - cp's OEIS frontend

A262626 Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.

1, 1, 1, 3, 2, 2, 2, 2, 2, 1, 1, 2, 7, 3, 1, 1, 3, 3, 3, 3, 2, 2, 3, 12, 4, 1, 1, 1, 1, 4, 4, 4, 4, 2, 1, 1, 2, 4, 15, 5, 2, 1, 1, 2, 5, 5, 3, 5, 5, 2, 2, 2, 2, 5, 9, 9, 6, 2, 1, 1, 1, 1, 2, 6, 6, 6, 6, 3, 1, 1, 1, 1, 3, 6, 28, 7, 2, 2, 1, 1, 2, 2, 7, 7, 7, 7, 3, 2, 1, 1, 2, 3, 7, 12, 12, 8, 3, 1, 2, 2, 1, 3, 8, 8, 8, 8, 8, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Sep 26 2015

Also the rows of both triangles A237270 and A237593 interleaved.
Also, irregular triangle read by rows in which T(n,k) is the area of the k-th region (from left to right in ascending diagonal) of the n-th symmetric set of regions (from the top to the bottom in descending diagonal) in the two-dimensional diagram of the perspective view of the infinite stepped pyramid described in A245092 (see the diagram in the Links section).
The diagram of the symmetric representation of sigma is also the top view of the pyramid, see Links section. For more information about the diagram see also A237593 and A237270.
The number of cubes at the n-th level is also A024916(n), the sum of all divisors of all positive integers <= n.
Note that this pyramid is also a quarter of the pyramid described in A244050. Both pyramids have infinitely many levels.
Odd-indexed rows are also the rows of the irregular triangle A237270.
Even-indexed rows are also the rows of the triangle A237593.
Lengths of the odd-indexed rows are in A237271.
Lengths of the even-indexed rows give 2*A003056.
Row sums of the odd-indexed rows gives A000203, the sum of divisors function.
Row sums of the even-indexed rows give the positive even numbers (see A005843).
Row sums give A245092.
From the front view of the stepped pyramid emerges a geometric pattern which is related to A001227, the number of odd divisors of the positive integers.
The connection with the odd divisors of the positive integers is as follows: A261697 --> A261699 --> A237048 --> A235791 --> A237591 --> A237593 --> A237270 --> this sequence.

			Irregular triangle begins:
  1;
  1, 1;
  3;
  2, 2;
  2, 2;
  2, 1, 1, 2;
  7;
  3, 1, 1, 3;
  3, 3;
  3, 2, 2, 3;
  12;
  4, 1, 1, 1, 1, 4;
  4, 4;
  4, 2, 1, 1, 2, 4;
  15;
  5, 2, 1, 1, 2, 5;
  5, 3, 5;
  5, 2, 2, 2, 2, 5;
  9, 9;
  6, 2, 1, 1, 1, 1, 2, 6;
  6, 6;
  6, 3, 1, 1, 1, 1, 3, 6;
  28;
  7, 2, 2, 1, 1, 2, 2, 7;
  7, 7;
  7, 3, 2, 1, 1, 2, 3, 7;
  12, 12;
  8, 3, 1, 2, 2, 1, 3, 8;
  8, 8, 8;
  8, 3, 2, 1, 1, 1, 1, 2, 3, 8;
  31;
  9, 3, 2, 1, 1, 1, 1, 2, 3, 9;
  ...
Illustration of the odd-indexed rows of triangle as the diagram of the symmetric representation of sigma which is also the top view of the stepped pyramid:
.
   n  A000203    A237270    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
   1     1   =      1      |_| | | | | | | | | | | | | | | |
   2     3   =      3      |_ _|_| | | | | | | | | | | | | |
   3     4   =    2 + 2    |_ _|  _|_| | | | | | | | | | | |
   4     7   =      7      |_ _ _|    _|_| | | | | | | | | |
   5     6   =    3 + 3    |_ _ _|  _|  _ _|_| | | | | | | |
   6    12   =     12      |_ _ _ _|  _| |  _ _|_| | | | | |
   7     8   =    4 + 4    |_ _ _ _| |_ _|_|    _ _|_| | | |
   8    15   =     15      |_ _ _ _ _|  _|     |  _ _ _|_| |
   9    13   =  5 + 3 + 5  |_ _ _ _ _| |      _|_| |  _ _ _|
  10    18   =    9 + 9    |_ _ _ _ _ _|  _ _|    _| |
  11    12   =    6 + 6    |_ _ _ _ _ _| |  _|  _|  _|
  12    28   =     28      |_ _ _ _ _ _ _| |_ _|  _|
  13    14   =    7 + 7    |_ _ _ _ _ _ _| |  _ _|
  14    24   =   12 + 12   |_ _ _ _ _ _ _ _| |
  15    24   =  8 + 8 + 8  |_ _ _ _ _ _ _ _| |
  16    31   =     31      |_ _ _ _ _ _ _ _ _|
  ...
The above diagram arises from a simpler diagram as shown below.
Illustration of the even-indexed rows of triangle as the diagram of the deployed front view of the corner of the stepped pyramid:
.
.                                 A237593
Level                               _ _
1                                 _|1|1|_
2                               _|2 _|_ 2|_
3                             _|2  |1|1|  2|_
4                           _|3   _|1|1|_   3|_
5                         _|3    |2 _|_ 2|    3|_
6                       _|4     _|1|1|1|1|_     4|_
7                     _|4      |2  |1|1|  2|      4|_
8                   _|5       _|2 _|1|1|_ 2|_       5|_
9                 _|5        |2  |2 _|_ 2|  2|        5|_
10              _|6         _|2  |1|1|1|1|  2|_         6|_
11            _|6          |3   _|1|1|1|1|_   3|          6|_
12          _|7           _|2  |2  |1|1|  2|  2|_           7|_
13        _|7            |3    |2 _|1|1|_ 2|    3|            7|_
14      _|8             _|3   _|1|2 _|_ 2|1|_   3|_             8|_
15    _|8              |3    |2  |1|1|1|1|  2|    3|              8|_
16   |9                |3    |2  |1|1|1|1|  2|    3|                9|
...
The number of horizontal line segments in the n-th level in each side of the diagram equals A001227(n), the number of odd divisors of n.
The number of horizontal line segments in the left side of the diagram plus the number of the horizontal line segment in the right side equals A054844(n).
The total number of vertical line segments in the n-th level of the diagram equals A131507(n).
The diagram represents the first 16 levels of the pyramid.
The diagram of the isosceles triangle and the diagram of the top view of the pyramid shows the connection between the partitions into consecutive parts and the sum of divisors function (see also A286000 and A286001). - _Omar E. Pol_, Aug 28 2018
The connection between the isosceles triangle and the stepped pyramid is due to the fact that this object can also be interpreted as a pop-up card. - _Omar E. Pol_, Nov 09 2022

Robert Price, Table of n, a(n) for n = 1..16048 (n=1..412 rows)
Omar E. Pol, An infinite stepped pyramid (A237593, A237270, A262626)
Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Perspective view of the stepped pyramid (levels: 1..16)
Index entries for sequences related to sigma(n)

Famous sequences that are visible in the stepped pyramid:
Cf. A000040 (prime numbers)......., for the characteristic shape see A346871.
Cf. A000079 (powers of 2)........., for the characteristic shape see A346872.
Cf. A000203 (sum of divisors)....., total area of the terraces in the n-th level.
Cf. A000217 (triangular numbers).., for the characteristic shape see A346873.
Cf. A000225 (Mersenne numbers)...., for a visualization see A346874.
Cf. A000384 (hexagonal numbers)..., for the characteristic shape see A346875.
Cf. A000396 (perfect numbers)....., for the characteristic shape see A346876.
Cf. A000668 (Mersenne primes)....., for a visualization see A346876.
Cf. A001097 (twin primes)........., for a visualization see A346871.
Cf. A001227 (# of odd divisors)..., number of subparts in the n-th level.
Cf. A002378 (oblong numbers)......, for a visualization see A346873.
Cf. A008586 (multiples of 4)......, perimeters of the successive levels.
Cf. A008588 (multiples of 6)......, for the characteristic shape see A224613.
Cf. A013661 (zeta(2))............., (area of the horizontal faces)/(n^2), n -> oo.
Cf. A014105 (second hexagonals)..., for the characteristic shape see A346864.
Cf. A067742 (# of middle divisors), # cells in the main diagonal in n-th level.
Other sequences that are visible in the stepped pyramid: A000096, A001065, A001359, A001747, A002939, A002943, A003056, A004125, A004277, A004526, A005279, A006512, A007606, A007607, A082647, A008438, A008578, A008864, A010814, A014106, A014107, A014132, A014574, A016945, A019434, A024206, A024916, A028552, A028982, A028983, A034856, A038550, A047836, A048050, A052928, A054735, A054844, A062731, A065091, A065475, A071561, A071562, A071904, A092506, A100484, A108605, A139256, A139257, A144396, A152677, A152678, A153485, A155085, A161680, A161983, A162917, A174905, A174973, A175254, A176810, A224880, A235791, A237270, A237271, A237591, A237593, A238005, A238524, A244049, A245092, A259176, A259177, A261348, A278972, A317302, A317303, A317304, A317305, A317307, A319529, A319796, A319801, A319802, A327329, A336305, (and several others).
Apart from zeta(2) other constants that are related to the stepped pyramid are A072691, A353908, A354238.
Cf. A054844, A131507, A196020, A236104, A237048, A239660, A244050, A259179, A261350, A261697, A261699, A262612, A280850, A286000, A286001, A296508.

A139256 Twice even perfect numbers. Also a(n) = M(n)*(M(n)+1), where M(n) is the n-th Mersenne prime A000668(n).

Original entry on oeis.org

12, 56, 992, 16256, 67100672, 17179738112, 274877382656, 4611686016279904256, 5316911983139663489309385231907684352, 383123885216472214589586756168607276261994643096338432

Offset: 1

Omar E. Pol, Apr 22 2008

nonn

Also, twice perfect numbers, if there are no odd perfect numbers.
If there are no odd perfect numbers, essentially the same as A065125. - R. J. Mathar, May 23 2008
The sum of reciprocals of even divisors of a(n) equals 1. Proof: Let n = (2^m - 1)*2^m where 2^m - 1 is a Mersenne prime. The sum of reciprocals of even divisors of n is s1 + s2 where: s1 = 1/2 + 1/4 + ... + 1/2^m = (2^m - 1)/2^m and s2 = s1/(2^m - 1) => s1 + s2 = 1. - Michel Lagneau, Jul 17 2013

			a(3) = 992 because the third Mersenne prime A000668(3) is 31 and 31*(31+1) = 31*32 = 992.
a(3) = 992 because the sum of the divisors of the third perfect number is 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 + 496 = 992. - _Omar E. Pol_, Dec 05 2016
From _Omar E. Pol_, Aug 13 2021: (Start)
Illustration of initial terms in which a(n) is represented as the sum of the divisors of the n-th even perfect number P(n).
-------------------------------------------------------------------------
  n  P(n) a(n)  Diagram:   1                                           2
-------------------------------------------------------------------------
                           _                                           _
                          | |                                         | |
                          | |                                         | |
                       _ _| |                                         | |
                      |    _|                                         | |
                 _ _ _|  _|                                           | |
  1    6   12   |_ _ _ _|                                             | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                             _ _ _ _ _| |
                                                            |  _ _ _ _ _|
                                                            | |
                                                         _ _| |
                                                     _ _|  _ _|
                                                    |    _|
                                                   _|  _|
                                                  |  _|
                                             _ _ _| |
                                            |  _ _ _|
                                            | |
                                            | |
                                            | |
                 _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  2   28   56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) equals the area (also the number of cells) in the n-th diagram.
For n = 3, P(3) = 496 and a(3) = 992, the diagram is too large to include here. To draw that diagram note that the lengths of the line segments of the smallest Dyck path are [248, 83, 42, 25, 17, 13, 9, 7, 6, 5, 5, 3, 4, 2, 3, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 3, 5, 5, 6, 7, 9, 13, 17, 25, 42, 83, 248] and the lengths of the line segments of the largest Dyck path are [249, 83, 42, 25, 17, 13, 9, 7, 6, 5, 5, 3, 4, 2, 3, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 3, 5, 5, 6, 7, 9, 13, 17, 25, 42, 83, 249]. For a definition of these numbers related to partitions into consecutive parts see A237591. (End)

Walter A. Kehowski, Power-spectral Numbers, ResearchGate (2024); also available at vixra.org.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000203, A000396, A000668, A001065, A065125, A139257, A237591, A237593, A245092.

DeleteCases[2 Map[(# (# + 1))/2 &, Select[2^Range[100] - 1, PrimeQ]], k_ /; OddQ@ k] (* Michael De Vlieger, Dec 05 2016, after Harvey P. Dale at A000396 *)

a(n) = A000668(n)*(A000668(n)+1).
a(n) = 2*A000396(n), if there are no odd perfect numbers.
a(n) = A000203(A000396(n)) = A001065(A000396(n)) + A000396(n), assuming there are no odd perfect numbers. - Omar E. Pol, Dec 04 2016

More terms from Omar E. Pol, Jun 07 2012

A381750 Nonprime-powers k such that, for any prime p dividing k and m = 1+floor(log k/log p), rad(p^m (mod k)) divides k, where rad = A007947.

Original entry on oeis.org

6, 12, 14, 24, 39, 56, 62, 112, 120, 155, 254, 992, 1984, 3279, 5219, 16256, 16382, 19607, 32512, 70643, 97655, 208919, 262142, 363023, 402233, 712979, 1040603, 1048574, 1508597, 2265383, 2391483, 4685519, 5207819, 6728903, 21243689, 25239899, 56328959, 61035155, 67100672

Offset: 1

Michael De Vlieger, Mar 27 2025

nonn

The number p^m is the smallest power of p dividing k that exceeds k, where m = 1+floor(log k/log p).
Let S(n,p) be the set of distinct power residues r (mod n) beginning with empty product and recursively multiplying by prime p | n. For example, S(10,2) = {1,2,4,8,6}.
Prime powers k = p^m, m >= 0 have omega(k) = 1 and all r in S(n,p) are such that rad(r) | n.
Numbers k in this sequence have omega(k) > 1 and all r in S(n,p) are such that rad(r) | n.
A139257 is a proper subset since 2^m is congruent to 2 (mod 2^m-2).
Intersection of this sequence and A381525 is {6}.
Row a(n) of A381799 only contains powers of primes, i.e., row a(n) of A381799 is a proper subset of A000961.

			Table of a(n) for n = 1..10, showing prime decomposition (facs(a(n))), and S(n,p_x), where x = 1 denotes the smallest prime factor, x = 2, the second smallest prime factor, etc.
                         Numbers in row n of A381799:
 n   a(n)  facs(a(n))    S(n,p_1)            S(n,p_2)        S(n,p_3)
---------------------------------------------------------------------
 1     6   2 * 3         {1,2,4},            {1,3}
 2    12   2^2 * 3       {1,2,4,8},          {1,3,9}
 3    14   2 * 7         {1,2,4,8},          {1,7}
 4    24   2^3 * 3       {1,2,4,8,16},       {1,3,9}
 5    39   3 * 13        {1,3,9,27},         {1,13}
 6    56   2^3 * 7       {1,2,4,8,16,32},    {1,7,49}
 7    62   2 * 31        {1,2,4,8,16,32},    {1,31}
 8   112   2^4 * 7       {1,2,4,8,16,32,64}, {1,7,49}
 9   120   2^3 * 3 * 5   {1,2,4,8,16,32,64}, {1,3,9,27,81}, {1,5,25}
10   155   5 * 31        {1,5,25,125},       {1,31}
.
a(1) = 6, the smallest number that is not a prime power, since 2^3 mod 6 = 2, and 3^2 mod 6 = 3, both divide 6.
10 is not in the sequence since 2^4 mod 10 = 6, rad(6) does not divide 10.
a(2) = 12 since 2^4 mod 12 = 4, rad(4) | 12, and 3^3 mod 12 = 3, rad(3) | 12.
a(3) = 14 since 2^4 mod 14 = 2 and 7^2 mod 14 = 7, both divide 14.
15 is not in the sequence since 3^3 mod 15 = 12, rad(12) does not divide 15, etc.

Cf. A000961, A007947, A024619, A139257, A381525, A381799.

nn = 10^5;
Monitor[Reap[Do[
  If[! PrimePowerQ[n],
    If[AllTrue[
      Map[PowerMod[#, 1 + Floor@ Log[#, n], n] &, FactorInteger[n][[All, 1]] ],
        Divisible[n, rad[#]] &],
      Sow[n] ] ], {n, 2, nn}] ][[-1, 1]], n]

A318145 Numbers m such that 2^phi(m) mod m is a perfect power other than 1.

Original entry on oeis.org

6, 12, 14, 20, 24, 28, 30, 40, 48, 56, 60, 62, 70, 72, 80, 84, 96, 112, 120, 124, 126, 132, 140, 144, 168, 176, 192, 198, 208, 224, 240, 248, 252, 254, 260, 272, 286, 288, 320, 336, 340, 344, 384, 390, 396, 408, 430, 448, 456, 480, 496, 504, 508, 510, 532

Offset: 1

Peter Luschny, Sep 01 2018

nonn

All terms are even, as 2^phi(m) == 1 (mod m) if m is odd. - Robert Israel, Sep 02 2018

Perfect power terms are 144, 576, 900, 1600, 3136, 9216, 12544, 20736, 36864, 57600, 63504, ... - Altug Alkan, Sep 04 2018

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

Cf. A000010, A001597, A318623. Contains A139257.

ispow:= proc(n) local F;
  F:= map(t -> t[2], ifactors(n)[2]);
  igcd(op(F)) > 1
end proc:
select(m -> ispow(2 &^ numtheory:-phi(m) mod m), [seq(i,i=2..1000,2)]); # Robert Israel, Sep 02 2018

okQ[n_] := GCD @@ FactorInteger[PowerMod[2, EulerPhi[n], n]][[All, 2]] > 1;
Select[Range[2, 1000, 2], okQ] (* Jean-François Alcover, Aug 02 2019 *)

def isA318145(n):
    m = power_mod(2, euler_phi(n), n)
    return m > 0 and m.is_perfect_power()
def A318145_list(search_bound):
    return [n for n in range(2, search_bound + 1, 2) if isA318145(n)]
print(A318145_list(532))

Definition corrected by Robert Israel, Sep 02 2018

A067612 Numbers n such that sigma(n) = 3*phi(sigma(n)).

Original entry on oeis.org

5, 6, 10, 11, 14, 15, 17, 22, 23, 30, 33, 34, 35, 42, 46, 47, 51, 53, 55, 62, 66, 69, 70, 71, 77, 85, 94, 102, 105, 106, 107, 110, 115, 119, 138, 141, 142, 154, 155, 159, 161, 165, 170, 186, 187, 191, 210, 213, 214, 230, 231, 235, 238, 253, 254, 255, 265, 282, 310

Offset: 1

Benoit Cloitre, Feb 22 2002

From Robert Israel, Feb 26 2017: (Start)
Numbers n such that sigma(n) is in A033845.
Contains A139257. (End)

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

Cf. A033845, A139257.

filter:= proc(n) local s; s:= numtheory:-sigma(n); s mod 6 = 0 and s = 2^padic:-ordp(s,2)*3^padic:-ordp(s,3) end proc:
select(filter, [$1..10000]); # Robert Israel, Feb 26 2017

s3pQ[n_]:=Module[{s=DivisorSigma[1,n]},s==3*EulerPhi[s]]; Select[ Range[ 400],s3pQ] (* Harvey P. Dale, May 28 2015 *)

for(n=1,500,if(sigma(n)==3*eulerphi(sigma(n)),print1(n,", "))) \\ Derek Orr, Feb 26 2017

is(n)=my(s=sigma(n)); s%3==0 && s==3*eulerphi(s) \\ Charles R Greathouse IV, Feb 27 2017

A252541 Numbers k such that A146076(A000593(k)) = k.

Original entry on oeis.org

6, 14, 62, 254, 756, 16382, 262142, 1048574, 39606840, 4294967294

Offset: 1

Michel Lagneau, Dec 18 2014

All integers of the form 2*(2^p-1) where 2^p-1 is prime are terms (see A139257). The terms that are not of this form are 756, 39606840. Are there any other? [Edited by Michel Marcus, Nov 22 2022]
All terms are even because all terms of A146076 are even. - Michel Marcus, Nov 22 2022
a(11) > 10^10. - Michel Marcus, Nov 22 2022
a(11) > 10^11. - Amiram Eldar, May 19 2024

			14 is in the sequence because the divisors are {1, 2, 7, 14} => sum of odd divisors 1 + 7 = 8. The divisors of 8 are {1, 2, 4, 8} => sum of even divisors = 2 + 4 + 8 = 14. That is, A146076(A000593(14)) = A146076(8) = 14.

Cf. A000593 (sum of odd divisors), A139257, A146076 (sum of even divisors), A252540.

f[n_]:= Plus @@ Select[ Divisors@ n, OddQ];g[n_]:= Plus @@ Select[ Divisors@ n, EvenQ];Do[If[g[f[n]]==n,Print[n]],{n,1,10^8}]

sod(n) = sigma(n>>valuation(n, 2)); \\ A000593
sed(n) = if (n%2, 0, 2*sigma(n/2)); \\ A146076
isok(n) = sed(sod(n)) == n;
lista(nn) = forstep(n=2, nn, 2, if(isok(n), print1(n, ", "))); \\ Michel Marcus, Nov 22 2022

a(10) from Michel Marcus, Nov 22 2022

A279389 3 times Mersenne primes A000668.

Original entry on oeis.org

9, 21, 93, 381, 24573, 393213, 1572861, 6442450941, 6917529027641081853, 1856910058928070412348686333, 486777830487640090174734030864381, 510423550381407695195061911147652317181

Offset: 1

Omar E. Pol, Dec 20 2016

nonn

Also sum of n-th Mersenne prime and the radical of n-th even perfect number.

The binary representation of a(n) has only two zeros, starting with "10" and ending with "01". The sequence begins: 1001, 10101, 1011101, 101111101, 101111111111101,...

Cf. A000010, A000203, A000396, A000668, A051027, A147757.
Subsequence of A001748, and of A147758, and of A174055, and possibly of other sequences, see below:
Cf. A060482, A068156, A072087, A136252, A144482, A144856, A249780, A269633.

a(n) = 3*A000668(n) = A000668(n) + A139257(n).

a(n) = phi(M(n)) + sigma(sigma(M(n))) = A000010(A000668(n)) + A000203(A000203(A000668(n))) = A000010(A000668(n)) + A051027(A000668(n)).

A382438 Numbers k in A024619 such that all residues r (mod k) in row k of A381801 are such that rad(r) divides k, where rad = A007947.

Original entry on oeis.org

6, 12, 14, 24, 39, 62, 155, 254, 3279, 5219, 16382, 19607, 70643, 97655, 208919, 262142, 363023, 402233, 712979, 1040603, 1048574, 1508597, 2265383, 2391483, 4685519, 5207819, 6728903, 21243689, 25239899, 56328959, 61035155, 67977559, 150508643

Offset: 1

Michael De Vlieger, Mar 27 2025

nonn

Numbers k in A024619 such that A381804(k) = 0.
Let S(n,p) be the set of distinct power residues r (mod n) beginning with empty product and recursively multiplying by prime p | n. For example, S(10,2) = {1,2,4,8,6}.
This sequence builds on A381750, taking the tensor product T(k) (mod k) of S(k,p), p | k. If all products r (mod k) are such that rad(r) | k, then k is in this sequence. Distinct residues r (mod k) in T(k) are listed in row k of A381801.
Proper subset of A381750.
A139257 is a proper subset since 2^m is congruent to 2 (mod 2^m-2).
Conjecture: 12 and 24 are the only nonsquarefree numbers in this sequence, i.e., in A126706.

			Table of a(n) for n = 1..10, showing prime decomposition (facs(a(n))), row a(n) of A381801:
                        Row a(n) of A381801
 n    a(n)  facs(a(n))  k (mod a(n)) such that rad(k) | a(n).
-------------------------------------------------------------
 1      6   2 * 3       {0, 1, 2, 3, 4}
 2     12   2^2 * 3     {0, 1, 2, 3, 4, 6, 8, 9}
 3     14   2 * 7       {0, 1, 2, 4, 7, 8}
 4     24   2^3 * 3     {0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18}
 5     39   3 * 13      {0, 1, 3, 9, 13, 27}
 6     62   2 * 31      {0, 1, 2, 4, 8, 16, 31, 32}
 7    155   5 * 31      {0, 1, 5, 25, 31, 125}
 8    254   2 * 127     {0, 1, 2, 4, 8, 16, 32, 64, 127, 128}
 9   3279   3 * 1093    {0, 1, 3, 9, 27, 81, 243, 729, 1093, 2187}
10   5219   17 * 307    {0, 1, 17, 289, 307, 4913}
Let b = A381750.
a(1) = 6 since T(6) (mod 6) = {1,2,4} X {1,3} = {{1,2,4},{3,0,0}}; all residues r (mod 6) in T(6) (i.e., in row 6 of A381801) are such that rad(r) | 6.
a(2) = 12 since T(12) (mod 12) = {1,2,4,8} X {1,3,9} = {{1,2,4,8},{3,6,0,0},{9,6,0,0}}; all residues r (mod 12) in T(12) are such that rad(r) | 12.
a(3) = 14 since T(14) (mod 14) = {1,2,4,8} X {1,7} = {{1,2,4,8},{7,0,0,0}}; all residues r (mod 14) in T(14) are such that rad(r) | 14.
a(4) = 24 since T(24) (mod 24) = {1,2,4,8,16} X {1,3,9} = {{1,2,4,8,16},{3,6,12,0,0},{9,18,0,0,0}}; all residues r (mod 24) in T(24) are such that rad(r) | 24.
b(6) = 56 is not in the sequence since 49*2 = 98 = 42 (mod 56), rad(42) does not divide 56.
b(8) = 112 is not in the sequence since 49*4 = 196 = 84 (mod 112), rad(84) does not divide 112, etc.

Michael De Vlieger, Efficient Mathematica code for this sequence (2025).

Cf. A007947, A024619, A381750, A381801.

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