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A386221 Numbers which can be expressed as the product of a number and its binary reversal in at least three different ways.

2371610879733375, 8379443074856875, 103889625367330285, 162508095102648823, 2143169709271976875, 2481725627762299375, 4055619414785589625, 8167773178498814075, 9027536760163222895, 133527604616779133915, 133893081609954481115, 137216105281788994475, 457495296809227508125

Offset: 1

Zhao Hui Du, Aug 12 2025

It appears that most numbers that can be expressed in three different ways can also be expressed in four different ways. For a(n) < 2^88, only 11 numbers can be expressed in exactly three ways while 691 numbers can be expressed in exactly four ways.

			2371610879733375 = 51606261*45955875 = 64244529*36915375 = 64338225*36861615 while 51606261 = 11000100110111001011110101_2, 45955875 = 10101111010011101100100011_2 and reverse(11000100110111001011110101) = 10101111010011101100100011.
8379443074856875 = 101377465*82655875 = 102886105*81443875 = 114021425*73490075 = 115718225*72412475.

Shou'en Wang, Binary reversal (Chinese)

Cf. A066531, A066598.

A386213 Integers t having at least one nonempty subset of the set of its proper divisors for which the equation sigma(t) + r = m*t (m is any integer > 1, r is the sum of elements of such subset) is true.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 15, 16, 18, 20, 21, 24, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 70, 72, 75, 78, 80, 84, 88, 90, 96, 99, 100, 102, 104, 105, 108, 112, 114, 117, 120, 126, 128, 130, 132, 135, 136, 138, 140, 144, 150, 152, 153, 154, 156, 160, 162, 165

Offset: 1

Lechoslaw Ratajczak, Aug 12 2025

The following table lists sequences which give k-deficient-m-perfect numbers:
------------------------------------------------------------
 k/m   |     any m     |       2       |         3         |
------------------------------------------------------------
 any k | this sequence | A331627 \ {1} |         -         |
------------------------------------------------------------
 1     | A385462       | A271816 \ {1} | A364977 \ A000396 |
------------------------------------------------------------
 2     |       -       | A331628       |         -         |
------------------------------------------------------------
 3     |       -       | A331629       |         -         |
------------------------------------------------------------
This sequence contains all, and only, (any k)-deficient-m-perfect numbers (m = 2,3,4,...), equivalently it contains all, and only, k-deficient-(any m)-perfect numbers (k = 1,2,3,...).

			24 is a term because for 24 the set of proper divisors is {1, 2, 3, 4, 6, 8, 12} and it has exactly 6 subsets which sum up to r satisfying the equation sigma(24) + r = k*24:
  (1) sigma(24) + d_7(24) = 60 + 12 = 72 and 72 = 3*24,
  (2) sigma(24) + (d_4(24) + d_6(24)) = 60 + (4 + 8) = 72 and 72 = 3*24,
  (3) sigma(24) + (d_2(24) + d_4(24) + d_5(24)) = 60 + (2 + 4 + 6) = 72 and 72 = 3*24,
  (4) sigma(24) + (d_1(24) + d_3(24) + d_6(24)) = 60 + (1 + 3 + 8) = 72 and 72 = 3*24,
  (5) sigma(24) + (d_1(24) + d_2(24) + d_3(24) + d_5(24)) = 60 + (1 + 2 + 3 + 6) = 72 and 72 = 3*24,
  (6) sigma(24) + (d_1(24) + d_2(24) + d_3(24) + d_4(24) + d_5(24) + d_6(24) + d_7(24)) = 60 + (1 + 2 + 3 + 4 + 6 + 8 + 12) = 96 and 96 = 4*24.
So 24 is (1, 2, 3 (in 2 variants), 4)-deficient-3-perfect and 7-deficient-4-perfect number.

Cf. A000005, A000203, A007691, A331627, A385462.

n = 1;l={};Do[x = 1;s=DivisorSigma[1,t];A=Most[Divisors[t]];B=Subsets[A];  Do[r=Total[B[[i]]];If[Mod[s+r,t]==0,x=x+1],{i,2,2^Length[A]}];  If[x>1,AppendTo[l,t];n=n+1],{t,1,165}];l (* James C. McMahon, Aug 25 2025 *)

(n:1, for t:1 thru 300 do (x:1, s:divsum(t), A:delete(t, divisors(t)), B:args(powerset(A)),
              for i:2 thru 2^(length(args(A))) do (r:apply("+", args(B[i])),
                      if mod(s+r, t)=0 then (x:x+1)),
                                       if x>1 then (print(n, "", t), n:n+1)));

A386287 Values of w in the quartets (3, u, v, w) of type 2; i.e., values of v for solutions to 3(3 + u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see A386285.

Original entry on oeis.org

4, 11, 14, 20, 2, 10, 23, 26, 1, 13, 29, 32, 16, 35, 38, 1, 19, 41, 4, 44, 2, 8, 22, 47, 50, 25, 53, 56, 4, 7, 11, 28, 59, 2, 62, 5, 31, 65, 68, 1, 6, 14, 34, 71, 10, 74, 7, 37, 77, 80, 5, 8, 17, 40, 83, 86, 1, 9, 13, 43, 89, 92, 4, 10, 20, 46, 95, 2, 98, 11

Offset: 1

Clark Kimberling, Aug 12 2025

Cf. A386285.

A386982 Values of w in the quartets (2, u, v, w) of type 2; i.e., values of v for solutions to 2(2 + u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see A385884.

Original entry on oeis.org

5, 9, 11, 13, 15, 3, 17, 1, 19, 21, 5, 23, 25, 3, 27, 1, 7, 29, 4, 31, 33, 5, 9, 35, 37, 3, 6, 39, 1, 11, 41, 7, 43, 45, 8, 13, 47, 5, 49, 9, 51, 3, 15, 53, 1, 10, 55, 57, 4, 7, 11, 17, 59, 61, 12, 63, 5, 19, 65, 13, 67, 3, 9, 69, 1, 6, 14, 21, 71, 73, 15

Offset: 1

Clark Kimberling, Aug 12 2025

nonn

Cf. A385884.

A386841 Triangle read by rows: T(n,k) is the number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on the n-dimensional probability simplex and of degree 2n-k (n>=1, 1<=k<=n).

Original entry on oeis.org

1, 1, 3, 2, 4, 12, 4, 10, 38, 82, 2, 24, 88, 254, 602, 4, 32, 198, 643, 2421, 6710, 8, 56, 332, 1442, 6445, 23285, 83906, 4

Offset: 1

Carlos Améndola, Aug 12 2025

The range of k is precisely chosen so that T(n,k) is positive. That is, whenever the degree is higher than 2n-1 or lower than n, there are no fundamental models.

			When n=1 then k=1 and the unique model T(1,1)=1 corresponds to the model described by a Bernoulli random variable that assigns probabilities 1-t and t to two possible states, 0<=t<=1. This line segment parametrizes the 1-dimensional probability simplex.
When n=2 we have 1<=k<=2. The T(2,1)=1 unique fundamental model with degree 3 corresponds to the parametrization t -> ((1-t)^3, 3t(1-t), t^3) and the T(2,2)=3 fundamental models of degree 2 correspond to the parametrizations ((1-t)^2, 2t(1-t), t^2) , (1-t, t(1-t), t^2) and ((1-t)^2, t(1-t), t).
Continuing in this way, the first five rows (1<=n<=5) of the fundamental models triangle are:
  1
  1 3
  2 4 12
  4 10 38 82
  2 24 88 254 602

Carlos Améndola, Viet Duc Nguyen, and Janike Oldekop, One-dimensional discrete models of maximum likelihood degree one, arXiv:2507.18686 [math.ST], 2025. See p. 20 (Figure 9).
Arthur Bik and Orlando Marigliano, Classifying one-dimensional discrete models with maximum likelihood degree one, Adv. Appl. Math., 170 (2025), 102928.

Columns 1..4 are A143107, A143108, A387029, A386840.

A387012 Number of ternary strings of length 2*n that have fewer 0's than the combined number of 1's and 2's.

Original entry on oeis.org

0, 4, 48, 496, 4864, 46464, 436992, 4068096, 37601280, 345733120, 3166363648, 28910051328, 263320698880, 2393742770176, 21726260035584, 196938517118976, 1783247797223424, 16132649384411136, 145839570932465664, 1317564543167102976, 11896996193604993024, 107375816824319901696

Offset: 0

Enrique Navarrete, Aug 12 2025

			a(2) = 48 since the strings of length 4 are the following (number of permutations in parentheses): 1110 (4), 1120 (12), 1220 (12), 2220 (4), 1111 (1), 1112 (4), 1122 (6), 1222 (4), 2222 (1).

Cf. A001019, A128418, A385252.

a[n_] := 9^n - Sum[2^(n-k) * Binomial[2*n, n-k], {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, Aug 16 2025 *)

a(n) = 9^n - Sum_{k=0..n} 2^(n-k)*binomial(2*n,n-k).
G.f.: (sqrt(1-8*x)*(sqrt(1-8*x)+12*x-1)-8*x*(1-9*x))/((1-9*x)*sqrt(1-8*x)*(sqrt(1-8*x)+12*x-1)).
a(n) = A001019(n) - A128418(n).
D-finite with recurrence n*a(n) +(-29*n+28)*a(n-1) +12*(23*n-41)*a(n-2) +432*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Aug 26 2025

A386288 Values of u in the quartets (4, u, v, w) of type 2; i.e., values of u for solutions to 4(4 + u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 10, 11, 11, 11, 11, 12, 12, 13, 13, 14, 14, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24

Offset: 1

Clark Kimberling, Aug 12 2025

nonn

A 4-tuple (m, u, v, w) is a quartet of type 2 if m, u, v, w are distinct positive integers such that m < v and m*(m + u) = v*(v - w). Here, the values of u are arranged in nondecreasing order. When there is more than one solution for given m and u, the values of v are arranged in increasing order. Here, m = 4.

			First 20 quartets (4,u,v,w) of type 2:
   m   u    v    w
   4   1   10    8
   4   1   20   19
   4   2    8    5
   4   2   12   10
   4   2   24   23
   4   3   14   12
   4   3   28   27
   4   5   12    9
   4   5   18   16
   4   5   36   35
   4   6    8    3
   4   6   20   18
   4   6   40   39
   4   7   22   20
   4   7   44   43
   4   8   16   13
   4   8   24   22
   4   8   48   47
   4   9   26   24
   4   9   52   51
4(4+2) = 8(8-5), so (4,2,8,5) is in the list.

Cf. A385182 (type 1, m=1), A386630 (type 3, m=1).

solnsB[t_, u_] := Module[{n = t*(t + u)},
Cases[Select[Divisors[n], # < n/# &],
d_ :> With[{v = n/d, w = n/d - d}, {t, u, v, w} /;
Length[DeleteDuplicates[{t, u, v, w}]] == 4]]];
TableForm[solns = Flatten[Table[Sort[solnsB[4, u]], {u, 26}], 1],
TableHeadings -> {None, {"m", "u", "v", "w"}}]
u1 = Map[#[[2]] &, solns] (*u, A386288 *)
v1 = Map[#[[3]] &, solns] (*v, A386628 *)
w1 = Map[#[[4]] &, solns] (*w, A386629 *)
(* Peter J. C. Moses, Aug 17 2025  *)

A385315 Smallest number k such that both k^n - 1 and k^n + 1 have n prime factors, counted with repetitions.

Original entry on oeis.org

4, 12, 66, 920, 26, 132, 79, 17958, 53, 693, 4181, 122160, 29791, 32318, 971

Offset: 1

Jean-Marc Rebert, Aug 12 2025

			a(1) = 4, because 4^1 - 1 = 3 and 4^1 + 1 = 5, and no lesser number has this property.
See the Links section for more examples.

Jean-Marc Rebert, Factorizations of k^n-1 and k^n+1

Cf. A001222, A001358, A014612, A078840, A385591, A368162, A368163.

a(n) = my(k=2, kn=k^n); while ((bigomega(kn-1)!=n) || (bigomega(kn+1)!=n), k++;kn=k^n); k; \\ Michel Marcus, Aug 18 2025

a(n) >= max(A368162(n), A368163(n)). - Daniel Suteu, Sep 02 2025

A386990 Decimal expansion of Sum_{k>=0} 2/(k!*(k! + 1)).

Original entry on oeis.org

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

Offset: 1

Kelvin Voskuijl, Aug 12 2025

Sum of reciprocals of A055555 (triangular numbers of factorials).

			2.3844273879714288211644804923804481846149857064...

Cf. A000217, A070910 (of n!^2), A055555, A091131 (of n!).

evalf(sum(2/(n!*(n!+1)),n=0..infinity), 120);  # Alois P. Heinz, Aug 13 2025

PARI
```
suminf(k=0, 2/(k!*(k!+1)))
```

sumpos(k=0,1/binomial(k!+1,2)) \\ Charles R Greathouse IV, Aug 19 2025

Equals Sum_{k>=0} 1/A055555(k).

A386989 Irregular triangle read by rows: T(n,k) is the product of terms in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 1, 3, 8, 1, 5, 36, 1, 7, 64, 1, 3, 9, 2, 50, 1, 11, 1728, 1, 13, 2, 98, 1, 15, 15, 1024, 1, 17, 5832, 1, 19, 8000, 1, 3, 7, 21, 2, 242, 1, 23, 331776, 1, 5, 25, 2, 338, 1, 3, 9, 27, 21952, 1, 29, 810000, 1, 31, 32768, 1, 3, 11, 33, 2, 578, 1, 35, 35, 10077696, 1, 37, 2, 722, 1, 3, 13, 39, 2560000

Offset: 1

Omar E. Pol, Aug 12 2025

In a sublist of divisors of n the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of n.
The 2-dense sublists of divisors of n are the maximal sublists whose terms increase by a factor of at most 2.
It is conjectured that row lengths are given by A237271.

			Triangle begins:
   1;
   2;
   1,  3;
   8;
   1,  5;
  36;
   1,  7;
  64;
   1,  3,  9;
   2, 50;
  ...
For n = 10 the list of divisors of 10 is [1, 2, 5, 10]. There are two 2-dense sublists of divisors of 10, they are [1, 2] and [5, 10]. The product of terms are 1*2 = 2 and 5*10 = 50 respectively, so the row 10 of the triangle is [2, 50].

Paolo Xausa, Table of n, a(n) for n = 1..10607 (rows 1..3500 of triangle, flattened).

Row products give A007955.

Cf. A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384928, A384930, A384931, A386984.

A386989row[n_] :=Times @@@ Split[Divisors[n], #2/# <= 2 &];
Array[A386989row, 50] (* Paolo Xausa, Aug 29 2025 *)

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A386221 Numbers which can be expressed as the product of a number and its binary reversal in at least three different ways.

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A386213 Integers t having at least one nonempty subset of the set of its proper divisors for which the equation sigma(t) + r = m*t (m is any integer > 1, r is the sum of elements of such subset) is true.

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A386287 Values of w in the quartets (3, u, v, w) of type 2; i.e., values of v for solutions to 3(3 + u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see A386285.

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A386982 Values of w in the quartets (2, u, v, w) of type 2; i.e., values of v for solutions to 2(2 + u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see A385884.

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A386841 Triangle read by rows: T(n,k) is the number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on the n-dimensional probability simplex and of degree 2n-k (n>=1, 1<=k<=n).

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A387012 Number of ternary strings of length 2*n that have fewer 0's than the combined number of 1's and 2's.

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Formula

A386288 Values of u in the quartets (4, u, v, w) of type 2; i.e., values of u for solutions to 4(4 + u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

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A385315 Smallest number k such that both k^n - 1 and k^n + 1 have n prime factors, counted with repetitions.

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A386990 Decimal expansion of Sum_{k>=0} 2/(k!*(k! + 1)).

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A386989 Irregular triangle read by rows: T(n,k) is the product of terms in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

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