16 - cp's OEIS frontend

A385476 Values of u in the quartets (1, u, v, w) of type 3; i.e., values of u for solutions to (1 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

5, 7, 7, 9, 9, 10, 11, 11, 13, 13, 13, 13, 15, 15, 16, 16, 17, 17, 17, 19, 19, 19, 19, 21, 21, 21, 21, 22, 22, 23, 23, 25, 25, 25, 25, 25, 25, 26, 27, 27, 28, 28, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 33, 33, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 37, 37

Offset: 1

Clark Kimberling, Aug 16 2025

A 4-tuple (m, u, v, w) is a quartet of type 3 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 = 1.

			First 20 quartets (1,u,v,w) of type 3:
   m    u    v    w
   1    5    2    4
   1    7    2    5
   1    7    3    5
   1    9    2    6
   1    9    4    6
   1   10    3    6
   1   11    2    7
   1   11    5    7
   1   13    2    8
   1   13    3    7
   1   13    4    7
   1   13    6    8
   1   15    2    9
   1   15    7    9
   1   16    3    5
   1   16    3    8
   1   17    2   10
   1   17    4    8
   1   17    8   10
   1   19    2   11
1(1-11) = 5(5-7), so (1, 11, 5, 7) is in the list.

Cf. A385182 (type 1), A386218 (type 2), A386631, A385246.

solnsM[m_Integer?Positive, u_Integer?Positive] :=
  Module[{n = m  (m - u), nn, sgn, ds, tups}, If[n == 0, Return[{}]];
   sgn = Sign[n]; nn = Abs[n];
   ds = Divisors[nn];
   If[sgn > 0, ds = Select[ds, # < nn/# &]];
   tups = ({m, u, nn/#, nn/# - sgn  #} & /@ ds);
   Select[tups, #[[3]] > 1 && #[[4]] > 0 && #[[2]] =!= #[[4]](*&&
     Length@DeleteDuplicates[#]==4*)&]];
(solns =
   Sort[Flatten[Map[solnsM[1, #] &, Range[2, 30]], 1]]) // ColumnForm
Map[#[[2]] &, solns] (*A385476*)
Map[#[[3]] &, solns] (*A163870*)
Map[#[[4]] &, solns] (*A385246*)
(* Peter J. C. Moses, Aug 22 2025 *)

A387100 a(n) is the least number that can be written in exactly n ways as s_1^x_1 + ... + s_t^x_t, with 1 < s_1 < ... < s_t and {s_1,..., s_t} = {x_1,..., x_t}.

Original entry on oeis.org

4, 545, 23506, 331979, 5260225, 10630307

Offset: 1

David A. Corneth, Aug 16 2025

			a(5) = 5260225 via
5260225 = 2^22 + 3^8 + 4^5 + 5^2 + 7^7 + 8^3 + 22^4
        = 2^21 + 3^8 + 4^10 + 7^3 + 8^7 + 10^4 + 21^2
        = 2^7 + 3^14 + 4^5 + 5^8 + 6^6 + 7^3 + 8^2 + 14^4
        = 2^15 + 3^10 + 4^9 + 5^5 + 6^4 + 7^6 + 9^7 + 10^3 + 15^2
        = 2^11 + 3^7 + 4^10 + 5^9 + 6^8 + 7^3 + 8^5 + 9^6 + 10^4 + 11^2,
and no positive integer smaller than 5260225 can be written as such in exactly five ways.

Cf. A385969, A386968.

A387099 Numbers that can be written in exactly five ways as s_1^x_1 + ... + s_t^x_t, with 1 < s_1 < ... < s_t and {s_1,..., s_t} = {x_1,..., x_t} for some t > 0.

Original entry on oeis.org

5260225, 7923882, 11054875, 11224211, 11870046, 15466174, 16859617, 16911017, 17276523, 17326946, 18664520, 18668302, 18908170, 19375153, 19706896, 19854394, 20050965, 20757873, 21468249, 24723272, 26689657, 26925803, 26974782, 27214122, 27336893, 28055974

Offset: 1

David A. Corneth, Aug 16 2025

			5260225 = 2^22 + 3^8 + 4^5 + 5^2 + 7^7 + 8^3 + 22^4
        = 2^21 + 3^8 + 4^10 + 7^3 + 8^7 + 10^4 + 21^2
        = 2^7 + 3^14 + 4^5 + 5^8 + 6^6 + 7^3 + 8^2 + 14^4
        = 2^15 + 3^10 + 4^9 + 5^5 + 6^4 + 7^6 + 9^7 + 10^3 + 15^2
        = 2^11 + 3^7 + 4^10 + 5^9 + 6^8 + 7^3 + 8^5 + 9^6 + 10^4 + 11^2.
7923882 = 2^8 + 3^5 + 4^11 + 5^9 + 6^3 + 8^4 + 9^2 + 11^6
        = 2^12 + 3^9 + 4^5 + 5^6 + 6^2 + 7^8 + 8^7 + 9^3 + 12^4
        = 2^14 + 3^13 + 4^4 + 5^5 + 6^7 + 7^8 + 8^6 + 13^2 + 14^3
        = 2^18 + 3^14 + 4^6 + 5^9 + 6^3 + 7^7 + 9^5 + 14^4 + 18^2
        = 2^19 + 3^14 + 4^8 + 5^9 + 6^6 + 8^2 + 9^4 + 14^5 + 19^3.
11054875 = 2^3 + 3^6 + 4^10 + 5^5 + 6^2 + 7^4 + 10^7
         = 2^15 + 3^8 + 4^6 + 5^2 + 6^9 + 7^7 + 8^3 + 9^5 + 15^4
         = 2^22 + 3^12 + 4^2 + 5^6 + 6^7 + 7^8 + 8^5 + 12^3 + 22^4
         = 2^9 + 3^13 + 4^10 + 5^3 + 6^5 + 7^8 + 8^7 + 9^6 + 10^4 + 13^2
         = 2^11 + 3^12 + 4^8 + 5^7 + 6^9 + 7^6 + 8^3 + 9^2 + 11^5 + 12^4.

David A. Corneth, Table of n, a(n) for n = 1..152

Cf. A385969, A386968.

A386698 Values of w in the quartets (4, u, v, w) of type 2; i.e., values of w for solutions to 4(4 + u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see A386288.

Original entry on oeis.org

8, 19, 5, 10, 23, 12, 27, 9, 16, 35, 3, 18, 39, 20, 43, 13, 22, 47, 24, 51, 1, 26, 55, 7, 17, 28, 59, 30, 63, 32, 67, 1, 6, 21, 34, 71, 36, 75, 2, 38, 79, 5, 8, 25, 40, 83, 3, 42, 87, 44, 91, 10, 29, 46, 95, 15, 48, 99, 5, 50, 103, 3, 12, 33, 52, 107, 6, 9

Offset: 1

Clark Kimberling, Aug 16 2025

Cf. A386288.

A387121 Array read by antidiagonals: T(n,k) is the number of sets of k integer sided cuboids with distinct volumes that fill an n X n X n cube.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 0, 4, 3, 2, 1, 0, 0, 2, 11, 8, 2, 1, 0, 0, 1, 26, 47, 11, 3, 1, 0, 0, 0, 55, 206, 77, 19, 3, 1, 0, 0, 0, 48, 793, 442, 183, 23, 4, 1, 0, 0, 0, 23, 2653, 2451, 1531, 259, 35, 4, 1, 0, 0, 0, 0, 6706, 13022, 12178

Offset: 1

Janaka Rodrigo, Aug 16 2025

The partitions here must be valid packings of the n X n X n cube, hence T(n,k) is generally less than the number of partitions of n^3 into distinct cuboids (x,y,z) with 1 <= x,y,z <= n and no pair of triplets having equal volume x*y*z.

			Array begins:
  1     0     0     0     0
  1     0     0     0     0
  1     1     2     4     2
  1     1     3    11    26
  1     2     8    47   206
  1     2    11    77   442
  1     3    19   183  1531
  1     3    23   259  2661
  1     4    35   457  5574
  1     4    40   599  8514
  ...

Columns are A004526 (k=2), A381847 (k=3), A385580 (k=4), A387040 (k=5).

T(n,1) = 1, T(n,k) = 0 for k > n^3.

More terms from Sean A. Irvine, Aug 25 2025

A387083 Expansion of e.g.f. (2(1-x)^2)/(2-4x+x^2).

Original entry on oeis.org

1, 0, 1, 6, 42, 360, 3690, 44100, 602280, 9253440, 157966200, 2966317200, 60765843600, 1348539192000, 32229405608400, 825285553092000, 22541609025936000, 654175871661312000, 20101465198839024000, 651991603501798560000, 22260385752292527840000

Offset: 0

Enrique Navarrete, Aug 16 2025

For n > 0, a(n) is the number of ways to sit n people around circular tables, choose 2 people from each table, and linear order the tables.

			a(6)=3690 since for 6 people the number of ways to do the combined tasks in the comment are: 1800 using one table, 1080 using two tables with 4 and 2 people, 720 using two tables with 3 people each, and 90 using three tables with 2 people each.

Cf. A001286.

With[{m = 20}, CoefficientList[Series[(2*(1 - x)^2)/(2 - 4*x + x^2), {x, 0, m}], x] * Table[n!, {n, 0, m}]] (* Amiram Eldar, Aug 16 2025 *)

For n > 0, a(n) = ((1 + sqrt(2))^(n-1) - (sqrt(2) - 1)^(n-1)) * n! / 2^(n/2 + 1). - Vaclav Kotesovec, Aug 18 2025
D-finite with recurrence 2*a(n) -4*n*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Aug 26 2025
a(n) = A060995(n)*n!/2^n. - R. J. Mathar, Aug 26 2025

A387094 An infinite variant of the EKG sequence (A064413) such that for any n > 0, a(2n) > a(2n-1) and a(2n) > a(2n+1).

Original entry on oeis.org

1, 4, 2, 6, 3, 12, 8, 10, 5, 15, 9, 18, 14, 20, 16, 22, 11, 33, 21, 27, 24, 26, 13, 39, 30, 32, 28, 34, 17, 51, 36, 38, 19, 57, 42, 44, 40, 45, 25, 35, 7, 56, 46, 50, 48, 54, 52, 58, 29, 87, 60, 62, 31, 93, 63, 66, 55, 70, 49, 84, 64, 72, 68, 74, 37, 111, 69

Offset: 1

Rémy Sigrist, Aug 16 2025

For the sequence to be infinite, when computing a term with even index, say a(2*n) for some n > 1, we must ensure that some missing value v < a(2*n) such that gcd(a(2*n), v) != 1 exists.

Will every integer appear in the sequence?

			Sequence begins:
  n   a(n)
  --  ----
   1     1
   2     4
   3     2
   4     6
   5     3
   6    12
   7     8
   8    10
   9     5
  10    15
  11     9
  12    18
  13    14
  14    20
  15    16
  16    22
  17    11

Rémy Sigrist, Table of n, a(n) for n = 1..10001
Rémy Sigrist, PARI program
Index entries for sequences related to EKG sequence

Cf. A064413.

PARI
```
\\ See Links section.
```

A387107 a(n) = psi(2^n-1), where psi = A002322.

Original entry on oeis.org

1, 2, 6, 4, 30, 6, 126, 16, 72, 30, 88, 12, 8190, 126, 150, 256, 131070, 72, 524286, 120, 1008, 2728, 178480, 240, 1800, 8190, 262656, 1008, 39672, 1650, 2147483646, 65536, 2397912, 131070, 1106280, 216, 616318176, 524286, 12743640, 61680, 26815350376, 43344, 1186422030

Offset: 1

Emanuele Pace, Aug 16 2025

Cf. A002322, A000225.

a[n_] := CarmichaelLambda[2^n - 1]; Array[a, 50] (* Amiram Eldar, Aug 16 2025 *)

a(n) = lcm(znstar(2^n-1)[2]); \\ Michel Marcus, Aug 22 2025

a(n) = A002322(A000225(n)).

A387095 a(1) = 1 for the single prime 3; for n>=2, a(n) is the number of primes between 2^n and 2^(n+1) whose pairs lay symmetrically at each side of the center 3*2^(n-1) of that interval.

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 8, 22, 26, 42, 92, 128, 218, 416, 750, 1300, 2342, 4136, 7440, 13572, 24820, 45420, 82922, 152964, 282626, 522354, 972388, 1809744, 3379508, 6318652, 11855790, 22277960, 41917480

Offset: 1

Tamas Sandor Nagy, Aug 16 2025

These primes form the sequence A092570.
With a different approach by seeking symmetries in the distribution of primes, we arrive at the same sequence as A092570. Here, the numbers of primes that fit those symmetries in their ranges form this sequence. Those ranges, interestingly, lay between the powers of 2.
The sum of each prime pair in a range is 3*2^n. (Sequence A007283).
In rows 2, 3, and 7, the middle primes (5, 7), (11, 13) and (191, 193) respectively are just next to the centers of their ranges, and with gaps of size 2 in between them, they are twin prime pairs.
If it could be proved that a derived sequence, 0, 1, 1, 5, 5, 7, 1, ..., that is the distances of the closest primes to the centers 3*2^n of the ranges contains infinitely many 1's, which always correspond to twin primes, then that would also prove the infinitude of twin primes. Of that particular form, A181494 shows the only known eight primes.

			Intervals:                         Primes                       Centers       Count
                                in A092570:                  of intervals:    a(n):
  [2,4]                              3                             3            1
  [4,8]                           5     7                          6            2
  [8,16]                         11    13                         12            2
  [16,32]                  17    19    29    31                   24            4
  [32,64]                  37    43    53    59                   48            4
  [64,128]           79    83    89   103   109   113             96            6
  [128,256]   151   157   173   191   193   211   227   233      192            8
  ...
In row 5 for the interval 2^5 = 32 to 2^6 = 64, the prime pair 37 and 59 lay symmetrically to each side of the center of that interval, 3*2^4 = 48 as abs(48-37) = abs(48-59) = 11, and their sum 37 + 59 = 96, which is 3*2^5. So are 43 and 53 in similarly symmetrical positions in that range, with their sum being 96 also.

Cf. A000040, A007283, A092570, A181494.

a[n_] := Module[{c = 0, r = r1 = 3*2^(n-1)}, While[(r1 = NextPrime[r1, -1]) > 2^n, If[PrimeQ[2*r - r1], c += 2]]; c]; a[1] = 1; Array[a, 20] (* Amiram Eldar, Aug 16 2025 *)

a(13)-a(33) from Amiram Eldar, Aug 16 2025

A387106 Number of iterations of seven segments count x -> A074458(x) to go from n to a fixed point.

Original entry on oeis.org

1, 2, 1, 1, 0, 0, 0, 1, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 2, 4, 4, 2, 4, 2, 2, 3, 4, 2, 2, 4, 4, 2, 4, 2, 2, 3, 4, 4, 1, 2, 2, 3, 2, 4, 3, 2, 2, 2, 2, 4, 4, 2, 4, 2, 2, 3, 4, 3, 3, 2, 2, 4, 2, 3, 4, 3, 2, 4, 1, 2, 2, 3, 2, 4, 3, 2, 2, 3, 2, 3, 3, 2, 3, 3

Offset: 0

Marco Ripà, Aug 16 2025

A074458 is a strictly decreasing function A063720(x) < x whenever x >= 10 and all single digit x reach a fixed point A063720(x) = x with x in {4, 5}.

This sequence is unbounded and the first occurrence of a(n) = k is at n = A338255(k + 2) for any k >= 3.

			For n = 10, the a(10) = 3 steps are 10 -> 8 -> 7 -> 4 segments, and 4 is a fixed point A074458(4) = 4.

Wikipedia, Seven-segment display.

Cf. A074458, A328330, A385249, A386910.

Cf. A006942, A010371, A063720, A277116 (segments variation).

cp's OEIS Frontend

User: 16

A385476 Values of u in the quartets (1, u, v, w) of type 3; i.e., values of u for solutions to (1 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A387100 a(n) is the least number that can be written in exactly n ways as s_1^x_1 + ... + s_t^x_t, with 1 < s_1 < ... < s_t and {s_1,..., s_t} = {x_1,..., x_t}.

Views

Author

Keywords

Examples

Crossrefs

A387099 Numbers that can be written in exactly five ways as s_1^x_1 + ... + s_t^x_t, with 1 < s_1 < ... < s_t and {s_1,..., s_t} = {x_1,..., x_t} for some t > 0.

Views

Author

Keywords

Examples

Links

Crossrefs

A386698 Values of w in the quartets (4, u, v, w) of type 2; i.e., values of w for solutions to 4(4 + u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see A386288.

Views

Author

Keywords

Crossrefs

A387121 Array read by antidiagonals: T(n,k) is the number of sets of k integer sided cuboids with distinct volumes that fill an n X n X n cube.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A387083 Expansion of e.g.f. (2*(1-x)^2)/(2-4*x+x^2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A387094 An infinite variant of the EKG sequence (A064413) such that for any n > 0, a(2*n) > a(2*n-1) and a(2*n) > a(2*n+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A387107 a(n) = psi(2^n-1), where psi = A002322.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A387095 a(1) = 1 for the single prime 3; for n>=2, a(n) is the number of primes between 2^n and 2^(n+1) whose pairs lay symmetrically at each side of the center 3*2^(n-1) of that interval.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A387106 Number of iterations of seven segments count x -> A074458(x) to go from n to a fixed point.

Views

Author

A387083 Expansion of e.g.f. (2(1-x)^2)/(2-4x+x^2).

A387094 An infinite variant of the EKG sequence (A064413) such that for any n > 0, a(2n) > a(2n-1) and a(2n) > a(2n+1).