A007583 -id:A007583 - cp's OEIS frontend

A250291 Numbers k such that (2^k+1)/3 is a semiprime.

29, 37, 41, 47, 49, 53, 67, 71, 73, 103, 107, 109, 139, 151, 179, 223, 229, 251, 269, 277, 311, 349, 353, 433, 457, 487, 503, 599, 601, 613, 619, 643, 739, 757, 827, 839, 1031, 1061, 1117, 1123, 1217

Offset: 1

Eric Chen, Dec 24 2014

If (2^k+1)/3 is a semiprime, k must be prime or the square of a prime; the only known square of a prime in this sequence is 49.

a(42) >= 1259.

			a(1) = 29 so (2^29+1)/3 = 178956971 = 59 * 3033169 is a semiprime.

Samuel Wagstaff, The Cunningham Project.

Cf. A000978, A085724, A007583.

a(40)-a(41) from Max Alekseyev, Feb 25 2025

A297180 a(n) is the smallest positive integer of length (distance from origin) n in the Cayley graph of the integers generated by all powers of 7.

Original entry on oeis.org

1, 2, 3, 4, 11, 18, 25, 74, 123, 172, 515, 858, 1201, 3602, 6003, 8404, 25211, 42018, 58825, 176474, 294123, 411772, 1235315, 2058858, 2882401, 8647202, 14412003, 20176804, 60530411, 100884018, 141237625, 423712874, 706188123, 988663372, 2965990115, 4943316858

Offset: 1

N. J. A. Sloane, Dec 28 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 1..1000
G. Bell, A. Lawson, N. Pritchard, and D. Yasaki, Locally infinite Cayley graphs of the integers, arXiv:1711.00809 [math.GT], 2017.

Cf. A007583, A007051, A294566, A297181, A297182 for the sequences obtained if "7" is replaced by a different prime.

Conjectures from Colin Barker, Dec 28 2017: (Start)
G.f.: x*(1 + x + x^2 - 6*x^3) / ((1 - x)*(1 - 7*x^3)).
a(n) = a(n-1) + 7*a(n-3) - 7*a(n-4) for n>4.
(End)
The second conjecture by Colin Barker is true up to n=1000. - Lars Blomberg, Dec 29 2017

Terms a(21) and beyond from Lars Blomberg, Dec 29 2017

A306696 Lexicographically earliest sequence of nonnegative terms such that for any n > 0 and k > 0, if a(n) >= a(n+k), then a(n+2*k) <> a(n+k).

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Mar 05 2019

nonn

This sequence has graphical features in common with A286326.

			For n=1:
- a(1) = 0 is suitable.
For n=2:
- a(2) = 0 is suitable.
For n=3:
- a(1) = 0 >= a(2) = 0, so a(3) <> 0,
- a(3) = 1 is suitable.
For n=4:
- a(2) = 0 < a(3) = 1,
- a(4) = 0 is suitable.
For n=5:
- a(3) = 1 >= a(4) = 0, so a(5) <> 0,
- a(1) = 0 < a(3) = 1,
- a(5) = 1 is suitable.
For n=6:
- a(4) = 0 < a(5) = 1,
- a(2) = 0 >= a(4) = 0, so a(6) <> 0,
- a(6) = 1 is suitable.
For n=7:
- a(5) = 1 >= a(6) = 1, so a(7) <> 1,
- a(3) = 1 >= a(5) = 1, so a(7) <> 1,
- a(1) = 0 >= a(4) = 0, so a(7) <> 0,
- a(7) = 2 is suitable.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C++ program for A306696
Rémy Sigrist, Colored scatterplot of (n, a(n)) for n = 1..100000 (where the color is function of the least k >= 0 such that (1+a(n))/n >= 2^k/A007583(k))

Cf. A000079, A007583, A164095, A181497, A286326.

Empirically:
- a(n) = 0 iff n is a power of 2 (A000079),
- a(n) = 1 iff n = 3 or belongs to A164095,
- a(2*n) = a(n),
- A181497(n) is the least k such that a(k) = n.

A318236 a(n) = (32^(4n+3) + 1)/5.

Original entry on oeis.org

5, 77, 1229, 19661, 314573, 5033165, 80530637, 1288490189, 20615843021, 329853488333, 5277655813325, 84442493013197, 1351079888211149, 21617278211378381, 345876451382054093, 5534023222112865485, 88544371553805847757, 1416709944860893564109, 22667359117774297025741

Offset: 0

Jianing Song, Aug 21 2018

a(n) is the smallest positive multiplicative inverse of 5 modulo 2^(4*n+3).

In binary, a(n) is written as 10011001...1001101 where "1001" appears n times. When n approaches infinity we get the 2-adic expansion of 1/5: ...10011001101. Similarly, the 2-adic expansion of 1/3 is ...101010101011.

			The smallest solution to 5*x == 1 (mod 8) is x = (3*2^3 + 1)/5 = 5.
The smallest solution to 5*x == 1 (mod 128) is x = (3*2^7 + 1)/5 = 77.

Jianing Song, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (17,-16).

A007583 gives the smallest positive multiplicative inverse of 3 modulo 2^(2*n) and 2^(2*n+1), A299960 gives the smallest positive multiplicative inverse of 5 modulo 2^(4*n), 2^(4*n+1) and 2^(4*n+2).

[(3*2^(4*n + 3) + 1)/5: n in [0..20]]; // Vincenzo Librandi, Aug 24 2018

Table[(3 2^(4 n + 3) + 1) / 5, {n, 0, 20}] (* Vincenzo Librandi, Aug 24 2018 *)
LinearRecurrence[{17,-16},{5,77},20] (* Harvey P. Dale, Sep 25 2020 *)

PARI
```
a(n) = (3*2^(4*n + 3) + 1)/5
```

def A318236(n): return (3*(1<<(n<<2)+3)+1)//5 # Chai Wah Wu, Jul 29 2022

O.g.f.: (5 - 8*x)/((1 - x)*(1 - 16*x)).
E.g.f.: (24*exp(16*x) + exp(x))/5.
a(0) = 5, a(1) = 77; for n>1, a(n) = 17*a(n-1) - 16*a(n-2).

A334638 Three-column array pPT read by rows: subsequence of primitive Pythagorean triples (x, y, z) with x = A153893^2 - A000079^2, y = 2A153893A000079, z = A153893^2 + A000079^2, ordered by increasing z.

Original entry on oeis.org

3, 4, 5, 21, 20, 29, 105, 88, 137, 465, 368, 593, 1953, 1504, 2465, 8001, 6080, 10049, 32385, 24448, 40577, 130305, 98048, 163073, 522753, 392704, 653825, 2094081, 1571840, 2618369, 8382465, 6289408, 10479617, 33542145, 25161728, 41930753, 134193153, 100655104, 167747585, 536821761, 402636800, 671039489, 2147385345, 1610579968, 2684256257

Offset: 0

Ralf Steiner, May 07 2020

Let [h21] = {{1, 3}, {0, 2}} be the matrix [h_2]*[h_1] in Firstov's notation, from eqs. (24) and (39). Then primitive Pythagorean triples (pPT) (x(n), y(n), z(n)) = (u(n)^2 - v(n)^2, 2*u(n)*v(n), u(n)^2 + v(n)^2), with u(n) and v(n) of different parity, gcd(u(n), v(n)) = 1, and u(n) > v(n) > 0, are generated by (u(n), v(n))^T = [h21]^n*(2,1)^T (T for transpose).
For n > 0: (x(n), y(n), z(n)) = (1, 0, 1) (mod 4). Thus some z are Pythagorean primes (A002144).
The triples converge to the proportion (4:3:5) with:
lim_{n->infinity} x(n)/y(n) = 4/3, lim_{n->infinity} y(n)/z(n) = 3/5.
Altitude h(n) = x(n)*y(n)/z(n) is an irreducible fraction because of primitivity.
From Wolfdieter Lang, Jun 13 2020: (Start)
[h21]^n = sqrt(2)^n*(S(n, 3/sqrt(2))*[1_3] + S(n-1, 3/sqrt(2))*(1/sqrt(2))*([h21] - 3*[1_3])) with the Chebyshev S polynomials (A049310).
u(n) = sqrt(2)^n*(2*S(n, 3/sqrt(2)) - (1/sqrt(2))*S(n-1, 3/sqrt(2)))
     = A153893(n),
v(n) = sqrt(2)^n*(S(n, 3/sqrt(2)) - (1/sqrt(2))*S(n-1, 3/sqrt(2)))
     = A000079(n). Proof from the recurrence, using the Cayley-Hamilton theorem.
With the monic Chebyshev T polynomials, called R in A127672:
x(n)/3 = 2^(n+1)*(R(2*(n+1), 3/sqrt(2)) - (sqrt(2)/3)*R(2*n+1,3/sqrt(2)) - 1) = A171477(n),
y(n)/4 = 3*2^(n-1)*(sqrt(2)*R(2*n+1,3/sqrt(2)) - R(2*n,3/sqrt(2)) - 1/3)
= A010036(n),
z(n) = 3*2^(n+1)*((3/sqrt(2))*R(2*n+1, 3/sqrt(2)) - (4/3)*R(2*n,3/sqrt(2)) - 1).
Using 2^n*Rnx(2*n, 3/sqrt(2)) = A052539(n) = 2^(2*n) + 1, and
  2^(n)*(sqrt(2)/3)*Rnx(2*n+1, 3/sqrt(2)) = A007583(n) = (2^(2*n + 1) + 1)/3,
produces the explicit formulas given by the author in the formula section.
G.f.s for {x(n)} G0(x) = 3/((1 - 4*x)*(1 - 2*x)*(1 - x)), for {y(n)} G1(x) =  4*(1-x)/((1 - 4*x)*(1 - 2*x)), and for {z(n)} = (5 - 6*x + 4*x^2)/((1 - 4*x)*(1 - 2*x)*(1 - x)). This produces the g.f. for the array, read as sequence {a(n)}: G(x) = G0(x^3) + x*G1(x^3) + x^2*G2(x^3) given in the formula section by Colin Barker.
(End)

			The three-column array pPT(n,k) begins:
n\k        0        1         2
-------------------------------
0:         3        4         5
1:        21       20        29
2:       105       88       137
3:       465      368       593
4:      1953     1504      2465
5:      8001     6080     10049
6:     32385    24448     40577
7:    130305    98048    163073
8:    522753   392704    653825
9:   2094081  1571840   2618369
10:  8382465  6289408  10479617
... - _Wolfdieter Lang_, Jun 13 2020

V. E. Firstov, A Special Matrix Transformation Semigroup of Primitive Pairs and the Genealogy of Pythagorean Triples; Mathematical Notes, volume 84, number 2, August 2008, pages 263-279; Link of the page (for the Russian article).
Ralf Steiner, Weitere spezielle Folge primitiver pythagoraischer Dreiecke, ResearchGate.
Wikipedia, Tree of primitive Pythagorean triples.
Index entries for linear recurrences with constant coefficients, signature (0,0,7,0,0,-14,0,0,8).

Cf. A000079, A010035, A049310, A083329, A093357, A010036, A134057, A153893, A171477.

h21={{1, 3}, {0, 2}}; l = {}; Do[v = MatrixPower[h21, n, {2, 1}]; p = v[[1]]; q = v[[2]];
a = p^2 - q^2; b = 2 p q; c = p^2 + q^2; l = AppendTo[l, {a, b, c}], {n, 0, 14}]; l // Flatten

Vec((3 + 4*x + 5*x^2 - 8*x^4 - 6*x^5 + 4*x^7 + 4*x^8) / ((1 - x)*(1 + x + x^2)*(1 - 2*x^3)*(1 - 4*x^3)) + O(x^35)) \\ Colin Barker, Jun 12 2020

The three-column array PT(n, k) is for k = 0, 1, 2:  x(n), y(n), z(n), for n >= 0, with
x(n) = a(3*n + 0) = A153893(n)^2 - A000079(n)^2 = 1 - 3*2^(n+1) + 2^(2*n+3) = binomial(2^(n+2) - 1, 2) = 3*A171477(n),
y(n) = a(3*n + 1) = 2*A153893(n)*A000079(n) = 2^(n+1)*(-1 + 3*2^n) = 4*A010036(n),
z(n) = a(3*n + 2) = A153893(n)^2 + A000079(n)^2 = 1 - 6*2^n + 10*2^(2*n).
From Colin Barker, May 08 2020: (Start)
G.f. (read as sequence {a(n)}): (3 + 4*x + 5*x^2 - 8*x^4 - 6*x^5 + 4*x^7 + 4*x^8) / ((1 - x)*(1 + x + x^2)*(1 - 2*x^3)*(1 - 4*x^3)).
a(n) = 7*a(n-3) - 14*a(n-6) + 8*a(n-9), for n > 8.
(End)

Edited, and corrected proportion by Wolfdieter Lang, Jun 13 2020

Minor grammatical edits. - N. J. A. Sloane, Sep 12 2020

A364213 The number of trailing 0's in the canonical representation of n as a sum of distinct Jacobsthal numbers (A280049).

Original entry on oeis.org

0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 6, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Jul 14 2023

The even terms of A007583.
This sequence is unbounded. The first position of 2*k is A007583(k) = (2^(2*k+1) + 1)/3.
The asymptotic density of the occurrences of (2*k) in this sequence is 3/4^(k+1).
The asymptotic mean of this sequence is 2/3 and its asymptotic standard deviation is 4/3.

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

Cf. A007583, A007814, A122840, A280049.

Select[IntegerExponent[Range[100], 2], EvenQ]

select(x->!(x%2), vector(100, i, valuation(i, 2)))

a(n) = A122840(A280049(n)).

a(n) = A007583(A003159(n)).

A101480 E.g.f. (1/cosh(x)+tanh(x))(exp(2x)-exp(-x))/3.

Original entry on oeis.org

0, 1, 3, 3, 3, 11, 63, 43, -837, 171, 34023, 683, -1800477, 2731, 132912783, 10923, -12927652917, 43691, 1603253204343, 174763, -246914125142157, 699051, 46232582930156703, 2796203, -10343022775699132197, 11184811, 2724715006195438297863, 44739243, -834839760935753154167037

Offset: 0

Paul Barry, Jan 21 2005

A transform of the Jacobsthal numbers.

a(2n)=A007583(n)=(2*4^n+1)/3.

Cf. A101473, A001045.

With[{nn=30},CoefficientList[Series[(1/Cosh[x]+Tanh[x]) (Exp[2x]- Exp[-x])/3, {x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Aug 22 2012 *)

Corrected by N. J. A. Sloane, Nov 05 2005

A174736 Number of nodes with index 2^2k in a dyadic tree build alternatively with the schema "l" between the index 2^2k and 2^(2k+1) - 1, and the schema "^" between the index 2^(2k+1) and 2^(2k+2) - 1.

Original entry on oeis.org

2, 8, 2048, 8796093022208, 2993155353253689176481146537402947624255349848014848

Offset: 1

Michel Lagneau, Mar 28 2010

nonn

This tree represent the set of the interval [0,1], and the number of nodes of rank n is a(n). The number nodes with index 2^2k is a(2^2k) = (2^(2k+1) + 1)/3= 2^A007583.
Proof :
Let n(k) such that a(2^2k) = 2^n(k). By recurrence we have n(k) = 2^2k - 2^2k-1 + n(k-1) = 2^2k-1 + n(k-1). With n(1) = 3 = 2+1 we obtain : n(k) = 1 + 2(1 + 2^2 + ... + 2^(2k-2)) = (2^(2k+1) + 1)/3.
Graphic representation :
...................^
...........l.................l
...........^.................^
......^........^........^........^
....l...l... l...l... l...l... l...l
....l...l... l...l... l...l... l...l
....l...l... l...l... l...l... l...l
....l...l... l...l... l...l... l...l
....^...^... ^...^... ^...^... ^...^
...^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
.^^^^^^^^ ^^^^^^^^ ^^^^^^^^ ^^^^^^^^

			k=0, a(1) = 2 ; k=1, a(4) = 8 ; k=2, a(16) = 2048 ; k=3, a(64) = 2^43 = 8796093022208.

J. E. Hutchinson, Fractal and self-similarity, Ind. U. Math. J. 30 (1981), 713-747.

for n from 0 to 5 do: p:=(2*4^n + 1)/3:q:=2^p:print(q):od:

a(2^2k) = 2^A007583.

A340547 Square array, read by ascending antidiagonals, where row n gives all solutions n > 0 to A000120(n+1) = A000120((n+1)*k), A000120 is the Hamming weight.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 3, 8, 1, 2, 4, 4, 16, 1, 2, 4, 8, 6, 32, 1, 2, 3, 8, 16, 8, 64, 1, 2, 3, 4, 13, 32, 11, 128, 1, 2, 4, 4, 6, 16, 64, 12, 256, 1, 2, 2, 8, 5, 8, 26, 128, 16, 512, 1, 2, 4, 8, 16, 6, 11, 32, 256, 22, 1024

Offset: 1

Thomas Scheuerle, Jan 11 2021

Solutions to related equation A000120(k) = A000120(k*n) are A340351.

The same sequence without leading ones and only odd solutions is A340441.

			Eight initial terms of rows 1-8 are listed below:
   1: 1, 2, 4, 8, 16, 32, 64, 128, ...
   2: 1, 2, 3, 4,  6,  8, 11,  12, ...
   3: 1, 2, 4, 8, 16, 32, 64, 128, ...
   4: 1, 2, 4, 8, 13, 16, 26,  32, ...
   5: 1, 2, 3, 4,  6,  8, 11,  12, ...
   6: 1, 2, 3, 4,  5,  6,  7,   8, ...
   7: 1, 2, 4, 8, 16, 32, 64, 128, ...
   8: 1, 2, 4, 8, 16, 32, 57,  64, ...
T(3,4) = 8 because: (3+1) in binary is 100 and (3*1)*8 = 32 in binary is 100000, both have 1 bit set to 1.

Cf. A000120, A340351, A340069, A340441.

Cf. A263132 (superset of 1st row), A007583 (1st row), A299960 (2nd row).

T(2n, ...)   = 2^{0,1,2,...}, 2^{0,1,2,...} * row n of A340441.
T(4n+1, ...) = 2^{0,1,2,...}, 2^{0,1,2,...} * row n of A340441.
T(2^n, ...)  = 2^{0,1,2,...}.

A349890 Triangle read by rows: T(n,k) = n * 2^e(n) - (4^e(n) - 1) / 3 - k * (k - 1) / 2 with e(n) = 1 + floor(log_2(n)) for n >= 1 and 1 <= k <= n.

Original entry on oeis.org

1, 3, 2, 7, 6, 4, 11, 10, 8, 5, 19, 18, 16, 13, 9, 27, 26, 24, 21, 17, 12, 35, 34, 32, 29, 25, 20, 14, 43, 42, 40, 37, 33, 28, 22, 15, 59, 58, 56, 53, 49, 44, 38, 31, 23, 75, 74, 72, 69, 65, 60, 54, 47, 39, 30, 91, 90, 88, 85, 81, 76, 70, 63, 55, 46, 36, 107, 106, 104, 101, 97, 92, 86, 79, 71, 62, 52, 41

Offset: 1

Werner Schulte, Dec 04 2021

Conjecture: The terms of the triangle yield a permutation of the positive integers (A000027).

			The triangle T(n, k) for 1 <= k <= n begins:
n\k:   1   2   3   4   5   6   7   8   9  10  11
================================================
01 :   1
02 :   3   2
03 :   7   6   4
04 :  11  10   8   5
05 :  19  18  16  13   9
06 :  27  26  24  21  17  12
07 :  35  34  32  29  25  20  14
08 :  43  42  40  37  33  28  22  15
09 :  59  58  56  53  49  44  38  31  23
10 :  75  74  72  69  65  60  54  47  39  30
11 :  91  90  88  85  81  76  70  63  55  46  36
etc.

Cf. A000027, A000217, A007583.

T(n,k) = my(e=1+logint(n,2)); n*2^e - (4^e-1)/3 - k*(k-1)/2;
row(n) = vector(n, k, T(n,k)); \\ Michel Marcus, Dec 05 2021

T(2^n, 1) = A007583(n) for n >= 0.
T(n, 1) - T(n, n) = A000217(n-1) for n > 0.
T(n, k) = T(n-1, k) + T(n-1, k-1) - T(n-1-2^(e(n-1)-e(n-2)), k-1) with e(n) = 1 + floor(log_2(n)) for n > 3 and 1 < k < n-1 (conjectured).

cp's OEIS Frontend

A250291 Numbers k such that (2^k+1)/3 is a semiprime.

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A297180 a(n) is the smallest positive integer of length (distance from origin) n in the Cayley graph of the integers generated by all powers of 7.

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A306696 Lexicographically earliest sequence of nonnegative terms such that for any n > 0 and k > 0, if a(n) >= a(n+k), then a(n+2*k) <> a(n+k).

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A318236 a(n) = (3*2^(4*n+3) + 1)/5.

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A334638 Three-column array pPT read by rows: subsequence of primitive Pythagorean triples (x, y, z) with x = A153893^2 - A000079^2, y = 2*A153893*A000079, z = A153893^2 + A000079^2, ordered by increasing z.

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A364213 The number of trailing 0's in the canonical representation of n as a sum of distinct Jacobsthal numbers (A280049).

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Mathematica

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A101480 E.g.f. (1/cosh(x)+tanh(x))*(exp(2*x)-exp(-x))/3.

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Mathematica

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A174736 Number of nodes with index 2^2k in a dyadic tree build alternatively with the schema "l" between the index 2^2k and 2^(2k+1) - 1, and the schema "^" between the index 2^(2k+1) and 2^(2k+2) - 1.

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A318236 a(n) = (32^(4n+3) + 1)/5.

A334638 Three-column array pPT read by rows: subsequence of primitive Pythagorean triples (x, y, z) with x = A153893^2 - A000079^2, y = 2A153893A000079, z = A153893^2 + A000079^2, ordered by increasing z.

A101480 E.g.f. (1/cosh(x)+tanh(x))(exp(2x)-exp(-x))/3.