A002064 -id:A002064 - cp's OEIS frontend

A156708 Triangle read by rows, binomial transform of A154325.

1, 2, 1, 4, 4, 1, 8, 11, 5, 1, 16, 26, 16, 6, 1, 32, 57, 42, 22, 7, 1, 64, 120, 99, 64, 29, 8, 1, 128, 247, 219, 163, 93, 37, 9, 1, 256, 502, 466, 382, 256, 130, 46, 10, 1, 512, 1013, 968, 848, 638, 386, 176, 56, 11, 1

Offset: 0

Gary W. Adamson, Feb 13 2009

Row sums = A002064: (1, 3, 9, 25, 65, 161,...).

			First few rows of the triangle =
1;
2, 1;
4, 4, 1;
8, 11, 5, 1;
16, 26, 16, 6, 1;
32, 57, 42, 22, 7, 1;
64, 120, 99, 64, 29, 8, 1;
128, 247, 219, 163, 93, 37, 9, 1;
256, 502, 466, 382, 256, 130, 46, 10, 1;
512, 1013, 968, 848, 638, 386, 176, 56, 11, 1;
1024, 2036, 1981, 1816, 1486, 1024, 562, 232, 67, 12, 1;
...

Cf. A154325, A002064

Triangle read by rows, A007318 * A154325

A195418 a(n) = phi(C(n)) / gcd(C(n)-1, phi(C(n))), where C(n) is the n-th Cullen number.

Original entry on oeis.org

1, 1, 3, 5, 3, 33, 5, 33, 341, 1045, 189, 1299, 891, 4437, 9477, 581, 3855, 105525, 27825, 23751, 173043, 10531345, 56511, 2386125, 380955, 256861, 24926139, 5108467, 32397379, 930343095, 930291, 36512775

Offset: 0

Alonso del Arte, Sep 20 2011

When C(n) is prime (or 1), then a(n) = 1; that is, n is in A005849.

On the penultimate page of their paper, Grau and Luca ask for "a good (large) lower bound on this quantity which is valid for all n and which tends to infinity with n."

			a(2) = 3 because the second Cullen number is 9; phi(9) = 6, therefore 6/gcd(8, 6) = 6/2 = 3.

Amiram Eldar, Table of n, a(n) for n = 0..848
José María Grau Ribas and Florian Luca, Cullen numbers with the Lehmer property, Proceedings of the American Mathematical Society, Vol. 140, No. 1 (2012), pp. 129-134, preprint, arXiv:1103.3578 [math.NT], Mar 18, 2011.

Cf. A000010, A002064, A005849, A160595.

cullen[n_] := n(2^n) + 1; Table[EulerPhi[cullen[n]]/GCD[cullen[n] - 1, EulerPhi[cullen[n]]], {n, 0, 39}]

a(n)=my(C=n<Charles R Greathouse IV, Feb 05 2013

A274602 Triangle read by rows: T(n,k) = k*(n-k+1)^2 + n - k, 0 <= k <= n.

Original entry on oeis.org

0, 1, 1, 2, 5, 2, 3, 11, 9, 3, 4, 19, 20, 13, 4, 5, 29, 35, 29, 17, 5, 6, 41, 54, 51, 38, 21, 6, 7, 55, 77, 79, 67, 47, 25, 7, 8, 71, 104, 113, 104, 83, 56, 29, 8, 9, 89, 135, 153, 149, 129, 99, 65, 33, 9, 10, 109, 170, 199, 202, 185, 154, 115, 74, 37, 10

Offset: 1

Juri-Stepan Gerasimov, Dec 01 2016

Mirrored version of a(n) is T(n,k) = (n-k)*(k+1)^2+k, 0 <= k <= n, read by rows:
0
 1
 5   2
 9  11   3
13  20  19   4
17  29  35  29   5
As an infinite square array (matrix) with comments:
 1   2   3   4   5                      A001477
 5  11  19  29  41                      A028387
 9  20  35  54  77                      A014107
13  29  51  79 113                      A144391
17  38  67 104 149                      A182868
21  47  83 129 185

			0; 1,1; 2,5,2; 3,11,9,3; 4,19,20,13,4; 5,29,35,29,17,5; ...
As an infinite triangular array:
0
1   1
2   5   2
3  11   9    3
4  19  20   13    4
5  29  35   29   17    5
As an infinite square array (matrix) with comments:
0   1   2    3    4    5                   A001477
1   5   9   13   17   21                   A016813
2  11  20   29   38   47                   A017185
3  19  35   51   67   83
4  29  54   79  104  129
5  41  77  113  149  185

Cf. A002064, A001477, A016813, A017185, A062158 (central column). A028387, A014107, A144391, A182868.

Cf. Triangle read by rows: T(n,k) = k*(n-k+1)^m+n-k, 0 <= k <= n: A003056 (m = 0), A059036 (m = 1), A278910 (m = k).

/* As triangle */ [[k*(n-k+1)^2+n-k: k in [0..n]]: n in [0..10]];

Table[k (n - k + 1)^(k + #) + n - k &[2 - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 02 2016 *)

A278907 a(n) = floor((n2^(n+1)+2)/(2n-(-1)^n+3)) - floor((n2^(n+1)-2)/(2n-(-1)^n+3)).

Original entry on oeis.org

2, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1

Offset: 0

Juri-Stepan Gerasimov, Nov 30 2016

If a(n) = b(n) - c(n), then
b(n) = 1, 1, 3, 5, 13, 23, 55, 99, 227, 419, 931, 1733, 3781, 7099, 15291, 28913, 61681, 117275, 248347, 474355, 998643, 1914791, 4011943, 7717519, 16106127, 31068918, 64623350, 124961333, 259179061, 502234079, 1039104991, ...
c(n) = -1, 0, 2, 4, 12, 22, 54, 99, 227, 418, 930, 1732, 3780, 7099, 15291, 28912, 61680, 117274, 248346, 474355, 998643, 1914790, 4011942, 7717519, 16106127, 31068918, 64623350, 124961332, 259179060, 502234078, 1039104990, ...

			a(0) = b(0) - c(0) = 1 - (-1) = 2,
a(1) = b(1) - c(1) = 1 - 0 = 1,
a(2) = b(2) - c(2) = 3 - 2 = 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A002064 (Cullen numbers).

[((n*2^(n+1)+2) div (2*n-(-1)^n+3))-((n*2^(n+1)-2) div (2*n-(-1)^n+3)): n in [0..100]];

a[n_] := Floor[(n*2^(n + 1) + 2)/(2*n - (-1)^n + 3)] - Floor[(n*2^(n + 1) - 2)/(2*n - (-1)^n + 3)]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Apr 20 2017 *)

for(n=0,50, print1(floor((n*2^(n+1)+2)/(2*n-(-1)^n+3)) - floor((n*2^(n+1)-2)/(2*n-(-1)^n+3)), ", ")) \\ G. C. Greubel, Apr 20 2017

Definition corrected by R. J. Mathar, Dec 02 2016

A340841 Decimal expansion of Sum_{n>=0} 1/(n*2^n + 1).

Original entry on oeis.org

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

Offset: 1

Marco Ripà, Jan 23 2021

This constant is the sum of reciprocals of Cullen numbers.

			1.5106374481309548528461974960802456940320940529869241186970756482342266...

Cf. A002064.

First[RealDigits[NSum[1/(n*2^n+1), {n, 0, Infinity}, NSumTerms -> 100, Method -> {"NIntegrate", "MaxRecursion" -> 10}, WorkingPrecision -> 100]]] (* Stefano Spezia, Jan 24 2021 *)

suminf(n=0, 1/(n*2^n + 1)) \\ Michel Marcus, Jan 24 2021

Equals Sum_{n>=0} 1/A002064(n).

A367010 a(n) is the sum of the divisors of n*2^n + 1.

Original entry on oeis.org

1, 4, 13, 31, 84, 192, 576, 1344, 2736, 5040, 13680, 24276, 56000, 153842, 308416, 538920, 1110276, 2909040, 5495040, 14446080, 31374720, 45955008, 106119552, 233997312, 527587200, 1184932800, 3247522560, 4365239040, 7784309910, 16265125632, 36250560000

Offset: 0

Sean A. Irvine, Oct 31 2023

nonn

The numbers n*2^n+1 are called Cullen numbers.

Amiram Eldar, Table of n, a(n) for n = 0..865

Cf. A002064, A000203, A063515, A367007, A367009.

DivisorSigma[1, Table[n*2^n+1, {n,0,30}]] (* Paul F. Marrero Romero, Dec 17 2023 *)

a(n) = sigma(n*2^n + 1) = A000203(A002064(n)).

A373398 Triangle read by rows: T(n, k) = number of k-element subobjects of an n-element set in the category of relations, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 9, 1, 1, 15, 55, 25, 1, 1, 31, 285, 395, 65, 1, 1, 63, 1351, 5045, 2555, 161, 1, 1, 127, 6069, 56931, 78685, 15211, 385, 1, 1, 255, 26335, 592725, 2091171, 1101021, 85099, 897, 1, 1, 511, 111645, 5834515, 50334765, 67590387, 14169405, 454315, 2049, 1

Offset: 0

Keith J. Bauer, Jun 03 2024

A subobject of an object A is an object S equipped with a monomorphism S -> A, up to isomorphism in the category of objects equipped with such morphisms. Objects in the category of relations are sets, morphisms are relations, and composition is relation composition.
Objects and morphisms in Rel can be re-characterized as free complete join-semilattices (the power set of a set with join being union) and join-equivariant maps, respectively. Therefore, subobjects in Rel can be re-characterized as injective n X k matrices of truth values. Because every injective matrix of truth values can be shown to have pivots, subobjects can be counted via Schubert cells and this results in a family of generating functions describing the entire triangle. Short proof: if a monomorphism does not have a row consisting of all 0's except for one column in particular, then consider where it sends the column vector containing all 1's and the column vector containing all 1's but with the corresponding row flipped to 0. It cannot possibly send these vectors to two different vectors. (Here 0 and 1 represent false and true, respectively. Note that addition is logical "or" and multiplication is logical "and".)
Because Rel is self-dual, this sequence also counts quotient objects.
Entries not in the triangle's range are equal to 0 because there is no monomorphism from a k-element set to an n-element set when k > n.
All monomorphisms in Rel are regular, i.e., the equalizer of a pair of morphisms. In some categories, subobjects are taken to only be regular monomorphisms, or are at least distinguished; for example, a normal subgroup is (the domain of) a regular monomorphism in the category of groups. Because all monomorphisms in Rel are regular, there is no ambiguity in what a subobject in Rel is. See the link for a proof of this fact.

			There are 9 2-element subobjects of a 3-element set in Rel. As truth matrices:
  [1 0] [1 0] [0 0] [1 0] [0 1] [0 1] [1 1] [1 0] [1 0]
  [0 1] [0 0] [1 0] [0 1] [1 0] [0 1] [1 0] [1 1] [0 1]
  [0 0] [0 1] [0 1] [0 1] [0 1] [1 0] [0 1] [0 1] [1 1]
To convert to relations, note that each entry corresponds to whether
  [(1,1) (2,1)]
  [(1,2) (2,2)]
  [(1,3) (2,3)]
is in the relation.
Triangle starts:
  1,
  1,   1,
  1,   3,      1,
  1,   7,      9,       1,
  1,  15,     55,      25,        1,
  1,  31,    285,     395,       65,        1,
  1,  63,   1351,    5045,     2555,      161,        1,
  1, 127,   6069,   56931,    78685,    15211,      385,      1,
  1, 255,  26335,  592725,  2091171,  1101021,    85099,    897,    1,
  1, 511, 111645, 5834515, 50334765, 67590387, 14169405, 454315, 2049, 1,
  ...

Keith J. Bauer, Every monomorphism in Rel is regular.
nLab, Rel

T(n, 0) = A000012(n).
T(n, 1) = A000225(n).
T(n, 2) = A016269(n - 2).
T(n, 3) = A028130(n - 3).
T(n, n) = A000012(n).
T(n, n - 1) = A002064(n - 1).
Analogous sequence in the category Set: A007318.

T[n_,k_]:=SeriesCoefficient[(1 / (1 - 2^k* x)) * Product[1 / (1 - (2^k - 2^i) * x),{i,0,k-1}],{x,0,n}]; Table[T[n-k,k],{n,0,9},{k,0,n}]//Flatten (* Stefano Spezia, Jun 04 2024 *)

dim = 10
def getGF(n):
    R. = PowerSeriesRing(ZZ, 'x', dim)
    f = 1 / (1 - 2^n * x)
    for k in range(n):
        f = f / (1 - (2^n - 2^k) * x)
    return f
for n in range(dim):
    print([getGF(k).list()[n - k] for k in range(n + 1)])

G.f.: Sum_{n>=0} T(n + k, k) * x^n = (1 / (1 - 2^k * x)) * Product_{i=0..k-1} (1 / (1 - (2^k - 2^i) * x)).

A382887 Numbers k such that (k2^d + 1)(d*2^k + 1) is semiprime for some divisor d of k.

Original entry on oeis.org

1, 2, 8, 12, 30, 51, 63, 141, 201, 209, 534, 4713, 5795, 6611, 7050, 18496, 24105, 32292, 32469, 52782, 59656, 80190, 90825

Offset: 1

Juri-Stepan Gerasimov, Apr 07 2025

a(24) > 10^5. - Michael S. Branicky, Apr 08 2025

			12 is in this sequence because (12*2^3 + 1)*(3*2^12 + 1) = 97*12289 is semiprime for divisor 3 of 12.

Supersequence of A005849.

Cf. A001358, A002064, A382646.

[n: n in [1..1000] | not #[d: d in Divisors(n) | IsPrime(d*2^n+1) and IsPrime(n*2^d+1)] eq 0];

isok(k) = fordiv(k, d, if (ispseudoprime(k*2^d+1) && ispseudoprime(d*2^k+1), return(1))); \\ Michel Marcus, Apr 16 2025

from itertools import count, islice
from sympy import isprime, divisors
def A382887_gen(): # generator of terms
    yield from filter(lambda k:any(isprime((k<A382887_list = list(islice(A382887_gen(),10)) # Chai Wah Wu, Apr 15 2025

a(10) inserted and a(15)-a(23) from Michael S. Branicky, Apr 08 2025

A086093 a(n) = 3^n + 2n4^(n-1).

Original entry on oeis.org

1, 5, 25, 123, 593, 2803, 13017, 59531, 268705, 1199331, 5301929, 23245819, 101194737, 437801939, 1883831161, 8067412587, 34402785089, 146158028227, 618862711113, 2612502377435, 10998603062161, 46189948719795, 193545427548185, 809334701221963, 3377982150064353

Offset: 0

Paul Barry, Jul 10 2003

Binomial transform of A084859. Second binomial transform of Cullen numbers A002064.

Index entries for linear recurrences with constant coefficients, signature (11,-40,48).

Cf. A002064, A084859.

CoefficientList[Series[(1 - 6*x + 10*x^2)/((1 - 3*x)*(1 - 4*x)^2), {x, 0, 30}], x] (* Wesley Ivan Hurt, Oct 30 2022 *)

G.f.: (1-6*x+10*x^2)/((1-3*x)*(1-4*x)^2).

a(n) = 11*a(n-1) - 40*a(n-2) + 48*a(n-3). - Wesley Ivan Hurt, Oct 30 2022

More terms from Wesley Ivan Hurt, Oct 30 2022

A119865 Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 1, 0, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 9, 6, 1, 8, 25, 26, 10, 1, 16, 65, 95, 60, 15, 1, 32, 161, 308, 279, 120, 21, 1, 64, 385, 917, 1099, 693, 217, 28, 1, 128, 897, 2566, 3856, 3256, 1526, 364, 36, 1

Offset: 0

Philippe Deléham, Jul 31 2006

			Triangle begins:
1;
1, 1;
2, 3, 1;
4, 9, 6, 1;
8, 25, 26, 10, 1;
16, 65, 95, 60, 15, 1;
32, 161, 308, 279, 120, 21, 1;
64, 385, 917, 1099, 693, 217, 28, 1;
128, 897, 2566, 3856, 3256, 1526, 364, 36, 1;

Cf. Diagonals : A011782, A002064 ; A000012, A000217.

Sum_{k =0..n}T(n,k)= A087944(n).
Sum_{k=0..n}(-1)^k*2^(n-k)*T(n,k)= n^2-n+1= A002061(n).
Sum_{k=0..n}(-1)^k*T(n,k)=0^n= A000007(n).
G.f.: (1-2*x-2*x*y++x^2+x^2*y+x^2*y^2)/(1-3*x-3*x*y+2*x^2+4*x^2*y+3*x^2*y^2-x^3*y^2-x^3*y^3). - Philippe Deléham, Nov 09 2013
T(n,k) = 3*T(n-1,k) + 3*T(n-1,k-1) - 2*T(n-2,k) - 4*T(n-2,k-1) - 3*T(n-2,k-2) + T(n-3,k-2) + T(n-3,k-3), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = 2, T(2,1) = 3, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 09 2013

cp's OEIS Frontend

A156708 Triangle read by rows, binomial transform of A154325.

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A195418 a(n) = phi(C(n)) / gcd(C(n)-1, phi(C(n))), where C(n) is the n-th Cullen number.

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A274602 Triangle read by rows: T(n,k) = k*(n-k+1)^2 + n - k, 0 <= k <= n.

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Magma

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A278907 a(n) = floor((n*2^(n+1)+2)/(2*n-(-1)^n+3)) - floor((n*2^(n+1)-2)/(2*n-(-1)^n+3)).

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A340841 Decimal expansion of Sum_{n>=0} 1/(n*2^n + 1).

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A367010 a(n) is the sum of the divisors of n*2^n + 1.

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A373398 Triangle read by rows: T(n, k) = number of k-element subobjects of an n-element set in the category of relations, n >= 0, 0 <= k <= n.

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A382887 Numbers k such that (k*2^d + 1)*(d*2^k + 1) is semiprime for some divisor d of k.

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A278907 a(n) = floor((n2^(n+1)+2)/(2n-(-1)^n+3)) - floor((n2^(n+1)-2)/(2n-(-1)^n+3)).

A382887 Numbers k such that (k2^d + 1)(d*2^k + 1) is semiprime for some divisor d of k.

A086093 a(n) = 3^n + 2n4^(n-1).