A006127 -id:A006127 - cp's OEIS frontend

A094968 Indices of Fibonacci numbers in Stern's diatomic series A049456 regarded as a single linear sequence.

1, 4, 7, 14, 25, 48, 91, 178, 349, 692, 1375, 2742, 5473, 10936, 21859, 43706, 87397, 174780, 349543, 699070, 1398121, 2796224, 5592427, 11184834, 22369645, 44739268, 89478511, 178956998, 357913969, 715827912, 1431655795, 2863311562

Offset: 0

Paul Barry, May 26 2004

By definition, A049456(a(n))=Fib(n+2).

The rank of Fib(n+2) in row n of A049456 (regarded as an irregular triangle read by rows) is A128209(n) = A001045(n)+1. [Comment edited by N. J. A. Sloane, Nov 23 2016]

Colin Barker, Table of n, a(n) for n = 0..1000
D. H. Lehmer, On Stern's Diatomic Series, Amer. Math. Monthly 36(1) 1929, pp. 59-67.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Cf. A000045, A001045, A049456, A128209.

Vec((1 + x - 4*x^2) / ((1 - x)^2*(1 + x)*(1 - 2*x)) + O(x^30)) \\ Colin Barker, Sep 29 2017

G.f. : (1+x-4*x^2) / ((1-x)*(1-x^2)*(1-2*x)).
a(n) = 2^n + n + Jacobsthal(n).
a(n) = A006127(n) + A001045(n).
From Colin Barker, Sep 29 2017: (Start)
a(n) = ((-1)^(1+n) + 2^(2+n) + 3*n) / 3.
a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4) for n>3.
(End)

A097339 2^n+n^3.

Original entry on oeis.org

1, 3, 12, 35, 80, 157, 280, 471, 768, 1241, 2024, 3379, 5824, 10389, 19128, 36143, 69632, 135985, 267976, 531147, 1056576, 2106413, 4204952, 8400775, 16791040, 33570057, 67126440, 134237411, 268457408, 536895301, 1073768824, 2147513439

Offset: 0

Paul Barry, Aug 05 2004

Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-9,2).

Cf. A006127, A001580.

Table[2^n+n^3,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011*)

G.f. : (1-3x+8x^2-11x^3-x^4)/((1-x)^4(1-2x)).

A100339 Primes of the form 2^q + q where q is not a prime.

Original entry on oeis.org

3, 521, 32783, 549755813927, 37778931862957161709643, 2417851639229258349412433

Offset: 1

Cino Hilliard, Jan 11 2005

nonn

The next term is 2^735+735 = 18073..35103, 222 digits long.

			For q = 9, 2^9+9 = 521 which is prime.

Amiram Eldar, Table of n, a(n) for n = 1..11 (terms 1..7 from N. J. A. Sloane)

Cf. A006127, A057664, A081296, A100556, A129962.

Select[Table[If[!PrimeQ[n],2^n+n,0],{n,1200}],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011*)

g1(p,n)=for(x=1,n,c=composite(x);y=p^c+c;if(gcd(y,c)==1,if(isprime(y),print1 (y",")))) composite(n) = \ the n-th composite number { local(c,x); c=1; x=0; while(c <= n, x++; if(!isprime(x),c++); ); return(x) }

a(n) = A006127(A100556(n-1)) for n >= 2. - Amiram Eldar, Jun 30 2024

A111049 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 9, 4, 1, 11, 27, 25, 8, 1, 20, 70, 100, 65, 16, 1, 37, 170, 330, 325, 161, 32, 1, 70, 399, 980, 1295, 966, 385, 64, 1, 135, 917, 2723, 4515, 4501, 2695, 897, 128, 1, 264, 2076, 7224, 14406, 17976, 14364, 7176, 2049, 256

Offset: 0

Philippe Deléham, Oct 07 2005

			Rows begin:
  1;
  1,   1;
  1,   3,    2;
  1,   6,    9,    4;
  1,  11,   27,   25,     8;
  1,  20,   70,  100,    65,    16;
  1,  37,  170,  330,   325,   161,    32;
  1,  70,  399,  980,  1295,   966,   385,   64;
  1, 135,  917, 2723,  4515,  4501,  2695,  897,  128;
  1, 264, 2076, 7224, 14406, 17976, 14364, 7176, 2049, 256;

Cf. A002064, A006127.

With[{m = 9}, CoefficientList[CoefficientList[Series[(1 - 2*x - 2*x*y + x^2 *y + x^2*y^2)/(1 - 3*x - 3*x*y + 2*x^2 + 4*x^2*y + 2*x^2*y^2), {x, 0 , m}, {y, 0, m} ], x], y]] // Flatten (* Georg Fischer, Feb 17 2020 *)

T(n, k) = if (k<=n, 2^(n-1)*binomial(n-1, k-1)+binomial(n-1, k));
matrix(10, 10, n, k, T(n-1, k-1)) \\ to see the triangle \\ Michel Marcus, Feb 17 2020

T(n, k) = 2^(n-1)binomial(n-1, k-1) + binomial(n-1, k).
Sum_{k=0..n} T(n, k) = 2^(n-1)*(1+2^(n-1)) = A063376(n-1) for n >= 1.
From Peter Bala, Mar 20 2013: (Start)
O.g.f.: (1 - 2*t + x*t*(t-2) + x^2*t^2)/((1 - t*(1+x))*(1 - 2*t*(1+x))) = 1 + (1+x)*t + (1+3*x+2*x^2)*t^2 + ....
E.g.f.: (x + 2*exp((1+x)*t) + x*exp(2*t*(1+x)))/(2*(1+x)) = 1 + (1+x)*t + (1+3*x+2*x^2)*t^2/2! + ....
Recurrence equation: for n >= 1, T(n+1,k) = 2*T(n,k) + 2*T(n,k-1) - binomial(n,k). (End)
From Philippe Deléham, Oct 18 2013: (Start)
G.f.: (1 - 2*x - 2*x*y + x^2*y + x^2*y^2)/(1 - 3*x - 3*x*y + 2*x^2 + 4*x^2*y + 2*x^2*y^2).
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) - 2*T(n-2,k-2), T(0,0) = T(1,1) = T(1,0) = T(2,0) = 1, T(2,1) = 3, T(2,2) = 2, T(n,k) = 0 if k > n or if k < 0. (End)

Wrong a(42) removed by Georg Fischer, Feb 17 2020

A131897 A130321 + A131821 - A000012.

Original entry on oeis.org

1, 3, 2, 6, 2, 3, 11, 4, 2, 4, 20, 8, 4, 2, 5, 37, 16, 8, 4, 2, 6, 70, 32, 16, 8, 4, 2, 7, 135, 64, 32, 16, 8, 4, 2, 8, 264, 128, 64, 32, 16, 8, 4, 2, 9, 521, 256, 128, 64, 32, 16, 8, 4, 2, 10

Offset: 0

Gary W. Adamson, Jul 25 2007

Left column = A006127: (1, 3, 6, 11, 20, 37, ...).

Row sums = A131898: (1, 5, 11, 21, 39, 73, ...).

			First few rows of the triangle:
   1;
   3,  2;
   6,  2,  3;
  11,  4,  2,  4;
  20,  8,  4,  2,  5;
  37, 16,  8,  4,  2,  6;
  70, 32, 16,  8,  4,  2,  7;
  ...

Cf. A130321, A131821, A006127, A131898.

A131321 + A131821 - A000012 as infinite lower triangular matrices.

A133545 (A000012 * A007318 - A007318 * A000012) - A000012.

Original entry on oeis.org

1, 3, 1, 6, 5, 1, 11, 12, 7, 1, 20, 24, 20, 9, 1, 37, 45, 45, 30, 11, 1, 70, 83, 91, 76, 42, 13, 1, 135, 154, 175, 168, 119, 56, 15, 1, 264, 290, 330, 344, 288, 176, 72, 17, 1, 521, 555, 621, 675, 633, 465, 249, 90, 19, 1

Offset: 1

Gary W. Adamson, Sep 15 2007

Row sums = A133546: (1, 4, 12, 31, 74, 169, ...).

Left border = A006127: (1, 3, 6, 11, 20, ...).

			First few rows of the triangle:
   1;
   3,  1;
   6,  5,  1;
  11, 12,  7,  1;
  20, 24, 20,  9,  1;
  37, 45, 45, 30, 11,  1;
  70, 83, 91, 76, 42, 13,  1;
  ...

Cf. A000012, A007318, A133546, A006127.

(A000012 * A007318 - A007318 * A000012) - A000012 as infinite lower triangular matrices.

A337455 Numbers of the form m + bigomega(m) with m a positive integer.

Original entry on oeis.org

1, 3, 4, 6, 8, 11, 12, 14, 15, 16, 17, 18, 20, 21, 23, 24, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 40, 41, 42, 44, 45, 47, 48, 51, 53, 54, 55, 57, 58, 59, 60, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 84, 85, 87, 88, 89, 90, 92, 93

Offset: 1

Nathan J. McDougall, Aug 27 2020

nonn

If a(n) = m + A001222(m) then (a(n) - m) <= log(a(n))/log(2).

It appears that a(n)/n may converge to a constant around ~ 1.49.

			a(7) = 10 + A001222(10) = 10 + 2 = 12

Petr Kucheriaviy, On numbers not representable as n + ω(n), arXiv preprint (2022). arXiv:2203.12006 [math.NT]

Cf. A001222 (bigomega), A064800, A358973.
Numbers of the form k^n+n where k is prime are subsequences: A006127 (k=2), A104743 (k=3), A104745 (k=5), A226199 (k=7), A226737 (k=11).
Subsequences include A008864, A101340, and A160649 (excluding the first term).

m = 100; Select[Union @ Table[n + PrimeOmega[n], {n, 1, m}], # <= m &] (* Amiram Eldar, Aug 28 2020 *)

upto(limit)=Set(select(t->t<=limit, apply(m->m+bigomega(m), [1..limit]))) \\ Andrew Howroyd, Aug 27 2020

list(lim)=my(v=List()); forfactored(n=1, lim\1-1, my(t=n[1]+bigomega(n)); if(t<=lim, listput(v, t))); Set(v) \\ Charles R Greathouse IV, Dec 07 2022

Kucheriaviy proves that a(n) << n log log n and conjectures that a(n) ≍ n, that is, these numbers have positive lower density. - Charles R Greathouse IV, Dec 07 2022

A351911 a(n) is the least integer m such that every m-element subset of {1,2,3,...,n} contains two nonempty and disjoint subsets whose sums are equal.

Original entry on oeis.org

3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 3

Thomas King, Feb 25 2022

a(n) grows like log n. Proof: The set of powers of 2 in {1,2,3,...,n} cannot contain two (nonempty) disjoint subsets with the same sum due to binary representation being unique, hence 1+log_2(n) < a(n). Let N be the least integer such that Sum_{i=0..N-1} (n-i) < 2^N-1. Then by the pigeonhole principle, we have a(n) <= N because the range of possible sums (for nonempty subsets of an N-element set) is at most Sum_{i=0..N-1} (n-i) and there are 2^N-1 nonempty subsets in total. Also N is O(log n).
Note that if we have two (distinct) subsets whose sums agree, then we can remove the integers in the intersection from both sets to get two disjoint (and nonempty) subsets with the same sum.
Also note that a 0-, 1- or 2-element subset cannot have (nonempty) disjoint subsets with the same sum.
The sequence of run lengths starts 1, 3, 6, 11, ... could this be A006127?

			For n=3 we only have to check the subset {1,2,3} for which 1+2 = 3, and therefore a(3) = 3.
For n=4 we first check the 3-element subsets,
  {1,2,3} : 1+2 = 3
  {1,3,4} : 1+3 = 4
  {1,2,4} : Fail.
So we check the only 4-element subset {1,2,3,4} for which 1+2 = 3, and therefore a(4) = 4.

Thomas King, Disjoint subsets with same sum, Math StackExchange, 2021.
Thomas King, Python program to compute the sequence by brute force

Cf. A006127.

Python
```
# See Thomas King link.
```

A359685 Greatest prime dividing 2^n + n.

Original entry on oeis.org

3, 3, 11, 5, 37, 7, 5, 11, 521, 47, 71, 79, 547, 911, 32783, 241, 307, 6899, 24967, 87383, 457, 4799, 270601, 7109, 3728273, 12497, 1201, 100613, 2017, 17318417, 859, 87211, 47491, 8589934609, 195329, 1483453, 320370521, 8191129, 549755813927, 478881371

Offset: 1

Philippe Deléham, Jan 11 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..406

Cf. A000040, A006127, A006530.

a:= n-> max(numtheory[factorset](2^n+n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jan 11 2023

a[n_] := FactorInteger[2^n + n][[-1, 1]]; Array[a, 40] (* Amiram Eldar, Mar 30 2023 *)

a(n) = vecmax(factor(2^n+n)[,1]); \\ Michel Marcus, Jan 11 2023

from sympy import primefactors
def A359685(n): return max(primefactors((1<Chai Wah Wu, Jan 11 2023

a(n) = A006530(A006127(n)).

A359735 Let f(s,n) = 2^n + s*n, with s in {-1, 1}. Let c be the number of primes out of the pair f(-1,n), f(1,n). If only f(-1,n) is prime, a(n) = -1, otherwise a(n) = c.

Original entry on oeis.org

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

Offset: 0

Jean-Marc Rebert, Jan 12 2023

sign

			f(-1,1) = 2^1 - 1 = 1, is not prime and f(1,1) = 2^1 + 1 = 3 is prime, so a(1) = 1.
f(-1,2) = 2^2 - 2 = 2, is prime and f(1,2) = 2^2 + 2 = 6 = 2 * 3 is not prime, so a(2) = -1.
f(-1,3) = 2^3 - 3 = 5, is prime and f(1,3) = 2^3 + 3 = 11 is prime, so a(3) = 2.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A006127, A048744, A052007, A129962.

f(s,n)=2^n+s*n
a(n)=my(a=isprime(f(-1,n)),b=isprime(f(1,n)),c=a+b); if(c==1&&a==1,return(-1),return(c))

a(n) can be = 2 only if n = 6*m + 3 for m >= 0 and m is not congruent to {0, 4} mod 5, not congruent to {2, 4} mod 7, not congruent to {6, 7} mod 11 and not congruent to {3, 9} mod 13. Does a(n) = 2 for n > 9 exist? - Thomas Scheuerle, Jan 12 2023

cp's OEIS Frontend

A094968 Indices of Fibonacci numbers in Stern's diatomic series A049456 regarded as a single linear sequence.

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A097339 2^n+n^3.

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A100339 Primes of the form 2^q + q where q is not a prime.

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A111049 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

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A131897 A130321 + A131821 - A000012.

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A133545 (A000012 * A007318 - A007318 * A000012) - A000012.

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A337455 Numbers of the form m + bigomega(m) with m a positive integer.

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A351911 a(n) is the least integer m such that every m-element subset of {1,2,3,...,n} contains two nonempty and disjoint subsets whose sums are equal.

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