Gary E. Davis - cp's OEIS frontend

A339267 Level of the Calkin-Wilf tree in which the n-th convergent of the continued fraction for e appears.

2, 3, 5, 6, 7, 11, 12, 13, 19, 20, 21, 29, 30, 31, 41, 42, 43, 55, 56, 57, 71, 72, 73, 89, 90, 91, 109, 110, 111, 131, 132, 133, 155, 156, 157, 181, 182, 183, 209, 210, 211, 239, 240, 241, 271, 272, 273, 305, 306, 307, 341, 342, 343, 379, 380, 381, 419, 420, 421, 461

Offset: 1

Gary E. Davis, Nov 29 2020

The depth level of a rational in the Calkin-Wilf tree is the sum of its continued fraction terms, with the root (1/1) as level 1 for this purpose. So the present sequence is partial sums of the continued fraction terms of e (A003417). Depth levels are the same in the related trees Stern-Brocot, Bird, etc. - Kevin Ryde, Dec 26 2020

			a(1) = 2 since 1st convergent 2, to e, appears at level 2 of the Calkin-Wilf tree.
a(2) = 3 since 2nd convergent 3 appears at level 3, a(3) = 5 since 3rd convergent 8/3 appears at level 5.

Wikipedia, Calkin-Wilf Tree.
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A002487, A003417 (continued fraction for e), A007676/A007677 (convergents).

children[{a_,b_}]:={{a,a+b},{a+b,b}};
frac[{a_,b_}]:=a/b;
L[1]={{1,1}};
L[n_]:=Flatten[Map[children,L[n-1]],1];
CWLevel[n_]:=Map[frac,If[n==1,L[1],Complement[L[n],L[n-1]]]];
WhereCW[{a0_,b0_}]:=Module[{a=a0,b=b0,steps},steps =1;While[a>1 || b>1,{a,b}=parent[{a,b}];steps++];steps];
fracpair[k_]:={Numerator[FromContinuedFraction[ContinuedFraction[E,k]]],Denominator[FromContinuedFraction[ContinuedFraction[E,k]]]};
Table[WhereCW[fracpair[k]],{k,1,60}]

a(n) = sqr(n\3) + n + 1; \\ Kevin Ryde, Dec 26 2020

a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n > 7.
a(n) = floor(n/3)^2 + n + 1. - Kevin Ryde, Dec 26 2020
G.f.: x*(2 + x + 2*x^2 - 3*x^3 - x^4 + x^6)/((1 - x)^3*(1 + x + x^2)^2). - Stefano Spezia, Dec 27 2020

A300158 Absolute value of product of nonzero eigenvalues of upper left (n+1)X(n+1) rank 2 submatrix of Wythoff array.

Original entry on oeis.org

1, 1, 4, 8, 20, 38, 77, 143, 267, 474, 856, 1540, 2703, 4749, 8204, 14233, 24714, 42234, 72495, 122930, 209534, 357733, 603816, 1023096, 1735667, 2915260, 4913350, 8216036, 13794118, 23198608, 38710749, 64802028, 108623872, 180780234, 301734372, 500717764, 833682438, 1390233453, 2304627170

Offset: 1

Gary E. Davis, Feb 26 2018

nonn

Empirical observation via computation.

			a(1) = 1 = |(4 + sqrt(17))*(4 - sqrt(17))|;
a(2) = 1 = |(12 + sqrt(145))*(1/(-12 - sqrt(145)))|;
a(3) = 4 = (1/2)*(63 + sqrt(3985))*(8/(-63 - sqrt(3985))).

Wikipedia, Wythoff array

Cf. A035513.

\[Phi] = (1 + Sqrt[5])/2;
A[m_, 1] := Floor[Floor[m*\[Phi]]*\[Phi]]
A[m_, 2] := Floor[Floor[m*\[Phi]]*\[Phi]^2]
A[m_, n_] := A[m, n] = A[m, n - 1] + A[m, n - 2]
M[n_] := Table[A[i, j], {i, 1, n}, {j, 1, n}]
X = Table[{n, -Simplify[Eigenvalues[M[n]][[1 ;; 2]][[1]]*Eigenvalues[M[n]][[1 ;; 2]][[2]]]}, {n, 2, 40}]

A276986 Numbers n for which there is a permutation p of (1,2,3,...,n) such that k+p(k) is a Catalan number for 1<=k<=n.

Original entry on oeis.org

0, 1, 3, 4, 9, 10, 12, 13, 28, 29, 31, 32, 37, 38, 40, 41, 90, 91, 93, 94, 99, 100, 102, 103, 118, 119, 121, 122, 127, 128, 130, 131, 297, 298, 300, 301, 306, 307, 309, 310, 325, 326, 328, 329, 334, 335, 337, 338, 387, 388, 390, 391, 396, 397, 399, 400, 415, 416

Offset: 1

Gary E. Davis, Sep 24 2016

nonn

A001453 is a subsequence. - Altug Alkan, Sep 29 2016

n>=1 is in the sequence if and only if there is a Catalan number c such that c/2 <= n < c and c-n-1 is in the sequence. - Robert Israel, Nov 20 2016

			3 is in the sequence because the permutation (1,3,2) added termwise to (1,2,3) yields (2,5,5) and both 2 and 5 are Catalan numbers.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000108, A073364.

S:= {0}:
for i from 1 to 8 do
  c:= binomial(2*i,i)/(i+1);
  S:= S union map(t -> c - t - 1, S);
od:
sort(convert(S,list)); # Robert Israel, Nov 20 2016

CatalanTo[n0_] :=
Module[{n = n0}, k = 1; L = {};
  While[CatalanNumber[k] <= 2*n, L = {L, CatalanNumber[k]}; k++];
  L = Flatten[L]]
perms[n0_] := Module[{n = n0, S, func, T, T2},
  func[k_] := Cases[CatalanTo[n], x_ /; 1 <= x - k <= n] - k;
  T = Tuples[Table[func[k], {k, 1, n}]];
  T2 = Cases[T, x_ /; Length[Union[x]] == Length[x]];
  Length[T2]]
Select[Range[41], perms[#] > 0 &]

a(i) + a(2^n+1-i) = A000108(n+1)-1 for 1<=i<=2^n. - Robert Israel, Nov 20 2016

More terms from Alois P. Heinz, Sep 28 2016

a(23)-a(58) from Robert Israel, Nov 18 2016

A276962 Numbers n such that n^17 - 1 is semiprime.

Original entry on oeis.org

20, 62, 84, 368, 410, 614, 720, 740, 762, 1230, 1280, 1988, 1998, 2064, 2100, 2268, 2312, 2468, 2678, 2940, 3002, 3324, 3392, 3462, 3768, 3848, 3968, 4178, 4244, 4680, 4968, 5022, 5024, 5198, 5304, 5382, 5624, 5822, 5850, 6048, 6248, 6338, 6354, 6398, 6428

Offset: 1

Gary E. Davis, Sep 22 2016

nonn

Least number such that n^17-1 and n^17+1 are both semiprime is 93888. - Altug Alkan, Sep 30 2016

			a(1) = 20 because 20^17-1 = 13107199999999999999999 = 19*689852631578947368421 is the first occurrence of n^17 - 1 as a product of two distinct primes.

Cf. A001358, A010805, A104494.

Select[Range[3000],PrimeOmega[#^17-1] == 2 &]

isok(n) = bigomega(n^17-1)==2; \\ Michel Marcus, Sep 23 2016

lista(nn) = forprime(p=2, nn, if(ispseudoprime(((p+1)^17-1)/p), print1(p+1, ", "))); \\ Altug Alkan, Sep 30 2016

More terms from Altug Alkan, Sep 30 2016

A276905 Numbers k such that k^5-1 and k^5+1 are semiprimes.

Original entry on oeis.org

12, 1452, 11352, 79398, 146520, 281622, 352110, 536778, 643302, 680988, 723492, 739200, 878988, 992112, 1115268, 1189650, 1397022, 1698378, 1698510, 1728540, 1806222, 2486220, 2873178, 3031578, 3571458, 3946140, 4467012, 4983858, 5064510, 5135658, 5567562, 5753352

Offset: 1

Gary E. Davis, Sep 21 2016

nonn

Cf. A001358, A108278, A124936, A268043.

Intersection of A104238 and A261435.

upper=600000;
Select[Range[upper],
PrimeOmega[#^5 - 1] == PrimeOmega[#^5 + 1] == 2 &]

isok(n) = (bigomega(n^5-1)==2) && (bigomega(n^5+1)==2); \\ Michel Marcus, Sep 22 2016

More terms from Altug Alkan, Sep 30 2016

A276735 Number of permutations p of (1..n) such that k+p(k) is a semiprime for 1 <= k <= n.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 5, 12, 87, 261, 324, 1433, 5881, 37444, 196797, 708901, 2020836, 12375966, 105896734, 955344450, 11136621319, 95274505723, 590283352231, 4285001635230, 36417581252044, 272699023606314

Offset: 0

Gary E. Davis, Sep 24 2016

nonn

			a(4) = 1 because the permutation (3,4,1,2) added termwise to (1,2,3,4) yields (4,6,4,6) - both semiprimes - and only that permutation does so. a(5) = 3 because exactly 3 permutations - (3,2,1,5,4), (3,4,1,2,5) & (5,4,3,2,1) - added termwise to (1,2,3,4,5) yield semiprime entries.
From _David A. Corneth_, Sep 28 2016 (Start):
The semiprimes up to 10 are 4, 6, 9 and 10. To find a(5), we add (1,2,3,4,5) to some p. Therefore, p(1) in {3, 5}, p(2) in {2, 4}, p(3) in {1, 3}, p(4) in {2, 5} and p(5) in {1, 4, 5}.
If p(1) = 3 then p(3) must be 1. Then {p(2), p(4), p(5)} = {2, 4, 5} for which there are two possibilities.
If p(1) = 5 then p(3) = 3 and p(4) = 2. Then p(2) = 4 and p(5) = 1. So there's one permutation for which p(1) = 5.
This exhausts the options for p(1) and we found 3 permutations. Therefore, a(5) = 3. (End)

Cf. A073364, A095986, A096904, A097082.

with(LinearAlgebra): with(numtheory):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)->
        `if`((s-> bigomega(s)=2)(i+j), 1, 0)))):
seq(a(n), n=0..16);  # Alois P. Heinz, Sep 28 2016

perms[n0_] :=
Module[{n = n0, S, func, T, T2},
  S = Select[Range[2, 2*n], PrimeOmega[#] == 2 &];
  func[k_] := Cases[S, x_ /; 1 <= x - k <= n] - k;
  T = Tuples[Table[func[k], {k, 1, n}]];
  T2 = Cases[T, x_ /; Length[Union[x]] == Length[x]];
  Length[T2]]
Table[perms[n], {n, 0, 12}]
(* Second program (version >= 10): *)
a[0] = 1; a[n_] := Permanent[Table[Boole[PrimeOmega[i + j] == 2], {i, 1, n}, {j, 1, n}]]; Table[an = a[n]; Print[an]; an, {n, 0, 20}] (* Jean-François Alcover, Jul 25 2017 *)

isok(va, vb)=my(v = vector(#va, j, va[j]+vb[j])); #select(x->(bigomega(x) == 2), v) == #v;
a(n) = my(vpo = numtoperm(n,1)); sum(k=1, n!, vp = numtoperm(n,k); isok(vp, vpo)); \\ Michel Marcus, Sep 24 2016

listA001358(lim)=my(v=List()); forprime(p=2, sqrtint(lim\1), forprime(q=p, lim\p, listput(v, p*q))); Set(v)
has(v)=for(k=1,#v, if(!setsearch(semi, v[k]+k), return(0))); 1
a(n)=local(semi=listA001358(2*n)); sum(k=1,n!, has(numtoperm(n,k))) \\ Charles R Greathouse IV, Sep 28 2016

matperm(M)=my(n=matsize(M)[1], innerSums=vectorv(n)); if(n==0, return(1)); sum(x=1,2^n-1, my(k=valuation(x,2), s=M[,k+1], gray=bitxor(x,x>>1)); if(bittest(gray,k), innerSums+=s,innerSums-=s); (-1)^hammingweight(gray)*factorback(innerSums))*(-1)^n
a(n)=matperm(matrix(n,n,x,y,bigomega(x+y)==2)) \\ Charles R Greathouse IV, Oct 03 2016

More terms from Alois P. Heinz, Sep 28 2016

A276983 Semiprimes n such that n-1 or n+1 is prime.

Original entry on oeis.org

4, 6, 10, 14, 22, 38, 46, 58, 62, 74, 82, 106, 158, 166, 178, 194, 226, 262, 278, 314, 346, 358, 382, 398, 422, 458, 466, 478, 502, 542, 562, 586, 614, 662, 674, 718, 734, 758, 838, 862, 878, 886, 982, 998, 1018, 1094, 1154, 1186, 1202, 1214, 1238, 1282, 1306, 1318, 1322

Offset: 1

Gary E. Davis, Sep 24 2016

nonn

Union of A077065 and A077068.

			a(3) = 10 = 2*5 is a product of 2 primes and 10+1 = 11 is prime.
a(4) = 14 = 2*7 is a product of 2 primes and 14-1 = 13 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A077065, A077068, A120628.

select(t -> isprime(t/2) and (isprime(t-1) or isprime(t+1)), [seq(i,i=2..10000,2)]); # Robert Israel, Sep 30 2016

func[n_] := PrimeOmega[n] == 2 && (PrimeQ[n + 1] || PrimeQ[n - 1])
Select[Range[1000], func[#] &]

isok(n) = (bigomega(n)==2) && (isprime(n-1) || isprime(n+1)); \\ Michel Marcus, Sep 24 2016

lista(nn) = forprime(p=2, nn, if(isprime(2*p+1) || isprime(2*p-1), print1(2*p, ", "))); \\ Altug Alkan, Sep 30 2016

from sympy import isprime, primerange
def aupto(N): return [t for t in (2*p for p in primerange(2, N//2+1)) if isprime(t-1) or isprime(t+1)]
print(aupto(1322)) # Michael S. Branicky, Aug 21 2022

a(n) = 2*A120628(n).

cp's OEIS Frontend

User: Gary E. Davis

A339267 Level of the Calkin-Wilf tree in which the n-th convergent of the continued fraction for e appears.

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A300158 Absolute value of product of nonzero eigenvalues of upper left (n+1)X(n+1) rank 2 submatrix of Wythoff array.

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A276986 Numbers n for which there is a permutation p of (1,2,3,...,n) such that k+p(k) is a Catalan number for 1<=k<=n.

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A276962 Numbers n such that n^17 - 1 is semiprime.

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Mathematica

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Extensions

A276905 Numbers k such that k^5-1 and k^5+1 are semiprimes.

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A276735 Number of permutations p of (1..n) such that k+p(k) is a semiprime for 1 <= k <= n.

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Maple

Mathematica

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A276983 Semiprimes n such that n-1 or n+1 is prime.

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