A092829 -id:A092829 - cp's OEIS frontend

A073666 Rearrangement of natural numbers such that a(k)*a(k+1) + 1 is a prime for all k.

1, 2, 3, 4, 7, 6, 5, 8, 9, 12, 13, 10, 15, 14, 17, 18, 11, 30, 19, 22, 16, 21, 20, 23, 26, 33, 34, 27, 28, 24, 25, 42, 31, 36, 32, 29, 38, 39, 48, 37, 40, 43, 46, 51, 50, 45, 52, 49, 54, 44, 47, 56, 41, 62, 59, 60, 53, 66, 35, 68, 57, 58, 55, 70, 61, 76, 63, 74, 69, 72, 71, 98

Offset: 1

Amarnath Murthy, Aug 10 2002

nonn

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A092842, A081942, A081943, A092829.
Cf. A073667.
Cf. A096100.

a[1]=1; a[n_]:=a[n]=(For[c=Sort[Table[a[k], {k, n-1}]]; d=Append[c, Last[c]+1]; m=First[Complement[Range[Last[d]], c]], MemberQ[c, m]||!PrimeQ[m*a[n-1]+1], m++ ]; m); Table[a[k], {k, 70}] (* Farideh Firoozbakht, Apr 14 2004 *)

A073666(n,show=1,a=1,u=[a])={for(n=2,n,show&&print1(a",");for(k=u[1]+1,9e9,!setsearch(u,k) && isprime(a*k+1) && (a=k) && break);u=setunion(u,[a]); while(#u>1&&u[2]==u[1]+1,u=u[2..-1]));a} \\ Use 2nd, 3rd or 4th optional arg to display intermediate terms, to use another starting value, to exclude some terms. - M. F. Hasler, Nov 24 2015

More terms from Jason Earls, Aug 26 2002

Offset changed to 1 by Ivan Neretin, Mar 06 2016

A112848 Expansion of eta(q)eta(q^2)eta(q^18)^2/(eta(q^6)*eta(q^9)) in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Sep 22 2005

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A033687, A033762, A092829, A093829, A097195, A248897, A255648 (absolute values).

QP = QPochhammer; s = QP[q]*QP[q^2]*(QP[q^18]^2/(QP[q^6]*QP[q^9])) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)
f[p_, e_] := If[Mod[p, 6] == 1, e+1, (1+(-1)^e)/2]; f[2, e_] := (-1)^e; f[3, e_]:= -2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 28 2024 *)

{a(n)=if(n<1, 0, if(n%3==0, n/=3; -2,1)* sumdiv(n,d,kronecker(-12,d) -if(d%2==0, 2*kronecker(-3,d/2))))}

{a(n)=local(A); if (n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)*eta(x^2+A)*eta(x^18+A)^2/ eta(x^6+A)/eta(x^9+A), n))}

Euler transform of period 18 sequence [ -1, -2, -1, -2, -1, -1, -1, -2, 0, -2, -1, -1, -1, -2, -1, -2, -1, -2, ...].
Moebius transform is period 18 sequence [1, -2, -3, 2, -1, 6, 1, -2, 0, 2, -1, -6, 1, -2, 3, 2, -1, 0, ...].
Multiplicative with a(2^e) = (-1)^e, a(3^e) = -2 if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} Kronecker(-3, k) x^k(1-x^(2k))^2/(1-x^(6k)) = x Product_{k>0} (1-x^k)(1-x^(2k))(1+x^(9k))(1+x^(6k)+x^(12k)).
a(3n) = -2*A092829(n). a(3n+1) = A093829(3n+1) = A033687(n). a(3n+2) = A093829(3n+2). a(6n)/2 = A093829(n). a(6n+1) = A097195(n). a(6n+3) = -2*A033762(n). a(6n+5) = 0.
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Jan 23 2024

A092842 a(n) is the solution of the equation A073666(x) = n (A073666(a(n)) = n).

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 5, 8, 9, 12, 17, 10, 11, 14, 13, 21, 15, 16, 19, 23, 22, 20, 24, 30, 31, 25, 28, 29, 36, 18, 33, 35, 26, 27, 59, 34, 40, 37, 38, 41, 53, 32, 42, 50, 46, 43, 51, 39, 48, 45, 44, 47, 57, 49, 63, 52, 61, 62, 55, 56, 65, 54, 67, 74, 88, 58, 75, 60, 69, 64

Offset: 1

Farideh Firoozbakht, Apr 15 2004

nonn

For each n number of solutions of the equation A073666(x) =n is less than 2 because the terms of the sequence A073666 are distinct. Conjecture: This sequence is infinite(see A073666).

Cf. A073666, A081942, A081943, A092829.

cp's OEIS Frontend

A073666 Rearrangement of natural numbers such that a(k)*a(k+1) + 1 is a prime for all k.

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A112848 Expansion of eta(q)eta(q^2)eta(q^18)^2/(eta(q^6)*eta(q^9)) in powers of q.

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A092842 a(n) is the solution of the equation A073666(x) = n (A073666(a(n)) = n).

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cp's OEIS Frontend

A073666 Rearrangement of natural numbers such that a(k)*a(k+1) + 1 is a prime for all k.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A112848 Expansion of eta(q)*eta(q^2)*eta(q^18)^2/(eta(q^6)*eta(q^9)) in powers of q.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A092842 a(n) is the solution of the equation A073666(x) = n (A073666(a(n)) = n).

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Author

Keywords

Comments

Crossrefs

A112848 Expansion of eta(q)eta(q^2)eta(q^18)^2/(eta(q^6)*eta(q^9)) in powers of q.