A079394 Compare t(n) and t(n+1) where t = Ramanujan's tau function (A000594); a(n) = pq where p and q are given below.

21, 31, 21, 31, 21, 11, 31, 21, 11, 31, 22, 11, 32, 41, 42, 21, 32, 41, 22, 12, 11, 31, 41, 21, 32, 21, 32, 41, 22, 11, 11, 32, 41, 22, 31, 21, 11, 12, 31, 42, 42, 22, 11, 12, 12, 31, 42, 21, 32, 21, 32, 21, 31, 41, 22, 31, 21, 11, 12, 31, 42, 41, 41, 21, 11, 11, 32, 42, 42

Offset

1,1

Comments

p = 1 if t(n) < 0, t(n+1) < 0; p = 2 if t(n) >= 0, t(n+1) < 0; p = 3 if t(n) < 0, t(n+1) >= 0; p = 4 if t(n) >= 0, t(n+1) >= 0.

q = 1 if |t(n)| <= t(n+1); q = 2 if |t(n)| > t(n+1).

Links

Table of n, a(n) for n=1..69.

Example

A000594(1)=1, A000594(2)=-24. Hence the sign changes from + to - and the modulus increasing. Consulting the key, this gives a(1)=21.

Mathematica

f[i_, j_] := Which[i < 0 && j < 0, 1, i >= 0 && j < 0, 2, i < 0 && j >= 0, 3, i >= 0 && j >= 0, 4]; g[i_, j_] := If[Abs[i] <= Abs[j], 1, 2]; h[i_, j_] := FromDigits[{f[i, j], g[i, j]}]; h @@@ Partition[Array[RamanujanTau, 70], 2, 1] (* Amiram Eldar, Jan 10 2025 *)

Prog

(PARI) T(n)=n*(n+1)/2;
rtau3(n)=local(y, j); y=0; j=1; while (T(j-1)<n, j++); j--; for (i=1, j, y=y-(-1)^i*(2*i-1)*x^(T(i-1))); y=y^8; polcoeff(y, n-1);
for (n=1, 100, r3=rtau3(n); r31=rtau3(n+1); rn=(sign(r3)+1)/2; rn1=sign(r31)+1; print1((rn+rn1+1), "", (abs(r3)>abs(r31))+1, ", "))

Crossrefs

Cf. A000594.

Sequence in context: A115433 A116096 A116116 * A112375 A376294 A067599

Adjacent sequences: A079391 A079392 A079393 * A079395 A079396 A079397

Keyword

nonn

Author

Jon Perry, Jan 06 2003

Extensions

Offset corrected by Amiram Eldar, Jan 10 2025

Status

approved