A331008 Numbers m such that (11*prime(m)) mod Pi > (11*prime(m+1)) mod Pi.

71, 179, 274, 367, 452, 539, 623, 705, 786, 869, 943, 1024, 1106, 1183, 1262, 1335, 1405, 1483, 1562, 1636, 1705, 1780, 1860, 1929, 2000, 2074, 2146, 2214, 2286, 2355, 2431, 2502, 2576, 2645, 2717, 2781, 2849, 2918, 2990, 3059, 3130, 3201, 3262, 3330, 3399, 3462, 3538

Offset: 1

Author

Andres Cicuttin, Jan 06 2020

Keywords

• nonn

Comments

The average distance between consecutive terms decreases very slowly, and this pattern can be observed in this sequence up to values of m as high as 2^42 where the average distance is about four times lower than at the beginning of the sequence.

It seems that sequences of the form b(n) = (k*prime(n)) mod x exhibit a quasi-periodic sawtooth-like trend with slightly decreasing period when x is a positive irrational and k is the numerator (or a multiple of it) of a convergent to x. The Mathematica program in Links allows an easy experimentation on this feature and similar patterns obtained with other irrational constants x, and integer factors k.

Examples

			a(1) is 71 because (11*prime(71)) mod Pi = ~3.133072, a larger value than (11*prime(72)) mod Pi = ~0.018034. For any other primes p and q such that p < q < prime(71) we can see that (11*prime(p)) mod Pi < (11*prime(q)) mod Pi.
a(2) is 179 because (11*prime(179)) mod Pi = ~3.133735, a larger value than (11*prime(180)) mod Pi = ~0.018697. For any other primes p and q such that prime(71) < p < q < prime(179) we can see that (11*prime(p)) mod Pi < (11*prime(q)) mod Pi.
		

Links

• Eric Weisstein's World of Mathematics, Convergent.
• Andres Cicuttin, Mathematica program for exploration of patterns related to A331008

Crossrefs

Cf. A000796, A002485, A046947, A276435.

Programs

Maple

q:= n-> (f-> is(f(11*ithprime(n))>f(11*ithprime(n+1))))(k-> k-floor(k/Pi)*Pi):
select(q, [$1..4000])[];  # Alois P. Heinz, Jun 12 2023

Mathematica

Flatten@Position[Differences[N[Mod[11*Prime[Range[2^13]], Pi], 24]],
   x_ /; x < 0]
Select[Range[3700],Mod[11Prime[#],Pi]>Mod[11Prime[#+1],Pi]&] (* Harvey P. Dale, Jan 30 2025 *)

PARI

isok(k) = 11*prime(k) % Pi > 11*prime(k+1) % Pi; \\ Michel Marcus, Jun 12 2023