cp's OEIS Frontend

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

%I A331008 #40 Jan 30 2025 18:29:42
%S A331008 71,179,274,367,452,539,623,705,786,869,943,1024,1106,1183,1262,1335,
%T A331008 1405,1483,1562,1636,1705,1780,1860,1929,2000,2074,2146,2214,2286,
%U A331008 2355,2431,2502,2576,2645,2717,2781,2849,2918,2990,3059,3130,3201,3262,3330,3399,3462,3538
%N A331008 Numbers m such that (11*prime(m)) mod Pi > (11*prime(m+1)) mod Pi.
%C A331008 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.
%C A331008 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.
%H A331008 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Convergent.html">Convergent</a>.
%H A331008 Andres Cicuttin, <a href="/A331008/a331008.txt">Mathematica program for exploration of patterns related to A331008</a>
%e A331008 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.
%e A331008 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.
%p A331008 q:= n-> (f-> is(f(11*ithprime(n))>f(11*ithprime(n+1))))(k-> k-floor(k/Pi)*Pi):
%p A331008 select(q, [$1..4000])[];  # _Alois P. Heinz_, Jun 12 2023
%t A331008 Flatten@Position[Differences[N[Mod[11*Prime[Range[2^13]], Pi], 24]],
%t A331008    x_ /; x < 0]
%t A331008 Select[Range[3700],Mod[11Prime[#],Pi]>Mod[11Prime[#+1],Pi]&] (* _Harvey P. Dale_, Jan 30 2025 *)
%o A331008 (PARI) isok(k) = 11*prime(k) % Pi > 11*prime(k+1) % Pi; \\ _Michel Marcus_, Jun 12 2023
%Y A331008 Cf. A000796, A002485, A046947, A276435.
%K A331008 nonn
%O A331008 1,1
%A A331008 _Andres Cicuttin_, Jan 06 2020