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A331008
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Numbers m such that (11*prime(m)) mod Pi > (11*prime(m+1)) mod Pi.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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Eric Weisstein's World of Mathematics, Convergent.
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EXAMPLE
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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.
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MAPLE
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q:= n-> (f-> is(f(11*ithprime(n))>f(11*ithprime(n+1))))(k-> k-floor(k/Pi)*Pi):
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MATHEMATICA
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Flatten@Position[Differences[N[Mod[11*Prime[Range[2^13]], Pi], 24]],
x_ /; x < 0]
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PROG
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(PARI) isok(k) = 11*prime(k) % Pi > 11*prime(k+1) % Pi; \\ Michel Marcus, Jun 12 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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