A095062 -id:A095062 - cp's OEIS frontend

A095326 Number of A095316-primes in range [2^n,2^(n+1)].

1, 2, 2, 5, 7, 13, 20, 42, 65, 122, 203, 412, 718, 1413, 2381, 4859, 8266, 16955, 28995, 60484, 105524, 216830, 376969, 785885, 1383287, 2867949, 5044969, 10544609, 18699214, 39034399, 69349061, 145210901, 259051224

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratios a(n)/A036378(n) converge as: 1, 1, 1, 1, 1, 1, 0.869565, 0.976744, 0.866667, 0.890511, 0.796078, 0.887931, 0.823394, 0.876551, 0.785809, 0.851112, 0.769002, 0.831535, 0.750485, 0.82195, 0.751938, 0.808416, 0.73382, 0.797191, 0.730306, 0.786657, 0.717911, 0.777517, 0.713512, 0.769947, 0.706327, 0.76292, 0.701421

A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence
Index entries for sequences related to occurrences of various subsets of primes in range [2^n,2^(n+1)]

a(n) = A036378(n)-A095327(n).

A095327 Number of A095317-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 1, 10, 15, 52, 52, 154, 199, 649, 850, 2483, 3435, 9640, 13102, 34812, 51386, 136739, 199933, 510833, 777795, 1982321, 3017298, 7508064, 11663138, 28833595, 45124684, 110272081

Offset: 1

Antti Karttunen

nonn

Ratios a(n)/A036378(n) converge as: 0, 0, 0, 0, 0, 0, 0.130435, 0.023256, 0.133333, 0.109489, 0.203922, 0.112069, 0.176606, 0.123449, 0.214191, 0.148888, 0.230998, 0.168465, 0.249515, 0.17805, 0.248062, 0.191584, 0.26618, 0.202809, 0.269694, 0.213343, 0.282089, 0.222483, 0.286488, 0.230053, 0.293673, 0.23708, 0.298579, ...

A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

a(n) = A036378(n)-A095326(n).

A095329 Number of A095319-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 2, 2, 4, 4, 12, 19, 30, 43, 102, 159, 340, 537, 1146, 1914, 3992, 6378, 14010, 23145, 49682, 82295, 180016, 303833, 654049, 1098521, 2387358, 4075501, 8820563, 15062515, 32732736, 56401388, 121964573, 210680037

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratios a(n)/A036378(n) converge as: 1, 1, 1, 0.8, 0.571429, 0.923077, 0.826087, 0.697674, 0.573333, 0.744526, 0.623529, 0.732759, 0.615826, 0.710918, 0.631683, 0.699247, 0.593358, 0.687102, 0.599068, 0.675156, 0.586414, 0.671161, 0.591451, 0.663458, 0.579964, 0.654834, 0.579953, 0.650393, 0.574745, 0.645647, 0.574454, 0.640787, 0.570449

Ratios a(n)/A095020(n) converge as: 1, 1, 1, 1, 0.8, 1.2,1.1875, 0.9375, 0.895833, 1.051546, 0.908571, 1.017964, 1.015123,1.014159, 1.034595, 1.009866, 1.016252, 1.007117, 0.99557, 1.002381,1.008245, 1.006182, 1.011728, 1.005142, 1.006148, 1.002926, 1.00328,1.004575, 1.002721, 1.003502, 1.004757, 1.002787, 1.003766

A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

a(n) = A036378(n)-A095328(n).

A095330 Number of A095320-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 2, 2, 5, 7, 13, 23, 42, 71, 122, 241, 412, 789, 1413, 2770, 4859, 9545, 16955, 34039, 60484, 121241, 216830, 441223, 785885, 1597803, 2867949, 5874665, 10544609, 21636090, 39034399, 80414166, 145210901, 299284792

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratios a(n)/A036378(n) converge as: 1, 1, 1, 1, 1, 1, 1, 0.976744, 0.946667, 0.890511, 0.945098, 0.887931, 0.904817, 0.876551, 0.914191, 0.851112, 0.88799, 0.831535, 0.881041, 0.82195, 0.863934, 0.808416, 0.858898, 0.797191, 0.84356, 0.786657, 0.835979, 0.777517, 0.825576, 0.769947, 0.819026, 0.76292, 0.81036

A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

a(n) = A036378(n)-A095331(n).

A095331 Number of A095321-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 4, 15, 14, 52, 83, 199, 260, 850, 1204, 3435, 4596, 13102, 19095, 51386, 72485, 199933, 296317, 777795, 1152625, 3017298, 4571188, 11663138, 17768490, 45124684, 70038513

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratios a(n)/A036378(n) converge as: 0, 0, 0, 0, 0, 0, 0, 0.023256, 0.053333, 0.109489, 0.054902, 0.112069, 0.095183, 0.123449, 0.085809, 0.148888, 0.11201, 0.168465, 0.118959, 0.17805, 0.136066, 0.191584, 0.141102, 0.202809, 0.15644, 0.213343, 0.164021, 0.222483, 0.174424, 0.230053, 0.180974, 0.23708, 0.18964

A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

a(n) = A036378(n)-A095330(n).

A095335 Number of A095315-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 1, 2, 2, 4, 8, 19, 22, 43, 68, 159, 235, 537, 844, 1914, 2849, 6378, 10138, 23145, 37023, 82295, 134477, 303833, 494625, 1098521, 1829183, 4075501, 6789809, 15062515, 25412867, 56401388, 95440507, 210680037

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratios a(n)/A036378(n) converge as: 1, 0.5, 1, 0.4, 0.571429, 0.615385, 0.826087, 0.511628, 0.573333, 0.49635, 0.623529, 0.506466, 0.615826, 0.523573, 0.631683, 0.499037, 0.593358, 0.497205, 0.599068, 0.503126, 0.586414, 0.501376, 0.591451, 0.501741, 0.579964, 0.501731, 0.579953, 0.500653, 0.574745, 0.501264, 0.574454, 0.501433, 0.570449

Ratios a(n)/A095296(n) converge as: 1, 1, 1, 0.666667, 0.8,1.6, 1.1875, 1.047619, 0.895833, 0.985507, 0.908571, 1.026201,1.015123, 1.098958, 1.034595, 0.996154, 1.016252, 0.98888, 0.99557,1.012581, 1.008245, 1.005518, 1.011728, 1.006987, 1.006148, 1.006948,1.00328, 1.002615, 1.002721, 1.00507, 1.004757, 1.005748, 1.003766

A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

a(n) = A036378(n)-A095334(n).

A095353 Sum of 1-fibits in Zeckendorf-expansion A014417(p) summed for all primes p in range [Fib(n+1),Fib(n+2)[ (where Fib = A000045).

Original entry on oeis.org

0, 1, 1, 3, 2, 7, 7, 14, 23, 35, 56, 94, 155, 243, 402, 614, 1061, 1656, 2689, 4295, 6938, 11176, 18095, 29102, 46907, 75703, 122174, 197494, 317987, 514611, 829595, 1340861, 2166008, 3497040, 5645418, 9120129, 14733126, 23803219, 38460014

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratio a(n)/A095354(n) (i.e. average number of 1-fibits in Zeckendorf-expansions of primes p which Fib(n+1) <= p < Fib(n+2)) grows as: 1, 1, 1, 1.5, 2., 2.333333, 2.333333, 2.8, 3.285714, 3.181818, 3.5, 3.916667, 4.189189, 4.418182, 4.785714, 4.873016, 5.358586, 5.575758, 5.871179, 6.100852, 6.382705, 6.676225, 6.954266, 7.223132, 7.489542, 7.773978, 8.045173, 8.331323, 8.598659, 8.886546, 9.161734, 9.440489, 9.71936, 9.995484, 10.266207, 10.54327, 10.820602, 11.096084, 11.374267.

Ratio of that average compared to A010049(n)/A000045(n) (the expected value of that same sum computed for all integers in the same range) converges as: 1, 1, 0.666667, 0.9, 1, 1.037037, 0.919192, 0.99661, 1.063946, 0.945946, 0.96142, 1, 0.999059, 0.988519, 1.008389, 0.970278, 1.011305, 1.000122, 1.003368, 0.995592, 0.996635, 0.999338, 0.999601, 0.998575, 0.997298, 0.998427, 0.997837, 0.999078, 0.998056, 0.99941, 0.999296, 0.999567, 0.999834, 0.999811, 0.999265, 0.999347, 0.999451, 0.999382, 0.999555.

			a(1) = a(2) = 0, as there are no primes in ranges [1,2[ and [2,3[. a(3)=1 as in [3,5[ there is prime 3 with Fibonacci-representation 100. a(4)=3, as in [5,8[ there are primes 5 and 7, whose Fibonacci-representations are 1000 and 1010 respectively and we have three 1-fibits in total. a(5)=2, as in [8,13[ there is only one prime 11, with Zeckendorf-representation 10100.

A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence

Cf. A095336, A095298 (similar sums and ratios computed in binary system).

a(2) corrected by Chai Wah Wu, Jan 16 2020

A095354 Number of primes p such that Fib(n+1) <= p < Fib(n+2), (where Fib = A000045).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 3, 5, 7, 11, 16, 24, 37, 55, 84, 126, 198, 297, 458, 704, 1087, 1674, 2602, 4029, 6263, 9738, 15186, 23705, 36981, 57909, 90550, 142033, 222855, 349862, 549903, 865019, 1361581, 2145191, 3381318, 5334509, 8419527, 13298631

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

			I.e. gives the number of primes whose Zeckendorf-expansion is n fibits long. a(1) = a(2) = 0, as there are no primes in ranges [1,2[ and [2,3[. a(3)=1 as in [3,5[ there is prime 3 with Fibonacci-representation 100. a(4)=2, as in [5,8[ there are primes 5 and 7. a(5)=1, as in [8,13[ there is only one prime 11 and a(6)=3 as in [13,21[ there are primes 13,17,19.

Chai Wah Wu, Table of n, a(n) for n = 1..100
A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence
Index entries for sequences related to numbers of primes in various ranges

Cf. A095353, A036378.

a(2) corrected by Chai Wah Wu, Jan 16 2020

A095731 Number of such primes p (A095730) such that Fib(n+1) <= p < Fib(n+2) (where Fib = A000045) and p's Zeckendorf-expansion A014417(p) is palindromic.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 3, 3, 0, 4, 8, 0, 15, 4, 0, 20, 42, 0, 44, 35, 0, 67, 147, 0, 231, 147, 0, 209, 538, 0, 833, 450, 0, 819, 2064, 0, 1701

Offset: 1

Antti Karttunen, Jun 12 2004

A. Karttunen, J. Moyer: C-program for computing the initial terms of this sequence

Cf. A095732, A095741.

A095732 Sum of A095734(p) for all primes p such that Fib(n+1) <= p < Fib(n+2) (where Fib = A000045).

Original entry on oeis.org

0, 0, 1, 3, 1, 3, 7, 10, 12, 23, 31, 58, 93, 171, 243, 422, 634, 1142, 1684, 2971, 4406, 7768, 11502, 20502, 30242, 53039, 79161, 138410, 207536, 362391, 544895, 947189, 1431794, 2473232, 3749944, 6459373, 9823917, 16879245, 25745781, 44112347

Offset: 1

Antti Karttunen, Jun 12 2004

nonn

Ratio a(n)/A095354(n) converges as follows: 1, 1, 1, 1.5, 1, 1, 2.333333, 2, 1.714286, 2.090909, 1.9375, 2.416667, 2.513514, 3.109091, 2.892857, 3.349206, 3.20202, 3.845118, 3.676856, 4.22017, 4.053358, 4.640382, 4.420446, 5.088608, 4.828676, 5.446601, 5.212762, 5.838853, 5.611963, 6.257939, 6.017615, 6.668795, 6.424778, 7.069164, 6.819283, 7.467319, 7.215081, 7.868411, 7.614126, 8.269242

			a(1) = a(2) = 0, as there are no primes in ranges [1,2[ and [2,3[. a(3)=1 as in [3,5[ there is prime 3 with Fibonacci-representation 100, which is just a one fibit-flip away from being a palindrome (i.e. A095734(3)=1). a(4)=3, as in [5,8[ there are primes 5 and 7, whose Fibonacci-representations are 1000 and 1010 respectively and the other needs one bit-flip and the other two to become palindromes and 1 + 2 = 3. a(5)=1, as in [8,13[ there is only one prime 11, with Zeckendorf-representation 10100, which needs to have just its least significant fibit flipped from 0 to 1 to become palindrome.

A. Karttunen, J. Moyer: C-program for computing the initial terms of this sequence

Cf. A095730, A095731, A095742 (sums of similar assymetricity measures for binary-expansion).

cp's OEIS Frontend

A095326 Number of A095316-primes in range [2^n,2^(n+1)].

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A095327 Number of A095317-primes in range ]2^n,2^(n+1)].

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A095329 Number of A095319-primes in range ]2^n,2^(n+1)].

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A095330 Number of A095320-primes in range ]2^n,2^(n+1)].

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A095331 Number of A095321-primes in range ]2^n,2^(n+1)].

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A095335 Number of A095315-primes in range ]2^n,2^(n+1)].

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A095353 Sum of 1-fibits in Zeckendorf-expansion A014417(p) summed for all primes p in range [Fib(n+1),Fib(n+2)[ (where Fib = A000045).

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A095354 Number of primes p such that Fib(n+1) <= p < Fib(n+2), (where Fib = A000045).

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Extensions

A095731 Number of such primes p (A095730) such that Fib(n+1) <= p < Fib(n+2) (where Fib = A000045) and p's Zeckendorf-expansion A014417(p) is palindromic.

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A095732 Sum of A095734(p) for all primes p such that Fib(n+1) <= p < Fib(n+2) (where Fib = A000045).

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