A095753 -id:A095753 - cp's OEIS frontend

A095759 Triangle T(row>=0, 0<= pos <=row) by rows: T(r,p) contains number of odd primes p in range [2^(r+1),2^(r+2)] for which A037888(p)=pos.

1, 2, 0, 0, 2, 0, 2, 3, 0, 0, 0, 5, 2, 0, 0, 3, 4, 6, 0, 0, 0, 0, 15, 4, 4, 0, 0, 0, 3, 18, 15, 7, 0, 0, 0, 0, 0, 32, 20, 16, 7, 0, 0, 0, 0, 7, 33, 63, 24, 10, 0, 0, 0, 0, 0, 0, 63, 62, 88, 33, 9, 0, 0, 0, 0, 0, 12, 81, 135, 154, 56, 26, 0, 0, 0, 0, 0, 0, 0, 119, 150, 314, 197, 72, 20, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen, Jun 12 2004

			a(0) = T(0,0) = 1 as there is one prime 3 (11 in binary) in range ]2^1,2^2[ whose binary expansion is palindromic. a(1) = T(1,0) = 2 as there are two primes, 5 and 7 (101 and 111 in binary) in range ]2^2,2^3[ whose binary expansions are palindromic. a(2) = T(1,1) = 0, as there are no other primes in that range. a(3) = T(2,0) = 0, as there are no palindromic primes in range ]2^3,2^4[, but a(4) = T(2,1) = 2 as in the same range there are two primes 11 and 13 (1011 and 1101 in binary), whose binary expansion needs a flip of just one bit to become palindrome.

Row sums: A036378. Bisection of the leftmost diagonal: A095741. Next diagonals: A095753, A095754, A095755, A095756. Central diagonal (column): A095760. The rightmost nonzero terms from each row: A095757 (i.e. central diagonal and next-to-central diagonal interleaved). The penultimate nonzero terms from each row: A095758. Cf. also A095749, A048700-A048704, A095742.

A095742 Sum of A037888(p) for all primes p such that 2^n < p < 2^(n+1).

Original entry on oeis.org

0, 0, 2, 3, 9, 16, 35, 69, 148, 271, 628, 1167, 2629, 4830, 10597, 20083, 42928, 81579, 174223, 331314, 701382, 1340756, 2825575, 5422454, 11361615, 21873923, 45673361, 88161666, 183458213, 354899159, 736343490, 1427495050, 2954560104

Offset: 1

Antti Karttunen, Jun 12 2004

nonn

Ratio a(n)/A036378(n) gives the average asymmetricity ratio for n-bit primes: 0, 0, 1, 0.6, 1.285714, 1.230769, 1.521739, 1.604651, 1.973333, 1.978102, 2.462745, 2.515086, 3.014908, 2.996278, 3.49736, 3.517779, 3.993674, 4.000932, 4.50946, 4.502405, 4.997877, 4.998792, 5.500352, 5.500462, 5.998361, 5.999852, 6.499427, 6.500684, 7.000277, 7.000323, 7.499731, 7.499885, 7.999929, etc. I.e. 2- and 3-bit odd primes are all palindromes, 4-bit primes need on average just a one-bit flip to become palindromes, etc.

Ratio (a(n)/A036378(n))/f(n), where f(n) is (n-1)/4 if n is odd and (n-2)/4 if n is even (i.e. it gives the expected asymmetricity for all odd numbers in range [2^n,2^(n+1)]) converges as follows: 1, 1, 2, 1.2, 1.285714, 1.230769, 1.014493, 1.069767, 0.986667, 0.989051, 0.985098, 1.006034, 1.004969, 0.998759, 0.999246, 1.00508, 0.998418, 1.000233, 1.002102, 1.000535, 0.999575, 0.999758, 1.000064, 1.000084, 0.999727, 0.999975, 0.999912, 1.000105, 1.00004, 1.000046, 0.999964, 0.999985, 0.999991, ...

			a(1)=0, as only prime in range ]2,4] is 3, which has palindromic binary expansion 11, i.e. A037888(3)=0. a(2)=0, as in range ]4,8] there are two primes 5 (101 in binary) and 7 (111 in binary) so A037888(5) + A037888(7) = 0. a(3)=2, as in range ]8,16] there are two primes, 11 (1011 in binary) and 13 (1101 in binary), thus A037888(11) + A037888(13) = 1+1 = 2.

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

Cf. A095298, A095732 (sums of similar asymmetricity measures for Zeckendorf-expansion), A095753.

A095743 Primes p for which A037888(p) = 1, i.e., primes whose binary expansion is almost symmetric, needing just a one-bit flip to become palindrome.

Original entry on oeis.org

2, 11, 13, 19, 23, 29, 37, 41, 47, 59, 61, 67, 89, 97, 103, 131, 137, 157, 167, 173, 181, 191, 193, 199, 211, 223, 227, 229, 239, 251, 277, 281, 317, 337, 349, 367, 373, 383, 401, 419, 431, 463, 467, 479, 487, 491, 503, 509, 521, 563, 569, 577

Offset: 1

Antti Karttunen, Jun 12 2004

Robert Israel, Table of n, a(n) for n = 1..10000
Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

The second row of array A095749. Cf. A095753, A095748.

f:= proc(n) local L,i;
  L:= convert(n,base,2);
  add(abs(L[i]-L[-i]),i=1..floor(nops(L)/2))
end proc:
select(t -> f(t) = 1, [seq(ithprime(i),i=1..1000)]); # Robert Israel, Dec 04 2023

A095758 Number of A095748-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 2, 5, 4, 4, 15, 16, 24, 33, 56, 72, 95, 149, 219, 322, 537, 655, 998, 1309, 1859, 2784, 3886, 5340, 8091, 10718, 16191, 22316, 30372, 43425, 63699, 88186

Offset: 1

Antti Karttunen, Jun 12 2004

nonn

Ratio a(n)/A036378(n) converges as follows: 0, 0, 0, 0.4, 0.714286, 0.307692, 0.173913, 0.348837, 0.213333, 0.175182, 0.129412, 0.12069, 0.082569, 0.058933, 0.049175, 0.03836, 0.029956, 0.026336, 0.016954, 0.013562, 0.009328, 0.006931, 0.005419, 0.003942, 0.002819, 0.002219, 0.001525, 0.001194, 0.000852, 0.000599, 0.000442, 0.000335, 0.000239

Ratio a(n)/A095753(n) converges as follows: 1, 1, 0, 0.666667, 1, 1, 0.266667, 0.833333, 0.5, 0.727273, 0.52381, 0.691358, 0.605042, 0.659722, 0.582031, 0.688679, 0.611006, 0.839063, 0.63654, 0.779079, 0.58542, 0.724474, 0.651533, 0.718299, 0.646411, 0.762582, 0.635404, 0.767928, 0.657455, 0.704621, 0.636562, 0.71982, 0.646795

The penultimate nonzero terms from each row of triangle A095759. Cf. A095757, A095742.

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A095759 Triangle T(row>=0, 0<= pos <=row) by rows: T(r,p) contains number of odd primes p in range [2^(r+1),2^(r+2)] for which A037888(p)=pos.

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A095742 Sum of A037888(p) for all primes p such that 2^n < p < 2^(n+1).

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A095743 Primes p for which A037888(p) = 1, i.e., primes whose binary expansion is almost symmetric, needing just a one-bit flip to become palindrome.

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A095758 Number of A095748-primes in range ]2^n,2^(n+1)].

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