A095732 -id:A095732 - cp's OEIS frontend

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

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.

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.

A095734 Asymmetricity-index for Zeckendorf-expansion A014417(n) of n.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 2, 1, 0, 2, 1, 0, 1, 0, 2, 2, 1, 2, 1, 3, 1, 0, 2, 2, 1, 1, 0, 2, 2, 1, 3, 1, 0, 1, 0, 2, 2, 1, 2, 1, 3, 2, 1, 3, 3, 2, 2, 1, 3, 1, 0, 3, 2, 4, 1, 0, 2, 2, 1, 2, 1, 3, 1, 0, 2, 2, 1, 2, 1, 3, 3, 2, 1, 0, 2, 2, 1, 3, 1, 0, 3, 2, 4, 2, 1, 3, 1, 0, 1, 0, 2, 2, 1, 2, 1, 3, 2, 1, 3, 3, 2, 2, 1, 3

Offset: 0

Antti Karttunen, Jun 05 2004

Least number of flips of "fibits" (changing either 0 to 1 or 1 to 0 in Zeckendorf-expansion A014417(n)) so that a palindrome is produced.

			The integers 0 and 1 look as '0' and '1' also in Fibonacci-representation,
and being palindromes, a(0) and a(1) = 0.
2 has Fibonacci-representation '10', which needs a flip of other 'fibit',
that it would become a palindrome, thus a(2) = 1. Similarly 3 has representation
'100', so flipping for example the least significant fibit, we get '101',
thus a(3)=1 as well. 7 (= F(3)+F(5)) has representation '1010', which needs
two flips to produce a palindrome, thus a(7)=2. Here F(n) = A000045(n).

a(n) = A037888(A003714(n)). A094202 gives the positions of zeros. Cf. also A095732.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Links

Crossrefs

A095734 Asymmetricity-index for Zeckendorf-expansion A014417(n) of n.

Views

Author

Keywords

Comments

Examples

Crossrefs