A247128 Positive numbers that are congruent to {0,5,9,13,17} mod 22.

5, 9, 13, 17, 22, 27, 31, 35, 39, 44, 49, 53, 57, 61, 66, 71, 75, 79, 83, 88, 93, 97, 101, 105, 110, 115, 119, 123, 127, 132, 137, 141, 145, 149, 154, 159, 163, 167, 171, 176, 181, 185, 189, 193, 198, 203, 207, 211

Offset

1,1

Comments

This sequence is the union of 22*n-17, 22*n-13, 22*n-9, and 22*n-5, and A008604(22*n), for n>0.

This sequence is the integer values of sqrt(4*k - ceiling(k/3) + 3 + k mod 2), for k>0; see example.

The sequence numbers with both odd first and last digits are either palindromes or they have corresponding reversed digit numbers, e.g., 105, 501. Prime numbers in this sequence are also in A007500 (reversal primes). Some examples are 13, 17, 31, 71, 79, 97, 101.

The sequence numbers with even first digits and last digits of 2, 4, 6 or 8, are either palindromes or they have corresponding reversed digit numbers in this sequence.

The candidate Lychrel numbers, 295, 493, 691, 1677, 1765, 1857, 1945, 1997, 3493, are in this sequence.

Links

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000

Eric Weisstein's World of Mathematics, 196 Algorithm.

Eric Weisstein's World of Mathematics, Lychrel Number

Eric Weisstein's World of Mathematics, Palindromic Number Conjecture

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Formula

a(n) = a(n-1) + a(n-5) - a(n-6). - Colin Barker, Nov 20 2014

G.f.: x*(5*x^4+4*x^3+4*x^2+4*x+5) / ((x-1)^2*(x^4+x^3+x^2+x+1)). - Colin Barker, Nov 20 2014

Proof that a(n) = a(n-1) + a(n-5) - a(n-6): the sequence a(n) is a concatenation of the sequences [5+22*i, 9+22*i, 13+22*i, 17+22*i, 22+22*i] for i = 0,1,2,..., so it is clear that a(n-1) = a(n-6) + 22 and a(n) = a(n-5) + 22. - Chai Wah Wu, Jan 01 2015

Example

Sequence consists of the integer values of sqrt(4*k - ceiling(k/3) + 3 + k mod 2), for k>0; e.g.,
for k =  5, sqrt( 20 -  2 + 3 + 1) = sqrt(22)  =  4.6904;
for k =  6, sqrt( 24 -  2 + 3 + 0) = sqrt(25)  =  5;
for k = 21, sqrt( 84 -  7 + 3 + 1) = sqrt(81)  =  9;
for k = 44, sqrt(176 - 15 + 3 + 0) = sqrt(164) = 12.8062;
for k = 45, sqrt(180 - 15 + 3 + 1) = sqrt(169) = 13.
Of these, the only integer values are 5, 9, 13, so they are in the sequence.

Mathematica

a247128[n_Integer] := Select[Range[n], MemberQ[{0, 5, 9, 13, 17}, Mod[#, 22]] &]; a247128[211] (* Michael De Vlieger, Nov 23 2014 *)

Prog

(Python)
from math import *
for n in range(0, 100001):
..if (sqrt(4*n-ceil(n/3)+3+n%2))%1==0:print(int(sqrt(4*n-ceil(n/3)+3+n%2)), end=", ")
(PARI) isok(n) = m = n % 22; (m==0) || (m==5) || (m==9) || (m==13) || (m==17);
select(x->isok(x), vector(200, i, i)) \\ Michel Marcus, Nov 28 2014
(Python)
A247128_list = [n for n in range(1, 10**5) if (n % 22) in {0, 5, 9, 13, 17}]
# Chai Wah Wu, Dec 31 2014
(Python)
A247128_list, l = [], [5, 9, 13, 17, 22]
for _ in range(10**5):
....A247128_list.extend(l)
....l = [x+22 for x in l] # Chai Wah Wu, Jan 01 2015

Crossrefs

Cf. A008604, A002113 (palindromes), A007500 (reversible primes).

Cf. A023108.

Sequence in context: A314683 A184479 A314684 * A314685 A088346 A314686

Adjacent sequences: A247125 A247126 A247127 * A247129 A247130 A247131

Keyword

nonn,easy

Author

Karl V. Keller, Jr., Nov 19 2014

Status

approved