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

Author

Karl V. Keller, Jr., Nov 19 2014

Keywords

• nonn
• easy

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.

Examples

			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.
		

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).

Crossrefs

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

Cf. A023108.

Programs

Mathematica

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

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

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=",")

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

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