A165135 The number of n-digit positive papaya numbers.

9, 90, 252, 1872, 4464, 29250, 62946, 393912, 809442, 4945140, 9899910, 59366286, 116999892, 692936460, 1349989992, 7919601912, 15299999856, 89099130960, 170999999838, 989995038012, 1889999872488, 10889990099100, 20699999999802, 118799939782206, 224999999981964

Offset

1,1

Comments

Papaya numbers are concatenations of two palindromes or palindromes themselves. All one-digit and two-digit numbers are papaya numbers.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Tanya Khovanova, Papaya Words and Numbers

Formula

a(n) = A052268(n)-A165611(n). - R. J. Mathar, Sep 25 2009

a(n) = 9*R(n,10)/10 - Sum_{d|n,d<n} phi(n/d)*a(d) where R(2*k,r)=k*(r+1)*r^k, R(2*k+1,r)=(2*k+1)*r^(k+1). - Andrew Howroyd, Mar 29 2016

Example

Three-digit papaya numbers are of four types: aaa (total of 9) and aab, aba, abb, (total of 81 for each). Hence a(3) = 252.

Prog

(PARI)
R(n, b)=if(n%2==0, n/2*(b+1)*b^(n/2), n*b^((n+1)/2));
a(n) = 9*R(n, 10)/10 - sumdiv(n, d, if(n<>d, eulerphi(n/d)*a(d))); \\ Andrew Howroyd, Oct 14 2017
(Python)
from functools import lru_cache
from sympy import totient, proper_divisors
@lru_cache(maxsize=None)
def A165135(n): return 9*(n*10**(n>>1) if n&1 else 11*(a:=n>>1)*10**(a-1))-sum(totient(n//d)*A165135(d) for d in proper_divisors(n, generator=True)) # Chai Wah Wu, Feb 19 2024

Crossrefs

Cf. A007055, A052268, A165610, A165611.

Sequence in context: A140160 A043960 A044641 * A277105 A377858 A180289

Adjacent sequences: A165132 A165133 A165134 * A165136 A165137 A165138

Keyword

base,nonn

Author

Sergei Bernstein and Tanya Khovanova, Sep 04 2009

Extensions

a(7)-a(8) from R. J. Mathar, Sep 25 2009

a(9)-a(25) from Andrew Howroyd, Mar 29 2016

Status

approved