A175524 A000120-deficient numbers.

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset

1,2

Comments

For a more precise definition, see comment in A175522.

All odd primes (A065091) are in the sequence. Squares of the form (2^n+3)^2, n>=3, where 2^n+3 is prime (A057733), are also in the sequence. [Proof: (2^n+3)^2 = 2^(2*n)+2^(n+2)+2^(n+1)+2^3+1. Thus, since n>=3, A000120((2^n+3)^2)=5. Also, for primes of the form 2^n+3, (2^n+3)^2 has only two proper divisors, 1 and 2^n+3, so A000120(1)+A000120(2^n+3) = 4, and in conclusion, (2^n+3)^2 is deficient. QED]

It is natural to assume that there are infinitely many primes of the form 2^n+3 (by analogy with the Mersenne sequence 2^n-1). If this is true, the sequence contains infinitely many composite numbers, because it contains all of the form (2^n+3)^2.

a(n) = A006005(n) for n <= 30;

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Formula

A192895(a(n)) < 0. - Reinhard Zumkeller, Jul 12 2011

Mathematica

Select[Range[270], DivisorSum[#, DigitCount[#, 2, 1] &] < 2*DigitCount[#, 2, 1] &] (* Amiram Eldar, Jul 25 2023 *)

Prog

(Sage) is_A175524 = lambda x: sum(A000120(d) for d in divisors(x)) < 2*A000120(x)
A175524 = filter(is_A175524, IntegerRange(1, 10**4)) # D. S. McNeil, Dec 04 2010
(Haskell)
import Data.List (findIndices)
a175524 n = a175524_list !! (n-1)
a175524_list = map (+ 1) $ findIndices (< 0) a192895_list
-- Reinhard Zumkeller, Jul 12 2011
(PARI) is(n)=sumdiv(n, d, hammingweight(d))<2*hammingweight(n) \\ Charles R Greathouse IV, Jan 28 2016

Crossrefs

Cf. A175522 (perfect version), A175526 (abundant version), A000120, A005100, A005101, A006005, A192895.

Sequence in context: A056912 A075763 A074918 * A073579 A006005 A065091

Adjacent sequences: A175521 A175522 A175523 * A175525 A175526 A175527

Keyword

nonn,base

Author

Vladimir Shevelev, Dec 03 2010

Extensions

More terms from Amiram Eldar, Feb 18 2019

Status

approved