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 A175524 A000120-deficient numbers. 14
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified April 17 18:43 EDT 2024. Contains 371765 sequences. (Running on oeis4.)