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A211338
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Numbers k for which the number of divisors, tau(k), is congruent to 2 modulo 3.
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9
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2, 3, 5, 7, 11, 13, 16, 17, 19, 23, 24, 29, 30, 31, 37, 40, 41, 42, 43, 47, 53, 54, 56, 59, 61, 66, 67, 70, 71, 73, 78, 79, 81, 83, 88, 89, 97, 101, 102, 103, 104, 105, 107, 109, 110, 113, 114, 127, 128, 130, 131, 135, 136, 137, 138, 139, 149, 151, 152, 154
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OFFSET
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1,1
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COMMENTS
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The product of any 2 terms a(i)*a(j) is not a member of the sequence.
Any term a(n) can be expressed as 1 term (required to be greater than 1) from A211485 times 1 nonzero term from A000578. - Douglas Latimer, Apr 20 2012
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 4, 37, 368, 3681, 36596, 365336, 3653499, 36537962, 365381169, 3653826361, ... . Conjecture: the asymptotic density of this sequence exists and equals 3*zeta(3)/Pi^2 = 0.3653814847007... (A346602), so, a(n) ~ k*n with k = Pi^2/(3*zeta(3)) = 2.73686555524... . This conjecture is true if this sequence and A211337 have the same density (see A059269). - Amiram Eldar, Jan 06 2024
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LINKS
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FORMULA
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Conjecture: a(n) ~ k*n where k = 2/prod(1 - (p-1)/(p^(3*k))) = 2.7290077... where p ranges over the primes and k ranges over the positive integers. - Charles R Greathouse IV, Apr 13 2012
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EXAMPLE
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The divisors of 16 are: 1, 2, 4, 8, 16 (5 divisors). 5 is congruent to 2 modulo 3. Thus 16 is a member of this sequence.
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MATHEMATICA
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Select[Range[154], Mod[DivisorSigma[0, #], 3] == 2 &] (* T. D. Noe, Apr 21 2012 *)
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PROG
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(PARI) {plnt=1 ; mxind=100 ; for(k=1, 10^6,
if(numdiv(k) % 3 == 2, print(k); plnt++; if(mxind+1 == plnt, break() )))}
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CROSSREFS
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This is an extension of A000040 (the prime numbers, which each have 2 divisors).
The union of A059269 and A211337 is the complementary sequence to this one.
The definition of this sequence uses A000005 (the number of divisors of n).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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