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A337814 a(n) is the smallest primitive nondeficient number that divides n or is a multiple of n.

%I A337814 #28 Dec 23 2024 14:53:46
%S A337814 6,6,6,20,20,6,28,88,945,20,88,6,104,28,945,272,272,6,304,20,945,88,
%T A337814 368,6,550,104,945,28,464,6,496,1184,3465,272,70,6,1184,304,4095,20,
%U A337814 1312,6,1376,88,945,368,1504,6,2205,550
%N A337814 a(n) is the smallest primitive nondeficient number that divides n or is a multiple of n.
%C A337814 The list of primitive nondeficient numbers (A006039) starts 6, 20, 28, 70, 88, 104, 272, ..., with 945 the first odd term.
%C A337814 When n is a primitive nondeficient number, a(n) = n; for other values of n, either the set of divisors of n or the set of multiples of n contains a primitive nondeficient number, but not both.
%C A337814 If n is a nondeficient number, a(n) is a divisor: we know n is a multiple of at least one primitive nondeficient number, as it follows directly from the definition of primitive nondeficient number.
%C A337814 For deficient n, a(n) is a multiple. We can always find a multiple that is a primitive nondeficient number by multiplying n by the product of successive primes starting with A007918(2n-1), the first prime >= 2n-1. The smallest nondeficient number that is generated this way will be primitive, therefore an upper bound for a(n). (Reaching a nondeficient number is guaranteed because the sum of the inverses of the primes is infinite.)
%C A337814 More extensive explanation, due to _M. F. Hasler_, summarized from SeqFan list posting: (Start)
%C A337814 Any deficient number N has abundant multiples; to reach a primitive nondeficient number it is sufficient to choose additional prime factors in such a way that you just get abundancy >= 2, but < 2 whatever factor you omit.
%C A337814 An additional prime factor p increases abundancy by a factor 1 + 1/sum_{k=1..m(p)} p^k if the new multiplicity of p is m(p) >= 1.
%C A337814 Let x = max { sum_{k=1..m(p)} p^k : p | N } so that 1+1/x is the smallest such contribution of any prime factor in N.
%C A337814 Since the infinite product (over primes p) of 1+1/p diverges, a satisfactory method is multiplying N by distinct prime factors greater than x until abundancy is >= 2.
%C A337814 (End)
%H A337814 M. F. Hasler and Peter Munn, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2020-October/020998.html">A property of k-almost primes, but also A006039?</a>, SeqFan list, Oct 09 2020.
%H A337814 Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes">Divergence of the sum of the reciprocals of the primes</a>.
%F A337814 For m >= 2, a(A000040(m)) = A338133(m, 1).
%F A337814 For m >= 1, a(6m+3) == 1 (mod 2).
%e A337814 6 is the smallest primitive nondeficient number, and is a multiple of 1, 2 and 3; so a(1) = a(2) = a(3) = 6.
%e A337814 For n = 4: we see 6 is not a multiple of 4, but the second smallest primitive nondeficient number, 20, is a multiple of 4; so a(4) = 20.
%e A337814 For n = 9: as 9 is deficient, we seek a suitable multiple. All even multiples of 9 are nondeficient, as they are multiples of nondeficient 6, but that also means they are not primitive. So we seek a qualifying odd multiple. The first odd nondeficient number is 945, which must therefore be primitive, and is also a multiple of 9. So a(9) = 945.
%e A337814 For n = 40: as a nondeficient number, we know 40 must have a primitive nondeficient divisor; the least such is 20, so a(40) = 20.
%Y A337814 Sequences with related definitions: A064162, A254572.
%Y A337814 Range of values: A006039.
%Y A337814 See A000203, A005100 and A023196 for definitions of deficient and nondeficient numbers.
%Y A337814 Cf. A000040, A007918, A308710, A338133.
%K A337814 nonn
%O A337814 1,1
%A A337814 _Peter Munn_, Sep 23 2020