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A107705 a(n) is the least number of prime factors in any non-deficient number that has the n-th prime as its least prime factor.

%I A107705 #15 Aug 08 2019 04:33:42
%S A107705 2,5,9,18,31,46,67,91,122,158,194,238,284,334,392,456,522,591,668,749,
%T A107705 835,929,1028,1133,1242,1352,1469,1594,1727,1869,2019,2163,2315,2471,
%U A107705 2636,2802,2977,3157,3342,3534,3731,3933,4145,4358,4581,4811,5053,5293
%N A107705 a(n) is the least number of prime factors in any non-deficient number that has the n-th prime as its least prime factor.
%C A107705 Barring unforeseen odd perfect numbers (which it has been proved must have at least 29 prime factors if they exist at all), if we replace "non-deficient" in the description with "abundant", the value of a(1) becomes 3 and all other values stay the same.
%C A107705 The above mentioned sequence is A108227, see there for a comment on the relation of this sequence to that of primitive abundant numbers (A006038) which are products of consecutive primes, i.e., of the form N = Product_{0<=i<r} prime(n+i) for some r. The corresponding non-deficient products are A007702. - _M. F. Hasler_, Jun 15 2017
%H A107705 Amiram Eldar, <a href="/A107705/b107705.txt">Table of n, a(n) for n = 1..500</a>
%F A107705 a(n) = A007684(n)-n+1. A007702(n) = Product_{0<=i<a(n)} prime(n+i). - _M. F. Hasler_, Jun 15 2017
%e A107705 a(2) is 5 since 1) there are abundant numbers with a(2)=5 prime factors of which p_2=3 is the least prime factor (such as 945 = 3^3.5.7); 2) there are no non-deficient numbers with fewer than 5 prime factors, of which 3 is the least prime factor.
%o A107705 (PARI) A107705(n,s=1+1/prime(n))=for(a=1,9e9,2>(s*=1+1/prime(n+a))||return(a+1)) \\ _M. F. Hasler_, Jun 15 2017
%Y A107705 Cf. A000040, A023196, A005101, A001222.
%Y A107705 Cf. A007684 (~ A007707), A007708, A007741.
%K A107705 nonn
%O A107705 1,1
%A A107705 _Hugo van der Sanden_, Jun 10 2005
%E A107705 Data corrected by _Amiram Eldar_, Aug 08 2019