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A262228 Deficiency sequence: a(0) = 1, a(n) is the smallest prime p > a(n-1) such that the product of a(i), 1 <= i < n, is deficient (belongs to A005100).

%I A262228 #47 Mar 04 2025 07:37:17
%S A262228 1,2,5,11,59,653,84761,2763189059,377406001499268899,
%T A262228 2638619515495963542360422694651593,
%U A262228 135435890329895562961039215198033899386421965445591860752412324961
%N A262228 Deficiency sequence: a(0) = 1, a(n) is the smallest prime p > a(n-1) such that the product of a(i), 1 <= i < n, is deficient (belongs to A005100).
%C A262228 The product of the first n+1 terms is the smallest deficient multiple of the product of the first n terms.
%C A262228 The product of any finite number of distinct terms of this sequence is deficient.
%C A262228 a(n) for n > 0 is the lexicographically earliest sequence of primes P, such that the asymptotic density of the squarefree numbers (A005117) which are not divisible by any prime in P is 3/Pi^2 (A104141), i.e., half the asymptotic density of all the squarefree numbers. - _Amiram Eldar_, Nov 30 2020
%H A262228 Amiram Eldar, <a href="/A262228/b262228.txt">Table of n, a(n) for n = 0..14</a>
%F A262228 a(n) = A151800(floor(1/(2*(Product_{i=2..n-1} a(i)/(a(i)+1))-1))), where A151800 is the "next larger prime" function.
%F A262228 Lim_{n->infinity} A001065(Product_{i=0..n} a(i))/(Product_{i=0..n} a(i)) = 1. [Corrected by _M. F. Hasler_, Dec 04 2017]
%F A262228 Conjecture: log(a(n)) ~ e^(an+b) where a and b are approximately 0.6 and -1.6 respectively.
%e A262228 a(3) = 11 because A001065(2*5*7) = A001065(70) = 74 > 70, and A001065(2*5*11) = A001065(110) = 106 < 110.
%e A262228 From _M. F. Hasler_, Dec 14 2017: (Start)
%e A262228 Let Q(x) = 1/(2x/sigma(x) - 1), P(n) = Product( a(k), k<n): P(0) = 1 (empty product). Then:
%e A262228 Q(P(0)) = 1, a(0) = nextprime(1) = 2 = P(1).
%e A262228 Q(P(1)) = 3, a(1) = 5. (2*3 is perfect, P(2) = 2*5 is deficient.)
%e A262228 Q(P(2)) = 9, a(2) = 11. (2*5*7 is weird, P(3) = 2*5*11 is deficient.)
%e A262228 Q(P(3)) = 54, a(3) = 59. (P(3)*53 is weird, P(4) = 2*5*11*59 is deficient.)
%e A262228 Q(P(4)) = 648, a(4) = 653. (P(4)*647 is weird, P(5) = 2*5*11*59*653 is deficient.)
%e A262228 Q(P(5)) = 84758.4, a(5) = 84761. (P(5)*84751 is abundant and semiperfect: sum of all proper divisors except {1, 2, 11, 22, 55, 59, 590}; P(6) = 2*5*11*59*653*84761 is deficient.) (End)
%t A262228 a[0]=1; a[n_] := a[n] = NextPrime[1/(2*Product[a[i],{i,1,n-1}]/Product[a[i]+1,{i,1,n-1}]-1)]; Array[a, 11, 0] (* _Amiram Eldar_, Jun 10 2019 *)
%o A262228 (PARI) lista(nn) = {print1(p=1, ", "); vp = [p]; for (n=2, nn, np = nextprime(1+floor(1/(2*prod(i=2, n-1, vp[i]/(vp[i]+1))-1))); vp = concat(vp, np); print1(np, ", "););} \\ _Michel Marcus_, Oct 16 2015
%o A262228 (PARI) a=List(); m=1; for(n=0, 13, listput(a, p=nextprime(1\(2/sigma(m,-1)-1)+1)); p>default(primelimit)&&addprimes(p); m*=p); a \\ _M. F. Hasler_, Dec 14 2017
%Y A262228 Cf. A001065, A005100, A005117, A104141, A151800 (nextprime).
%Y A262228 Cf. A002975 (primitive weird numbers), A000203 (sigma), A295001 (same definition but a(0) = 4).
%K A262228 nonn
%O A262228 0,2
%A A262228 _Chayim Lowen_, Sep 15 2015