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A354154 a(1) = 0; for n>1, a(n) = prime(n-1) - A090252(n).

%I A354154 #17 May 29 2022 01:38:34
%S A354154 0,0,0,0,3,4,4,6,6,6,21,12,14,16,22,18,22,22,20,24,24,20,63,24,28,30,
%T A354154 30,30,52,30,86,78,48,48,42,48,48,50,54,54,46,48,44,52,44,46,173,54,
%U A354154 60,60,56,54,58,50,58,60,64,58,186,156,58,56,236,78,150,80,78,90,86,90,86,84,88,90,92,96,90,82,86,88,92,88,84,84,84,86,84,82,84
%N A354154 a(1) = 0; for n>1, a(n) = prime(n-1) - A090252(n).
%C A354154 Theorem: a(n) >= 0 for all n.
%H A354154 N. J. A. Sloane, <a href="/A354154/b354154.txt">Table of n, a(n) for n = 1..10000</a>
%e A354154 A090252 begins
%e A354154   1, 2, 3, 5, 4,  7,  9, 11, 13, ...
%e A354154 and we subtract these numbers from
%e A354154   1, 2, 3, 5, 7, 11, 13, 17, 19, ...
%e A354154 to get
%e A354154   0, 0, 0, 0, 3,  4,  4,  6,  6, ...
%o A354154 (Python)
%o A354154 from math import gcd, prod
%o A354154 from sympy import isprime, nextprime
%o A354154 from itertools import count, islice
%o A354154 def agen(): # generator of terms
%o A354154     alst, aset, mink, p = [1], {1}, 2, 1
%o A354154     yield 0
%o A354154     for n in count(2):
%o A354154         k, s, p = mink, n - n//2, nextprime(p)
%o A354154         prodall = prod(alst[n-n//2-1:n-1])
%o A354154         while k in aset or gcd(prodall, k) != 1: k += 1
%o A354154         alst.append(k); aset.add(k); yield p - k
%o A354154         while mink in aset: mink += 1
%o A354154 print(list(islice(agen(), 89))) # _Michael S. Branicky_, May 28 2022
%Y A354154 Cf. A090252.
%K A354154 nonn
%O A354154 1,5
%A A354154 _N. J. A. Sloane_, May 28 2022