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%I #14 May 17 2021 01:53:02
%S 2,3,2,4,2,5,4,1,3,10,15,14,9,2,4,0,5,8,5,15,23,25,23,15,18,17,14,7,
%T 19,27,21,11,18,23,18,11,23,29,31,27,17,5,10,13,12,9,23,25,23,19,11,
%U 28,43,48,47,40,27,12,24,32,38,34,16,29,40,47,40,27,4,16
%N Let b be the lexicographically earliest weakly increasing sequence of squares such that Sum_{k = 1..m} prime(k) <= Sum_{k = 1..m} b(k) for any m > 0 (where prime(k) denotes the k-th prime number); a(n) = Sum_{k = 1..n} (b(k) - prime(k)).
%C This sequence is a variant of A338699; here we use squares, there powers of 2.
%H Rémy Sigrist, <a href="/A344324/b344324.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A344324/a344324.png">Density plot of the first 1000000 terms</a>
%e The first terms, alongside the prime numbers (p(n)) and their partial sums (pp(n)), b(n) and their partial sums (bb(n)), are:
%e n a(n) p(n) pp(n) b(n) bb(n)
%e -- ---- ---- ----- ---- -----
%e 1 2 2 2 4 4
%e 2 3 3 5 4 8
%e 3 2 5 10 4 12
%e 4 4 7 17 9 21
%e 5 2 11 28 9 30
%e 6 5 13 41 16 46
%e 7 4 17 58 16 62
%e 8 1 19 77 16 78
%e 9 3 23 100 25 103
%e 10 10 29 129 36 139
%e 11 15 31 160 36 175
%e 12 14 37 197 36 211
%e 13 9 41 238 36 247
%e 14 2 43 281 36 283
%e 15 4 47 328 49 332
%o (PARI) pp=0; bb=0; b=0; forprime (p=2, 349, pp+=p; while (bb+b^2<pp, b++); bb+=b^2; print1 (bb-pp", "))
%Y Cf. A000290, A007504, A338699.
%K nonn
%O 1,1
%A _Rémy Sigrist_, May 15 2021
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