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A308669 a(1) = 1, a(2) = 3; for n > 1, a(n) = sum of the next two smallest integers > a(n-1) which are coprime to the sum s = a(1) + ... + a(n-1). 1

%I #68 Jun 28 2019 03:17:13

%S 1,3,12,28,60,124,252,510,1032,2070,4146,8298,16602,33210,66426,

%T 132858,265722,531450,1062906,2125818,4251642,8503290,17006588,

%U 34013180,68026364,136052736,272105480,544210964,1088421932,2176843872,4353687750,8707375506,17414751018,34829502042,69659004090

%N a(1) = 1, a(2) = 3; for n > 1, a(n) = sum of the next two smallest integers > a(n-1) which are coprime to the sum s = a(1) + ... + a(n-1).

%H Petros Hadjicostas, <a href="/A308669/b308669.txt">Table of n, a(n) for n = 1..500</a>

%F Conjecture 1: a(n) ~ 2^(n+1) * 1.0136719224974731811...

%F Conjecture 2: There are infinitely many n >= 1 such that the ratio (a(n+1) - A131357(n+1))/(a(n) - A131357(n)) equals exactly 2 (for example, for n = 10, 11, 12, 13, 18, 19, 20, 21, 24, 26, etc.).

%e n = 3: a(1) = 1, a(2) = 3, s = 1 + 3 = 4; n1 = 5 and n2 = 7 (not 6) are both coprime to s, hence a(3) = n1 + n2 = 12;

%e n = 4: a(1) = 1, a(2) = 3, a(3) = 12, s = 1 + 3 + 12 = 16; n1 = 13 and n2 = 15 (not 14) are both coprime to s, hence a(4) = n1 + n2 = 28;

%e n = 5: s = 44; n1 = 29 and n2 = 31 are both coprime to s, hence a(5) = n1 + n2 = 60;

%e n = 6: s = 104; n1 = 61 and n2 = 63 are both comprime to s, hence a(6) = n1 + n2 = 124.

%o (PARI) findnext(last, s) = {my(x = last+1); while (gcd(x, s) != 1, x++); my(y = x+1); while (gcd(y, s) != 1, y++); x+y; }

%o lista(nn) = {my(v = [1, 3], val); print1(v[1], ", ", v[2], ", "); for (n=3, nn, val = findnext(v[#v], vecsum(v)); v = concat(v, val); print1(val, ", "); ); } \\ [Modification of _Michel Marcus_'s PARI program from sequence A131357]

%Y Cf. A131357.

%K nonn

%O 1,2

%A _Petros Hadjicostas_, Jun 25 2019

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