A333980 -id:A333980 - cp's OEIS frontend

A337490 a(0)=1; for n > 0, a(n) = the greatest common divisor (GCD) of n and the sum of all previous terms if the GCD is not already in the sequence; otherwise a(n) = a(n-1) + n.

1, 2, 4, 7, 11, 5, 6, 13, 21, 30, 10, 21, 33, 46, 14, 29, 45, 62, 18, 37, 57, 78, 22, 45, 69, 94, 26, 53, 81, 110, 140, 171, 203, 236, 270, 305, 341, 378, 416, 39, 79, 120, 162, 205, 249, 294, 340, 387, 3, 52, 102, 17, 69, 122, 176, 231, 287, 344, 402, 461, 521, 582, 644, 707, 771, 836, 902, 969

Offset: 0

Scott R. Shannon, Aug 29 2020

nonn

The sequence displays the unusual behavior of decreasing 53 times in the first 1975 terms, due to the existence of a GCD which has not previously appeared in the sequence, but then not decreasing again for n up to at least 100 million. In this period there are 37 repeated terms, the first being 21 at n=11 and the last 161202 at n=2054. In the same range many values do not appear, for example 16,23,28,32,36. It is unknown when the sequence decreases again, or if all values eventually appear. The 100 millionth term is 4999999948050717.

See the companion sequence A333980 for the sum of the terms from a(0) to a(n).

			a(2) = 4 as the sum of all previous terms is a(0)+a(1) = 3, and the GCD of 3 and 2 is 1, which has already appeared in the sequence. Therefore a(2) = a(1) + n = 2 + 2 = 4.
a(4) = 11 as the sum of all previous terms is a(0)+...+a(3) = 14, and the GCD of 14 and 4 is 2. However 2 has already appeared so a(4) = a(3) + n = 7 + 4 = 11.
a(5) = 5 as the sum of all previous terms is a(0)+...+a(4) = 25, and the GCD of 25 and 5 is 5, and as 5 has not previous appeared a(5) = 5.

Scott R. Shannon, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Graph of the terms for n=0..2500. This includes the last known decrease in the sequence, n(1974) = 42.
Scott R. Shannon, Graph of the terms for n=0..10000000.

Cf. A333980, A333826 (same rules but starting a(1)=1), A165430, A064814, A082299, A005132, A336957.

lista(nn) = {my(va = vector(nn), s=0); va[1] = 1; s += va[1]; for (n=2, nn, my(g = gcd(n-1, s)); if (#select(x->(x==g), va), va[n] = va[n-1]+n-1, va[n] = g); s += va[n];); va;} \\ Michel Marcus, Sep 05 2020

A333826 a(1)=1; for n>1, a(n) = the greatest common divisor (GCD) of n and the sum of all previous terms if the GCD is not already in the sequence; otherwise a(n) = a(n-1) + n.

Original entry on oeis.org

1, 3, 6, 2, 7, 13, 20, 4, 13, 23, 34, 46, 59, 73, 88, 8, 25, 43, 62, 10, 31, 53, 76, 100, 125, 151, 178, 206, 235, 15, 46, 78, 111, 145, 5, 41, 78, 116, 155, 195, 236, 278, 321, 365, 410, 456, 503, 551, 600, 50, 101, 153, 206, 260, 315, 371, 428, 486, 545, 605, 666, 728, 791, 855, 920, 986, 1053

Offset: 1

Scott R. Shannon, Sep 03 2020

nonn

This is a variation of A337490; here we start with an offset of 1, so a(1) = 1. See that sequence for further details.

In the first 4212 terms the sequence decreases 69 times while 45 terms are repeated, the first being 13 at n=9 and the last 399876 at n=4212. After n(4166)=84 the sequence does not decrease again for n up to at least 100 million. The lowest numbers that have not appeared in that range are 30,37,47,48,49,51. The 100 millionth term is 4999999941527298.

			a(2) = 3 as the sum of all previous terms is a(1) = 1, and the GCD of 1 and 2 is 1. However 1 has already appeared so a(2) = a(1) + n = 1 + 2 = 3.
a(4) = 2 as the sum of all previous terms is a(1)+a(2)+a(3) = 10, and the GCD of 10 and 4 is 2, and as 2 has not previous appeared a(4) = 2.
a(8) = 4 as the sum of all previous terms is a(1)+...+a(7) = 52, and the GCD of 52 and 8 is 4, and as 4 has not previous appeared a(8) = 4.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Graph of the terms for n=1..4500. This includes the last known decrease in the sequence, n(4166)=84.

Cf. A337490 (same sequence rules but starting a(0)=1), A333980, A165430, A064814, A082299, A005132, A336957.

lista(nn) = {my(va = vector(nn), s=0); va[1] = 1; s += va[1]; for (n=2, nn, my(g = gcd(n, s)); if (#select(x->(x==g), va), va[n] = va[n-1]+n, va[n] = g); s += va[n];); va;} \\ Michel Marcus, Sep 05 2020

cp's OEIS Frontend

A337490 a(0)=1; for n > 0, a(n) = the greatest common divisor (GCD) of n and the sum of all previous terms if the GCD is not already in the sequence; otherwise a(n) = a(n-1) + n.

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