A337490 -id:A337490 - cp's OEIS frontend

A333980 Partial sums of A337490.

1, 3, 7, 14, 25, 30, 36, 49, 70, 100, 110, 131, 164, 210, 224, 253, 298, 360, 378, 415, 472, 550, 572, 617, 686, 780, 806, 859, 940, 1050, 1190, 1361, 1564, 1800, 2070, 2375, 2716, 3094, 3510, 3549, 3628, 3748, 3910, 4115, 4364, 4658, 4998, 5385, 5388, 5440, 5542, 5559, 5628, 5750

Offset: 0

Scott R. Shannon, Sep 04 2020

nonn

See A337490 for an explanation of the sequence.

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Cf. A337490, A165430, A064814, A082299, A005132, A336957.

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

A333716 a(0)=1; for n>0, a(n) is the greatest common divisor (GCD) of n and the sum of the previous terms back to the last GCD term, if the GCD is not already in the sequence; otherwise a(n) = a(n-1) + n.

Original entry on oeis.org

1, 2, 4, 7, 11, 5, 11, 18, 26, 3, 13, 24, 36, 49, 63, 78, 94, 111, 129, 148, 168, 189, 211, 234, 258, 283, 309, 336, 364, 393, 423, 454, 486, 519, 553, 588, 624, 661, 699, 738, 778, 819, 861, 43, 87, 132, 178, 225, 273, 322, 10, 61, 113, 166, 220, 275, 331, 388, 446, 505, 565

Offset: 0

Scott R. Shannon, Sep 03 2020

nonn

This sequence is similar to A337490 except that here when a GCD term is added to the sequence the sum of previous terms is reset to the value of that GCD. Subsequent terms calculate the sum of previous terms back to this last GCD value. See the examples below.

Unlike A337490 this sequence shows numerous drops in value as the sum used when calculating the GCD with n is constantly being reset back to a smaller value anytime a unique GCD value greater than 1 is found. In the first one million terms the sequence drops in value 1516 times, the largest drop occurring from a(738133) = 45463489818 to a(738134) = 738134.

			a(4) = 11 as the sum of the 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 previously appeared a(5) = 5. As this term adds a GCD value to the sequence, the running sum of previous terms is now set to 5.
a(6) = 11 as the sum of previous terms is now just a(5) = 5, and as the GCD of 5 and 6 is 1, which already appears in the sequence, a(6) = a(5) + 6 = 5 + 6 = 11.
a(9) = 3 as the sum of previous terms back to the last GCD term is a(5)+...+a(8) = 60, and the GCD of 60 and 9 is 3, and as 3 has not previously appeared, a(9) = 3. As this term adds a GCD value to the sequence, the running sum of previous terms is now set to 3.

Scott R. Shannon, Graph of the terms for n=0 to n=1000000.

Cf. A337490, A165430, A064814, A082299, A005132, A336957.

Block[{k = 0}, Nest[Append[#, If[FreeQ[#1, #3], Set[k, #2]; #3, #1[[-1]] + #2]] & @@ {#1, #2, GCD[Total@ #1[[k + 1 ;; #2]], #2]} & @@ {#, Length@ #} &, {1}, 60]] (* Michael De Vlieger, Sep 20 2020 *)

cp's OEIS Frontend

A333980 Partial sums of A337490.

Views

Author

Keywords

Comments

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A333716 a(0)=1; for n>0, a(n) is the greatest common divisor (GCD) of n and the sum of the previous terms back to the last GCD term, if the GCD is not already in the sequence; otherwise a(n) = a(n-1) + n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica