Augusto Santi - cp's OEIS frontend

User: Augusto Santi

Augusto Santi has authored 3 sequences.

A354184 a(1) = a(2) = 1, a(n) = (A007947(31a(n-1)) + A007947(31a(n-2)))/31 for n >= 3, i.e., 31a(n) is the largest squarefree divisor of 31a(n-1) plus the largest squarefree divisor of 31*a(n-2).

1, 1, 2, 3, 5, 8, 7, 9, 10, 13, 23, 36, 29, 35, 64, 37, 39, 76, 77, 115, 192, 121, 17, 28, 31, 15, 16, 17, 19, 36, 25, 11, 16, 13, 15, 28, 29, 43, 72, 49, 13, 20, 23, 33, 56, 47, 61, 108, 67, 73, 140, 143, 213, 356, 391, 569, 960, 599, 629, 1228, 1243, 1857, 3100, 1867

Offset: 1

Augusto Santi, May 18 2022

After the first 5 terms, the sequence values repeat periodically with a cycle length of 207. The maximum value of a(n) is 1142300, whose first occurrence appears at n = 111.

			31*2 is the largest squarefree divisor of 31*a(6) = 31*8. 31*7 is the largest squarefree divisor of 31*a(7) = 31*7. So a(8) = (31*2 + 31*7)/31 = 9.

Michael De Vlieger, Table of n, a(n) for n = 1..2075
Augusto Santi, Periodic sequences of integers generated by a(n+1) = rad(a(n)) + rad(a(n-1)), Mathematics Stack Exchange.

Cf. A007947, A121369.

Nest[Append[#, (Times @@ FactorInteger[31 #[[-1]]][[All, 1]] + Times @@ FactorInteger[31 #[[-2]]][[All, 1]])/31] &, {1, 1}, 62] (* Michael De Vlieger, Jul 18 2022 *)

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
lista(nn) = my(va = vector(nn)); va[1] = 1; va[2] = 1; for (n=3, nn, va[n] = (rad(31*va[n-2]) + rad(31*va[n-1]))/31;); va; \\ Michel Marcus, May 21 2022

from sympy import primefactors
def rad(num):
    primes = primefactors(num)
    value = 1
    for p in primes:
        value *= p
    return value
a = [1, 1]
for n in range(2, 1000):
    a += [(rad(31*a[n-1]) + rad(31*a[n-2])) // 31]

For n >= 6, a(207 + n) = a(n).

A351871 a(1) = 1, a(2) = 2; a(n) = gcd(a(n-1), a(n-2)) + (a(n-1) + a(n-2))/gcd(a(n-1), a(n-2)).

Original entry on oeis.org

1, 2, 4, 5, 10, 8, 11, 20, 32, 17, 50, 68, 61, 130, 192, 163, 356, 520, 223, 744, 968, 222, 597, 276, 294, 101, 396, 498, 155, 654, 810, 250, 116, 185, 302, 488, 397, 886, 1284, 1087, 2372, 3460, 1462, 2463, 3926, 6390, 5160, 415, 1120, 312, 187, 500, 688, 301, 66, 368, 219, 588, 272, 219, 492, 240, 73, 314, 388

Offset: 1

Augusto Santi, Feb 22 2022

After the first 277 terms, the sequence values repeat periodically with a period of 2901. The maximum value of a(n) is 2269429312765395470820, whose first occurrence appears at n = 2006.
Changing the initial terms a(1) and a(2) generates other periodic sequences. The periods found empirically are 3, 9, 155, 2901. It is not known whether the number of possible periods is finite.
Manuel Valdivia informs me that the possible periods 53 and 84 mentioned earlier are in fact impossible. - N. J. A. Sloane, Sep 08 2022
From Robert Gerbicz, Sep 18 2022: (Start)
Let a(1), a(2) be the first two (positive) integers, and for n>2 define a(n)=g+(a(n-1)+(a-2))/g, where g=gcd(a(n-1),a(n-2)).
If a(1) and a(2) are odd then it is easy to see that all numbers in the sequence are odd.
If a(1) or a(2) is even, then by induction out of every two consecutive numbers in the sequence at least one of them is even.
This partitions the sequences into two groups.
Conjecture: In the first group the sequence always goes to infinity (as in A355898), and in the second group it always goes to a cycle (as in the present sequence).
Here are three more cycle lengths:
For a(1)=52, a(2)=378 the sequence starts with: 52, 378, 217, 92, 310, 203, 514, 718, 618, 670, 646, 660, 655, 268, 924, 302, 615, 918, 514, 718, ...  and has a cycle length of 12, starting at 514.
For a(1)=264, a(2)=1037 the sequence starts with: 264, 1037, 1302, 2340, 613, 2954, 3568, 3263, 6832, 10096, 1074, 5587, 6662, 12250, 9458, 10856, 10159, 21016, 31176, 6532, 9431, 15964, 25396, 10344, 8939, 19284, 28224, 3971, 32196, 36168, 5709, 1302, 2340, ... and has a cycle length of 29, starting at 1302.
For a(1)=542, a(2)=6017 the cycle has length 802 and the maximum term is 557981456058. (End)

			a(3) = gcd(1,2) + (1+2)/gcd(1,2) = 1 + 3/1 = 4.
a(4) = gcd(2,4) + (2+4)/gcd(2,4) = 2 + 6/2 = 5.
a(5) = gcd(4,5) + (4+5)/gcd(4,5) = 1 + 9/1 = 10.
a(6) = gcd(5,10) + (5+10)/gcd(5,10) = 5 + 15/5 = 8.
...
a(3179) = a(2901 + 278) = a(278) = 40.

Augusto Santi, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^15, with red showing records, demonstrating periodicity.
Augusto Santi, A singular variant of the OEIS sequence A349576, Mathematics Stack Exchange, 2022.
Index entries for linear recurrences with constant coefficients, order 2901.

Cf. A349576, A349982, A355898, A355914 (the successive gcds).

A351871 := proc(u,v,M) local n,r,s,g,t,a;
a:=[u,v]; r:=u; s:=v;
for n from 1 to M do g:=gcd(r,s); t:=g+(r+s)/g; a:=[op(a),t];
   r:=s; s:=t; od;
a;
end proc;
A351871(1,2,100); # N. J. A. Sloane, Sep 01 2022

a[1] = 1; a[2] = 2; a[n_] := a[n] = GCD[a[n - 1], a[n - 2]] + (a[n - 1] + a[n - 2])/GCD[a[n - 1], a[n - 2]]; Array[a, 50] (* Amiram Eldar, Feb 24 2022 *)

{a351871(N=65,A1=1,A2=2)= my(a=vector(N)); a[1]=A1; a[2]=A2; for(n=1,N,if(n>2,my(g=gcd(a[n-1],a[n-2])); a[n]=g+(a[n-1]+a[n-2])/g); print1(a[n],",")) } \\ Ruud H.G. van Tol, Sep 19 2022

from math import gcd
a, terms = [1, 2], 65
[a.append(gcd(a[-1], a[-2]) + (a[-1] + a[-2])//gcd(a[-1], a[-2])) for n in range(3, terms+1)]
print(a) # Michael S. Branicky, Sep 01 2022

For n >= 278, a(2901 + n) = a(n).

A349982 a(n) = 199 + (a(n-1)+a(n-2))/gcd(a(n-1),a(n-2)), a(0)=1 and a(1)=2.

Original entry on oeis.org

1, 2, 202, 301, 702, 1202, 1151, 2552, 3902, 3426, 3863, 7488, 11550, 3372, 2686, 3228, 3156, 731, 4086, 5016, 1716, 250, 1182, 915, 898, 2012, 1654, 2032, 2042, 2236, 2338, 2486, 2611, 5296, 8106, 6900, 2700, 231, 1176, 266, 302, 483, 984, 688, 408, 336, 230

Offset: 0

Augusto Santi, Jan 08 2022

The sequence repeats periodically with a period of 2920. The first repeating term is a(2785) = 288 = a(5705).

			a(2) = 199+(1+2)/gcd(1,2) = 199+3 = 202,
a(3) = 199+(2+202)/gcd(2,202) = 199+102 = 301.

A. Santi on math.stackexchange.com, Investigating the recurrence relation ...

Nest[Append[#, 199 + Total[#[[-2 ;; -1]]]/(GCD @@ #[[-2 ;; -1]])] &, {1, 2}, 45] (* Michael De Vlieger, Jan 08 2022 *)

cp's OEIS Frontend

User: Augusto Santi

A354184 a(1) = a(2) = 1, a(n) = (A007947(31a(n-1)) + A007947(31a(n-2)))/31 for n >= 3, i.e., 31a(n) is the largest squarefree divisor of 31a(n-1) plus the largest squarefree divisor of 31*a(n-2).

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A351871 a(1) = 1, a(2) = 2; a(n) = gcd(a(n-1), a(n-2)) + (a(n-1) + a(n-2))/gcd(a(n-1), a(n-2)).

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A349982 a(n) = 199 + (a(n-1)+a(n-2))/gcd(a(n-1),a(n-2)), a(0)=1 and a(1)=2.

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cp's OEIS Frontend

User: Augusto Santi

A354184 a(1) = a(2) = 1, a(n) = (A007947(31*a(n-1)) + A007947(31*a(n-2)))/31 for n >= 3, i.e., 31*a(n) is the largest squarefree divisor of 31*a(n-1) plus the largest squarefree divisor of 31*a(n-2).

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A351871 a(1) = 1, a(2) = 2; a(n) = gcd(a(n-1), a(n-2)) + (a(n-1) + a(n-2))/gcd(a(n-1), a(n-2)).

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Maple

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A349982 a(n) = 199 + (a(n-1)+a(n-2))/gcd(a(n-1),a(n-2)), a(0)=1 and a(1)=2.

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A354184 a(1) = a(2) = 1, a(n) = (A007947(31a(n-1)) + A007947(31a(n-2)))/31 for n >= 3, i.e., 31a(n) is the largest squarefree divisor of 31a(n-1) plus the largest squarefree divisor of 31*a(n-2).