Flávio V. Fernandes - cp's OEIS frontend

User: Flávio V. Fernandes

Flávio V. Fernandes has authored 4 sequences.

A351123 Irregular triangle read by rows: row n lists the partial sums of the number of divisions by 2 after each tripling step in the Collatz trajectory of 2n+1.

1, 5, 4, 1, 2, 4, 7, 11, 2, 3, 4, 6, 9, 13, 1, 3, 6, 10, 3, 7, 1, 2, 3, 8, 12, 2, 5, 9, 1, 4, 5, 7, 10, 14, 6, 1, 2, 7, 11, 2, 3, 6, 7, 9, 12, 16, 1, 3, 4, 5, 6, 7, 9, 11, 12, 14, 15, 16, 18, 19, 20, 21, 23, 26, 27, 28, 30, 31, 33, 34, 35, 36, 37, 38, 41, 42, 43, 44, 48, 50, 52, 56, 59, 60, 61, 66, 70

Offset: 1

Flávio V. Fernandes, Feb 01 2022

The terms in row n are T(n,0), T(n,1), ..., T(n, A258145(n)-2), and are the partial sums of the terms in row n of A351122.
In each row n, the terms also satisfy the equation 3* (3* (3* (3* ... (3* (2n+1) +1) + 2^T(n,0)) + 2^T(n,1)) + 2^T(n,2)) + ... = 2^T(n, A258145(n)-2); e.g., for n=4, and A258145(4)-2=5: 3* (3* (3* (3* (3* (3*9+1) +2^2) +2^3) +2^4) +2^6) +2^9 = 2^13.
For row n, the right-hand side of the equation above is 2^A166549(n+1). E.g., for the above example (n=4), the right-hand side is 2^A166549(4+1) = 2^13.

			Triangle starts at T(1,0):
n\k   0   1   2   3   4   5   6   7   8   9   10 ...
1:    1   5
2:    4
3:    1   2   4   7  11
4:    2   3   4   6   9  13
5:    1   3   6  10
6:    3   7
7:    1   2   3   8   12
8:    2   5   9
...
E.g., row 3 of A351122 is [1, 1, 2, 3, 4]; its partial sums are [1, 2, 4, 7, 11].

Cf. A351122, A256598, A258145, A087230, A166549, A075680.

orow(n) = my(m=2*n+1, list=List()); while (m != 1, if (m%2, m = 3*m+1, my(nb = valuation(m,2)); m/=2^nb; listput(list, nb));); Vec(list); \\ A351122
row(n) = my(v = orow(n)); vector(#v, k, sum(i=1, k, v[i])); \\ Michel Marcus, Jul 18 2022

Data corrected by Mohsen Maesumi, Jul 18 2022

Last row completed by Michel Marcus, Jul 18 2022

A351122 Irregular triangle read by rows in which row n lists the number of divisions by 2 after tripling steps in the Collatz 3x+1 trajectory of 2n+1 until it reaches 1.

Original entry on oeis.org

1, 4, 4, 1, 1, 2, 3, 4, 2, 1, 1, 2, 3, 4, 1, 2, 3, 4, 3, 4, 1, 1, 1, 5, 4, 2, 3, 4, 1, 3, 1, 2, 3, 4, 6, 1, 1, 5, 4, 2, 1, 3, 1, 2, 3, 4, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 2, 2, 4, 3, 1, 1, 5, 4, 3, 1, 2, 3, 4

Offset: 1

Flávio V. Fernandes, Feb 01 2022

			Triangle starts at T(1,0):
   n\k   0   1   2   3   4   5   6   7   8 ...
   1:    1   4
   2:    4
   3:    1   1   2   3   4
   4:    2   1   1   2   3   4
   5:    1   2   3   4
   6:    3   4
   7:    1   1   1   5   4
   8:    2   3   4
   9:    1   3   1   2   3   4
  10:    6
  11:    1   1   5   4
  12:    2   1   3   1   2   3   4
  13:    1   2   1   1   1   1   2   2   1   2   1   1   2  ... (see A372362)
  ...
For n=6, the trajectory of 2*n+1 = 13 is as follows. The tripling steps ("=>") are followed by runs of 3 and then 4 halvings ("->"), so row n=6 is 3, 4.
  13  =>  40 -> 20 -> 10 -> 5  =>  16 -> 8 -> 4 -> 2 -> 1
    triple   \------------/   triple  \---------------/
               3 halvings                4 halvings
Runs of halvings are divisions by 2^T(n,k). Row n=11 is 1, 1, 5, 4 and its steps starting from 2*n+1 = 23 reach 1 by a nested expression
  (((((((23*3+1)/2^1)*3+1)/2^1)*3+1)/2^5)*3+1)/2^4 = 1.

Cf. A075680 (row lengths), A166549 (row sums), A351123 (row partial sums).
Cf. A256598.
Cf. A020988 (where row is [2*n]).
Cf. A198584 (where row length is 2), A228871 (where row is [1, x]).
Cf. A372362 (row 13, the first 41 terms).

row(n) = my(m=2*n+1, list=List()); while (m != 1, if (m%2, m = 3*m+1, my(nb = valuation(m,2)); m/=2^nb; listput(list, nb));); Vec(list); \\ Michel Marcus, Jul 18 2022

T(n,k) = log_2( (3*A256598(n,k)+1) / A256598(n,k+1) ).

Corrected by Michel Marcus, Jul 18 2022

A350179 Primes of the form ( A349309(n) + 1 ) ^ (1/3).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 23, 29, 31, 43, 47, 53, 59, 61, 67, 71, 79, 83, 101, 103, 107, 131, 139, 149, 151, 157, 167, 173, 179, 191, 197, 211, 223, 227, 229, 239, 263, 269, 277, 283, 293, 311, 317, 331, 347, 349, 359, 367, 373, 383, 389, 419, 421, 431, 439, 443, 461, 463, 467

Offset: 1

Flávio V. Fernandes, Dec 23 2021

nonn

Equivalently, primes p such that A051903(p^3-1) < 3. - Amiram Eldar, Dec 26 2021

			5 is a term since (A349309(3) + 1) ^ (1/3) = 125 ^ (1/3) = 5.

Cf. A051903, A349309.

filter:= n -> isprime(n) and max(map(t -> t[2],ifactors(n^3-1)[2]))<3:
select(filter, [2,seq(i,i=3..1000,2)]); # Robert Israel, Dec 26 2021

q[p_] := PrimeQ[p] && AllTrue[FactorInteger[p^3 - 1][[;; , 2]], # < 3 &]; Select[Range[500], q] (* Amiram Eldar, Dec 26 2021 *)

isok(p) = isprime(p) && (vecmax(factor(p^3-1)[,2]) < 3); \\ Michel Marcus, Jul 18 2022

from itertools import count, islice
from sympy import prime, factorint
def A350179_gen(): return (p for p in (prime(n) for n in count(1)) if max(factorint(p**3-1).values()) < 3)
A350179_list = list(islice(A350179_gen(),30)) # Chai Wah Wu, Jan 24 2022

More terms from Michel Marcus, Dec 25 2021

A347977 Primes of the form 2^p * 3^q * 5^r * 7^s - 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 53, 59, 71, 79, 83, 89, 97, 107, 127, 139, 149, 167, 179, 191, 199, 223, 239, 251, 269, 293, 349, 359, 383, 419, 431, 449, 479, 499, 503, 587, 599, 647, 719, 809, 839, 863, 881, 971, 1049, 1151, 1249, 1259, 1279, 1399, 1439, 1499, 1511, 1567, 1619, 1889

Offset: 1

Flávio V. Fernandes, Sep 21 2021

nonn

Restricting to r = s = 0 gives A005105; s = 0 gives A293194.

Primes of the form A002473(k) - 1.

			251 = 2^2 * 3^2 * 5^0 * 7^1 - 1 and 839 = 2^3 * 3^1 * 5^1 * 7^1 - 1 are terms.

Flávio V. Fernandes, Table of n, a(n) for n = 1..10000

Cf. A002473, A005105, A293194.

With[{n = 2000}, Sort@ Select[Flatten@ Table[2^p * 3^q * 5^r * 7^s - 1, {p, 0, Log[2, n]}, {q, 0, Log[3, n/(2^p)]}, {r, 0, Log[5, n/(2^p * 3^q)]}, {s, 0, Log[7, n/(2^p * 3^q * 5^r)]}], PrimeQ]] (* Amiram Eldar, Sep 25 2021 after Michael De Vlieger at A293194 *)

isok(p) = isprime(p) && (vecmax(factor(p+1)[,1]) < 11); \\ Michel Marcus, Nov 10 2021

upto(limit)={my(P=[2,3,5,7]); local(L=List()); my(recurse(k,t) = if(t<=limit+1, if(k>#P, if(isprime(t-1), listput(L,t-1)), my(b=P[k]); for(e=0, logint(limit+1,b), self()(k+1, t*b^e))))); recurse(1, 1); vecsort(Vec(L))} \\ Andrew Howroyd, Nov 20 2021

from itertools import count, islice
from sympy import integer_log, isprime
def A347977_gen(): # generator of terms
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = x
        for i in range(integer_log(x,7)[0]+1):
            for j in range(integer_log(m:=x//7**i,5)[0]+1):
                for k in range(integer_log(r:=m//5**j,3)[0]+1):
                    c -= (r//3**k).bit_length()
        return c
    yield from filter(isprime,(bisection(lambda k:n+f(k),n,n)-1 for n in count(1)))
A347977_list = list(islice(A347977_gen(),30)) # Chai Wah Wu, Mar 31 2025

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User: Flávio V. Fernandes

A351123 Irregular triangle read by rows: row n lists the partial sums of the number of divisions by 2 after each tripling step in the Collatz trajectory of 2n+1.

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A351122 Irregular triangle read by rows in which row n lists the number of divisions by 2 after tripling steps in the Collatz 3x+1 trajectory of 2n+1 until it reaches 1.

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A350179 Primes of the form ( A349309(n) + 1 ) ^ (1/3).

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A347977 Primes of the form 2^p * 3^q * 5^r * 7^s - 1.

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