Ya-Ping Lu - cp's OEIS frontend

A385613 Number of steps that n requires to reach 0 under the map: x-> x^2 - 1 if x is an odd prime, floor(x/3) if x is even, otherwise x - 1. a(n) = -1 if 0 is never reached.

0, 1, 1, 3, 2, 4, 2, 7, 2, 3, 4, 9, 3, 6, 3, 4, 5, 8, 3, 10, 3, 4, 8, 14, 3, 4, 3, 4, 4, 10, 5, 15, 5, 6, 10, 11, 4, 10, 4, 5, 7, 8, 4, 14, 4, 5, 5, 10, 6, 7, 6, 7, 9, 15, 4, 5, 4, 5, 11, 9, 4, 18, 4, 5, 5, 6, 9, 10, 9, 10, 15, 9, 4, 22, 4, 5, 5, 6, 4, 11, 4

Offset: 0

Ya-Ping Lu, Jul 04 2025

sign

n = 122827 is the smallest starting number that ends up in a loop. The loop contains 33 elements: 122827 -> 15086471928 -> 5028823976 -> 1676274658 -> 558758219 -> 558758218 -> 186252739 -> 186252738 -> 62084246 -> 20694748 -> 6898249 -> 47585839266000 -> 15861946422000 -> 5287315474000 -> 1762438491333 -> 1762438491332 -> 587479497110 -> 195826499036 -> 65275499678 -> 21758499892 -> 7252833297 -> 7252833296 -> 2417611098 -> 805870366 -> 268623455 -> 268623454 -> 89541151 -> 89541150 -> 29847050 -> 9949016 -> 3316338 -> 1105446 -> 368482 -> 122827.
No other loop is found for n < 164901049.
Conjecture: a(n) = -1 occurs only when starting number n runs into a loop.

			a(5) = 4 because iterating the map on n = 5 results in 0 in 4 steps: 5 -> 5^2-1=24 -> floor(24/3)=8 -> floor(8/3)=2 -> floor(2/3)=0.
a(9949031) = -1 because iterating the map on n = 9949031 ends up in the 33-member loop in 5 steps: 9949031 -> 9949030 -> 3316343 -> 3316342 -> 1105447 -> 1105446.

Cf. A339991, A340801, A384713.

from sympy import isprime
def A385613(n):
    S = {n}
    while n != 0:
        n = n//3 if n%2 == 0 else n*n - 1 if isprime(n) else n - 1
        if n in S: return -1
        S.add(n)
    return len(S) - 1

A384713 The number of steps that n requires to reach 1 under the map: x-> x^2 - 1 if x is an odd prime, x/2 if x is even, x - lpf(x) otherwise where lpf(x) is the least prime factor of x. a(n) = -1 if 1 is never reached.

Original entry on oeis.org

0, 1, 4, 2, 8, 5, 9, 3, 6, 9, 11, 6, 12, 10, 7, 4, 12, 7, 14, 10, 8, 12, 14, 7, 11, 13, 8, 11, 15, 8, 14, 5, 9, 13, 9, 8, 16, 15, 9, 11, 16, 9, 17, 13, 10, 15, 17, 8, 10, 12, 9, 14, 18, 9, 13, 12, 10, 16, 17, 9, 19, 15, 10, 6, 10, 10, 20, 14, 11, 10, 18, 9, 20

Offset: 1

Ya-Ping Lu, Jun 23 2025

nonn

First 8 terms are the same as those in A339991.

Conjecture: a(n) != -1.

Ya-Ping Lu, Table of n, a(n) for n = 1..10000

Cf. A339991.

from sympy import isprime, primefactors
def A384713(n, c = 0):
    while n != 1: n = n//2 if n%2 == 0 else n*n-1 if isprime(n) else n-min(primefactors(n)); c += 1
    return c

A384874 a(n) is the first prime encountered when iterating the map x -> x/2 if x is even, x*lpf(x) + 1 otherwise, where lpf(x) is the least prime factor of x, on n >= 2; or -1 if a prime is never reached.

Original entry on oeis.org

2, 3, 2, 5, 3, 7, 2, 7, 5, 11, 3, 13, 7, 23, 2, 17, 7, 19, 5, 2, 11, 23, 3, 313, 13, 41, 7, 29, 23, 31, 2, 313, 17, 11, 7, 37, 19, 59, 5, 41, 2, 43, 11, 17, 23, 47, 3, 43, 313, 5869, 13, 53, 41, 13, 7, 43, 29, 59, 23, 61, 31, 313, 2, 163, 313, 67, 17, 13, 11

Offset: 2

Ya-Ping Lu, Jun 11 2025

nonn

Conjecture: a(n) != -1.

For n <= 24, this sequence is the same as A320028.

Ya-Ping Lu, Table of n, a(n) for n = 2..1000
Ya-Ping Lu, A plot of a(n) for n <= 1000

Cf. A006370, A320028, A384698.

from sympy import isprime, primefactors
def A384874(n):
    while not isprime(n): n = n*min(primefactors(n))+1 if n%2 else n//2
    return n

A384512 Record terms in A384698.

Original entry on oeis.org

2, 3, 13, 17, 37, 41, 61, 613, 829, 1861, 2269, 7333, 35149, 1008229, 909889549, 1423665384101, 10341624100573, 440171836495742615578609, 471206109194322691633610979351605854911441181, 4466501842784976704198682186832272945270823914876207595593007001786562643495541

Offset: 1

Ya-Ping Lu, May 31 2025

nonn

			41 is a term because iterating the map on 12 results in a prime in 3 steps: 12 -> 2*12+1=25 -> 25-5=20 -> 2*20+1=41 and 41 is a record prime for starting integers <= 12.

Cf. A020639 (lpf), A383777, A384698.

from sympy import isprime, primefactors; rec = 1
for n in range (2, 158163):
    while not isprime(n): n = n - min(primefactors(n)) if n%2 else 2*n + 1
    if n > rec: rec = n; print(n, end = ', ')

a(n) mod 4 = 1, n > 2.

A384698 The first prime number reached by iterating the map, x -> 2*x + 1 if x is even; x - lpf(x) otherwise where lpf(x) is the least prime factor of x, on n >= 2; or -1 if a prime is never reached.

Original entry on oeis.org

2, 3, 13, 5, 13, 7, 17, 13, 37, 11, 41, 13, 29, 41, 61, 17, 37, 19, 41, 37, 613, 23, 613, 41, 53, 613, 109, 29, 61, 31, 829, 61, 1861, 61, 73, 37, 277, 73, 157, 41, 613, 43, 89, 613, 181, 47, 97, 613, 101, 97, 401, 53, 109, 101, 113, 109, 229, 59, 829, 61, 241

Offset: 2

Ya-Ping Lu, Jun 09 2025

nonn

Conjecture: a(n) != -1.

			a(4) = 13 because iterating the map on n = 4 reaches a prime, 13, in three steps: 4 -> 2*4+1=9 -> 9-3=6 -> 2*6+1=13.

Cf. A020639 (lpf), A383777.

from sympy import isprime, primefactors
for n in range (2, 63):
    while not isprime(n): n = n - min(primefactors(n)) if n%2 else 2*n + 1
    print(n, end = ', ')

a(n) = n if n is a prime; > n otherwise.

a(n) mod 4 = 1 unless n is a prime and n mod 4 = 3.

A383777 a(n) is the number of steps that n requires to reach 0 under the map: x -> 2*x + 1 if x is even; 0 if x = 1; x - lpf(x) otherwise where lpf(x) is the least prime factor of x. a(n) = -1 if 0 is never reached.

Original entry on oeis.org

0, 1, 2, 1, 4, 1, 2, 1, 2, 3, 4, 1, 4, 1, 2, 5, 4, 1, 2, 1, 2, 3, 10, 1, 10, 3, 2, 11, 4, 1, 2, 1, 10, 3, 12, 3, 2, 1, 6, 3, 4, 1, 8, 1, 2, 9, 4, 1, 2, 9, 2, 3, 6, 1, 2, 3, 2, 3, 4, 1, 8, 1, 4, 9, 10, 9, 10, 1, 2, 11, 4, 1, 4, 1, 2, 5, 10, 5, 2, 1, 6, 3, 6, 1

Offset: 0

Ya-Ping Lu, May 17 2025

nonn

Conjecture: a(n) != -1.

			a(10) = 4 because it takes 4 steps for 10 to reach 1 by iterating the map: 10 -> 2*10+1=21 -> 21-3=18 -> 2*18+1=37 -> 37-37=0.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A005408, A006577, A046666.

A383777[n_] := Length[NestWhileList[If[OddQ[#], # - FactorInteger[#][[1,1]], 2*# + 1] &, n, # >0 &]] - 1;
Array[A383777, 100, 0] (* Paolo Xausa, May 22 2025 *)

from sympy import primefactors; mp = lambda x: (0 if x ==1 else x - min(primefactors(x)) if x%2 else 2*x+1)
def A383777(n, c = 0):
    while n != 0: n = mp(n); c += 1
    return c

A382669 Even numbers m such that both p = m^2 + 1 and q = (p^2 + 1)/2 are primes.

Original entry on oeis.org

2, 10, 150, 160, 230, 270, 400, 890, 910, 920, 1060, 1430, 1550, 1970, 2700, 2960, 3280, 3290, 3520, 3660, 4140, 4330, 4510, 4700, 4780, 4850, 4920, 5180, 5360, 5500, 5560, 5620, 5880, 5960, 6220, 6460, 6980, 7160, 7190, 7520, 7550, 7820, 9630, 9760, 9900

Offset: 1

Ya-Ping Lu, Apr 24 2025

nonn

Except 2, all terms are divisible by 10, and p-1 and q-1 are divisible by 100.

Numbers m such that p = m^2+1 and p + m^4/2 are both prime. - Chai Wah Wu, May 01 2025

			10 is a term because both 10^2 + 1 = 101 and (101^2 + 1)/2 = 5101 are primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002522, A005574, A048161.

filter:= proc(m) local p;
  p:= m^2 + 1;
  isprime(p) and isprime((p^2+1)/2)
end proc:
select(filter, [2,seq(i,i=10..10000,10)]); # Robert Israel, May 02 2025

Select[2*Range[5000], PrimeQ[#^2 + 1] && PrimeQ[#^4/2 + #^2 + 1] &] (* Amiram Eldar, Apr 24 2025 *)

from sympy import isprime
for n in range(2, 10000, 2): x = n*n + 1; ct = 0; print(n, end = ', ') if isprime(x) and isprime((x*x + 1)//2) else 0

from itertools import count, islice
from sympy import isprime
def A382669_gen(): # generator of terms
    yield 2
    yield from filter(lambda m: isprime(p:=m**2+1) and isprime(p+(m**4>>1)),(10*k for k in count(1)))
A382669_list = list(islice(A382669_gen(),45)) # Chai Wah Wu, May 02 2025

A383131 a(n) is the number of iterations that n requires to reach 1 under the map x -> -x/2 if x is even, 3x + 1 if x is odd; a(n) = -1 if 1 is never reached.

Original entry on oeis.org

0, 3, 12, 2, 5, 5, 8, 5, 11, 11, 50, 14, 14, 14, 53, 4, 17, 17, 43, 7, 7, 7, 20, 7, 46, 46, 59, 10, 10, 10, 23, 7, 49, 49, 62, 13, 13, 13, 13, 13, 26, 26, 39, 52, 52, 52, 65, 16, 16, 16, 78, 16, 16, 16, 29, 16, 42, 42, 55, 55, 55, 55, 68, 6, 19, 19, 19, 19, 19

Offset: 1

Ya-Ping Lu, Apr 17 2025

nonn

A plot of a(n) for non-zero integers n in the range of (-10000, 10000) is given in LINKS.

Conjecture: a(n) != -1 for any positive or negative integer n.

			a(3) = 12 because it takes 12 steps for n = 3 to reach 1: 3 -> 10 -> -5 -> -14 -> 7 -> 22 -> -11 -> -32 -> 16 -> -8 -> 4 -> -2 -> 1.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Ya-Ping Lu, A plot of a(n) for n in the range of (-10000, 10000)

Cf. A006577, A381055.

a:= proc(n) option remember; `if`(n=1, 0,
      1+a(`if`(n::even, -n/2, 3*n+1)))
    end:
seq(a(n), n=1..69);  # Alois P. Heinz, Aug 13 2025

js[n_] := If[EvenQ[n],-n/2,3n+1]; f[n_] := Length[ NestWhileList[js, n, # != 1 &]] - 1; Table[ f[n], {n, 69}] (* James C. McMahon, Apr 30 2025 *)

a(n) = { for (k = 0, oo, if (n==1, return (k), n = if (n%2, 3*n+1, -n/2));); } \\ Rémy Sigrist, Apr 19 2025

def A383131(n):
    ct = 0
    while n != 1: n = 3*n+1 if n%2 else -n>>1; ct += 1
    return ct

A381055 a(n) = -n/2 if n is even, 3n + 1 if n is odd.

Original entry on oeis.org

0, 4, -1, 10, -2, 16, -3, 22, -4, 28, -5, 34, -6, 40, -7, 46, -8, 52, -9, 58, -10, 64, -11, 70, -12, 76, -13, 82, -14, 88, -15, 94, -16, 100, -17, 106, -18, 112, -19, 118, -20, 124, -21, 130, -22, 136, -23, 142, -24, 148, -25, 154, -26, 160, -27, 166, -28, 172

Offset: 0

Ya-Ping Lu, Apr 14 2025

If negative indices are allowed, integers m = 0, 1, 2, 3, or 5 (mod 6) appear once (at n = -2*m) in the sequence, while integers m = 4 (mod 6) appear twice (at n = -2*m and (m-1)/3).

Conjecture: Iterating the map x -> a(x) on any negative or positive integer eventually reaches 1. Some iteration paths between -35 and 35 are illustrated in the plot of a(n) in LINKS.

			a(0) = -0/2 = 0; a(1) = 3*1 + 1 = 4; a(10) = -10/2 = -5.

Ya-Ping Lu, A plot of a(n)
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A006370, A383131.

a[n_]:=If[EvenQ[n],-n/2,3n+1];Array[a,58,0] (* James C. McMahon, Apr 30 2025 *)

def A381055(n): return 3*n+1 if n%2 else -n>>1

a(n) = (-1)^(n+1)*A006370(n).
a(n) = (1/4)*((-1)^(n+1)*(7*n+2)+5*n+2).
G.f.: x*(2*x^2-x+4)/((x-1)^2*(x+1)^2). - Alois P. Heinz, Aug 13 2025

A381169 List of twin prime averages (A014574) is partitioned by including as many elements as possible in the n-th partition, L_n, such that any gap in L_n is smaller than the gap between L_n and L_(n-1) but not bigger than the first gap in L_n. a(n) is the number of elements in L_n.

Original entry on oeis.org

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

Offset: 1

Ya-Ping Lu, Feb 15 2025

nonn

The partition method used here is the same as that in A348168.
Conjecture 1: lim_{n->oo} N_i/n = k_i, where N_i is the number of partitions with i elements and k_i is a constant, with k_2 > k_1 > k_3 > k_4 > .... The values of k_i are the same as those in A348168.
Conjecture 2: lim_{n->oo} Sum_{1..n} a(n)/n = lim_{i->oo} Sum_{1..i} i*k_i = e, or the average partition length approaches 2.71828... as n tends to infinity.
Numbers of twin prime pairs (N) and partitions with 1 through 6 twin prime pairs for n up to 10000000 are given in the table below.
n         N         N_1      N_2      N_3      N_4     N_5     N_6
--------  --------  -------  -------  -------  ------  ------  ------
1         1         1        0        0        0       0       0
10        15        6        3        1        0       0       0
100       209       30       45       16       5       3       1
1000      2536      286      416      145      64      29      19
10000     26474     2851     4331     1271     544     311     190
100000    271338    28034    43375    12923    5731    3002    1870
1000000   2725126   281837   434234   128190   56563   30074   18171
10000000  27120107  2815831  4352926  1276953  563128  302256  181612

			Twin prime pair averages in the first 10 partitions are: [4], [6], [12], [18], [30], [42], [60, 72], [102, 108], [138, 150], and [180, 192, 198]. Thus, a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1, a(7) = a(8) = a(9) = 2, and a(10) = 3.

Cf. A001097, A014574, A348168.

from sympy import isprime, nextprime; L = [4]
def nexttwin(x):
    p1 = nextprime(x); t1 = p1 + 2
    while isprime(t1) == 0: p1 = nextprime(t1); t1 = p1 + 2
    return p1+1
for _ in range(2, 89):
    print(len(L), end = ', ')
    t0 = L[-1]; t1 = nexttwin(t0); g0 = t1 - t0; M = [t1]; t = nexttwin(t1); g1 = t - t1
    while g1 < g0 and t - t1 <= g1: M.append(t); t1 = t; t = nexttwin(t)
    L = M

cp's OEIS Frontend

User: Ya-Ping Lu

A385613 Number of steps that n requires to reach 0 under the map: x-> x^2 - 1 if x is an odd prime, floor(x/3) if x is even, otherwise x - 1. a(n) = -1 if 0 is never reached.

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A384713 The number of steps that n requires to reach 1 under the map: x-> x^2 - 1 if x is an odd prime, x/2 if x is even, x - lpf(x) otherwise where lpf(x) is the least prime factor of x. a(n) = -1 if 1 is never reached.

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A384874 a(n) is the first prime encountered when iterating the map x -> x/2 if x is even, x*lpf(x) + 1 otherwise, where lpf(x) is the least prime factor of x, on n >= 2; or -1 if a prime is never reached.

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A384512 Record terms in A384698.

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A384698 The first prime number reached by iterating the map, x -> 2*x + 1 if x is even; x - lpf(x) otherwise where lpf(x) is the least prime factor of x, on n >= 2; or -1 if a prime is never reached.

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A383777 a(n) is the number of steps that n requires to reach 0 under the map: x -> 2*x + 1 if x is even; 0 if x = 1; x - lpf(x) otherwise where lpf(x) is the least prime factor of x. a(n) = -1 if 0 is never reached.

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A382669 Even numbers m such that both p = m^2 + 1 and q = (p^2 + 1)/2 are primes.

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A383131 a(n) is the number of iterations that n requires to reach 1 under the map x -> -x/2 if x is even, 3x + 1 if x is odd; a(n) = -1 if 1 is never reached.

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