A393525 Consider the following transformation of prime numbers: If b(n) == 1 (mod 4) then b(n+1) = (b(n) + 1)/2. If b(n) == 3 (mod 4) then b(n+1) = (b(n) - 1)/2. Continue this transformation until a nonprime is reached. a(n) is the least prime number such that exactly n primes appear in the chain of numbers created by this transformation.

3, 5, 11, 23, 47, 2879, 1065601, 1985902081, 21981381119, 13357981992959, 568000797112319, 1136001594224639, 3105111850422067201, 567357957277949460481, 1134715914555898920961

Offset

1,1

Comments

Sequence made at the suggestion of Thomas Ordowski and with the assistance of Amiram Eldar.

a(10) > 3 * 10^11.

Is this sequence strictly increasing?

Links

Table of n, a(n) for n=1..15.

Example

a(5) = 47 because under the described transformation 47 -> 23 -> 11 -> 5 -> 3 -> 1. This longest full chain contains 5 prime numbers and 47 is the least prime with this property.

Prog

(Python)
from sympy import isprime, nextprime
terms, x = {}, 3
while len(terms) < 7: # program is slow for a(8) and beyond
    y = x
    steps = 0
    while y > 0 and isprime(y):
        steps += 1
        if y % 4 == 1:
            y = (y+1)//2
        elif y % 4 == 3:
            y = (y-1)//2
    if steps not in terms:
        terms[steps] = x
    x = nextprime(x)
for z in terms:
    print(terms[z], end=", ")

Crossrefs

Cf. A110092, A110093, A110095.

Sequence in context: A030494 A246491 A084361 * A077229 A382216 A382214

Adjacent sequences: A393522 A393523 A393524 * A393526 A393527 A393528

Keyword

nonn,more

Author

David Consiglio, Jr., Mar 18 2026

Extensions

a(10)-a(15) from Bert Dobbelaere, Mar 31 2026

Status

approved