A350590 -id:A350590 - cp's OEIS frontend

A352973 Prime numbers p such that iterating the map m -> m^2 + 1 on p generates a number ending with p in binary format.

2, 5, 37, 421, 8101, 11771813, 10593030863298469, 17520588382079786917, 644709886888204541861, 126810635974586364597324276501890165253751178116964261, 281339171965861859345972453867311708147087370351598335047820025433137061

Offset: 1

Ya-Ping Lu, Apr 13 2022

			37 is a term because iterating the map on 37, which is '100101' in binary format, gives: 37 -> 1370 -> 1876901, which in binary format is '111001010001110100101' ending with '100101'.

Cf. A002522, A066872, A350590.

from sympy import isprime; R = []
for i in range(0, 1000):
    t = 2**i; L = []
    while t not in L: L.append(t); t = (t*t + 1) % 2**(i+1)
    {R.append(j) for j in {L[-1], L[-2]} if j not in R and isprime(j)}
R.sort(); print(*R, sep = ', ')

A354832 Integers m such that iterating the map f(x) = x^2 + 1 on m generates a number ending with m in binary format.

Original entry on oeis.org

0, 1, 2, 5, 10, 26, 37, 90, 165, 421, 933, 1957, 4005, 8101, 8282, 24666, 40869, 106405, 237477, 286810, 811098, 1286053, 3383205, 5005402, 11771813, 28549029, 38559834, 105668698, 239886426, 296984485, 833855397, 1313628250, 3461111898, 7756079194, 9423789989

Offset: 1

Ya-Ping Lu, Jun 07 2022

It seems that 2^(n-2) <= a(n) < 2^(n-1) for n > 1.
All terms are part of a cycle under x -> f(x) mod 2^L. For example, 5 = f(2), 10 = f(5) mod (2^4), 26 = f(5), 37 = f(10) mod (2^6), and 90 = f(5) mod (2^6).
It takes 2 iterations for a term in the sequence to generate a number ending with the term itself in binary format. Endings of the numbers in the 2 iterations, m1 -> m2 -> m1, for the number of binary digits (d) up to 10 are given below. Note that m1 and m2 are bit-by-bit complement to each other, due to the fact that f(f(x)) = x mod 2^L as pointed out by Kevin Ryde in Discussion.
d     m1 or m2 (bin)      m2 or m1 (bin)       m1 (decimal)
--  ------------------  ------------------  ------------------
1            0 (m1/m2)           1 (m2/m1)  a(1) = 0; a(2) = 1
2           10 (m1)             01 (m2)     a(3) = 2
3          010 (m2)            101 (m1)     a(4) = 5
4         1010 (m1)           0101 (m2)     a(5) = 10
5        11010 (m1)          00101 (m2)     a(6) = 26
6       011010 (m2)         100101 (m1)     a(7) = 37
7      1011010 (m1)        0100101 (m2)     a(8) = 90
8     01011010 (m2)       10100101 (m1)     a(9) = 165
9    001011010 (m2)      110100101 (m1)     a(10)= 421
10  0001011010 (m2)     1110100101 (m1)     a(11)= 933

			26 is a term because iterating the map on 26 gives, in binary format, 11010 -> 1010100101 -> 1101111111001011010, which ends with 11010.

Cf. A002522, A350130, A350590, A352973.

R = []
for i in range(0, 34):
    t = 2**i; L = []
    while t not in L: L.append(t); t = (t*t + 1) % 2**(i+1)
    {R.append(j) for j in {L[-1], L[-2]} if j not in R}
R.sort(); print(*R, sep = ', ')

cp's OEIS Frontend

A352973 Prime numbers p such that iterating the map m -> m^2 + 1 on p generates a number ending with p in binary format.

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