Minimal set of the strings for primes with at least two digits

[sweety439]

Integers b>=2 sorted by A062955(b):

2 (1), 3 (4), 4 (6), 6 (10), 5 (16), 8 (28), 7&10 (36), 12 (44), 9 (48), 14 (78), 11 (100), 18 (102), 15 (112), 16 (120), 13 (144), 20 (152), 24 (184), 22 (210), 30 (232), 21 (240), 17 (256), 26 (300), 19&28 (324), 36 (420), 27 (468), 25 (480), 23 (484), 42 (492), 32 (496), 34 (528), 40 (624), 33 (640), 38 (666), 48 (752), 29 (784), 35 (816), 44 (860), 31 (900), 39 (912), 60 (944), 54 (954), 50 (980), 46 (990), ...

Integers b>=2 sorted by number of minimal primes (starting with b+1) base b: (not sure if 26 and 28 are before 17 and 21)

2 (1), 3 (3), 4 (5), 6 (11), 5 (22), 7 (71), 8 (75), 10 (77), 12 (106), 9 (151), 18 (549), 14 (650), 11 (1068), 15 (1284), 16 (2346~2347), 30 (2619), 13 (3195~3197), 20 (3314), 24 (3409), 22 (8003), 17 (10405~10428), 21 (13373~13395), ...

Integers b>=2 sorted by length of largest minimal prime (starting with b+1) base b:

2 (2), 3&4 (3), 6 (5), 7 (17), 10 (31), 12 (42), 5 (96), 15 (157), 8 (221), 9 (1161), 18&20 (6271), 24 (8134), 14 (19699), 22 (22003), 30 (34206), 11 (62669), ...

These three sequences are conjectured to be similar, the integers b = 7 and b = 15 for the third sequence is relatively small since they (as well as b = 3) are high-weight bases (like CRUS bases R7, R15, S7, S15, they are high-weight bases), i.e. they are very "primeful", while b = 5 and b = 8 and b = 11 and b = 14 are relatively large, since they are low-weight bases (like CRUS, bases == 2 mod 3 are low-weight bases), although this does not hold for b = 20, which is also == 2 mod 3

[sweety439]

Originally Posted by sweety439

Number of totally digits of minimal primes (start with b+1) in base b

Sum of all minimal primes (start with b+1) in base b

Product of all minimal primes (start with b+1) in base b

Base 2:

1 primes, totally 2 digits, sum, product

Base 3:

3 primes, totally 7 digits, sum, product

Base 4:

5 primes, totally 11 digits, sum, product

Base 5:

22 primes, totally 169 digits, sum, product

Base 6:

11 primes, totally 29 digits, sum, product

Base 7:

71 primes, totally 288 digits, sum, product

Base 8:

75 primes, totally 523 digits, sum, product

Base 9:

151 primes, totally 3004 digits, sum, product

Base 10:

77 primes, totally 310 digits, sum, product

Base 11:

1068 primes, totally 75414 digits, sum, product

Base 12:

106 primes, totally 433 digits, sum, product

Base 14:

650 primes, totally 25404 digits, sum, product

Base 15:

1284 primes, totally 8286 digits, sum, product

Base 18:

Conjecture: the sum of all minimal primes (start with b+1) base b is always in https://oeis.org/A063538, i.e. it must have a prime factor >= its square root, this has been verified for bases 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, but this is very hard to prove or disprove, since proving or disproving this requires factoring large numbers.

Some interesting sequences: (since I have a conjecture that the sum of all minimal primes (start with b+1) base b is always in https://oeis.org/A063538, i.e. it must have a prime factor >= its square root, I have run the "greatest prime factor ^2-1" sequences for them, while it is meaningless for running Aliquot sequences and home prime sequences for them)

Base 5:

aliquot sequence starting with product home prime sequence starting with product inverse home prime sequence starting with product greatest prime factor ^2-1 sequence starting with product greatest prime factor ^2-1 sequence starting with sum

Base 7:

aliquot sequence starting with product home prime sequence starting with product inverse home prime sequence starting with product greatest prime factor ^2-1 sequence starting with product greatest prime factor ^2-1 sequence starting with sum

Base 8:

aliquot sequence starting with product home prime sequence starting with product inverse home prime sequence starting with product greatest prime factor ^2-1 sequence starting with product greatest prime factor ^2-1 sequence start with sum

Base 10:

aliquot sequence starting with product home prime sequence starting with product inverse home prime sequence starting with product greatest prime factor ^2-1 sequence starting with product greatest prime factor ^2-1 sequence start with sum

Base 12: (no inverse home prime sequence available)

aliquot sequence starting with product home prime sequence starting with product greatest prime factor ^2-1 sequence starting with product greatest prime factor ^2-1 sequence starting with sum

[sweety439]

For the interest of "greatest prime factor ^2-1" sequences:

* N-1 primality proving
* N+1 primality proving
* P-1 integer factorization method
* P+1 integer factorization method

https://oeis.org/A087713 (greatest prime factor of p^2-1)
https://oeis.org/A024710 (the same sequence (start with p=11) of the A087713, which is the greatest prime factor of A024702)
https://oeis.org/A024702 ((p^2-1)/24)
https://oeis.org/A084920 (p^2-1)
https://oeis.org/A001248 (p^2)
https://oeis.org/A001318 (generalized pentagonal numbers, (n^2-1)/24)