A389980 Positive integers (not multiples of 10) whose constant congruence speed is smaller than the maximum of the constant congruence speeds of all their prime factors (see A373387 for the definition of "constant congruence speed").

14, 21, 28, 42, 56, 63, 77, 84, 86, 91, 98, 112, 119, 129, 133, 147, 154, 161, 172, 189, 196, 202, 203, 214, 217, 231, 238, 252, 258, 259, 266, 273, 287, 294, 298, 302, 303, 308, 314, 321, 322, 329, 336, 344, 364, 371, 378, 386, 387, 392, 398, 404, 406, 413, 427

Offset

1,1

Comments

For the definition of "constant congruence speed", see Def. 1.2 (and also Def. 1.1) of "Number of stable digits of any integer tetration" in Links (for all positive integers m > 1 and not a multiple of 10, this corresponds to A373387(m)).

Furthermore, if the product of two positive integers m' and m'' is not divisible by 10, then the constant congruence speed of m'*m'' is necessarily greater than or equal to the minimum of the constant congruence speeds of m' and m'' (see Equation 2.4 of "A Compact Notation for Peculiar Properties Characterizing Integer Tetration" in Links).

By definition, this sequence contains no prime number.

References

Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6

Links

Table of n, a(n) for n=1..55.

Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43-61.

Marco Ripà and Gabriele di Pietro, A Compact Notation for Peculiar Properties Characterizing Integer Tetration, Zenodo, April 2025.

Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441-457.

Example

14 is a term since the constant congruence speeds of 2 and 14 is 1, while the constant congruence speed of 7 is 2, which is greater than the constant congruence speed of 2*7.

Prog

(Python)
from sympy import factorint
def upto(n):
    s=[]
    for k in range(2, n+1):
        if k%10==0:
            continue
        Vk=A373387(k)
        if Vk is None:
            continue
        pf=[p for p in factorint(k)]
        Vpf=[A373387(p) for p in pf if A373387(p) is not None]
        if Vpf and Vk<max(Vpf):
            s.append(k)
    return s
print(upto(10000))

Crossrefs

Cf. A317905, A373387, A389432, A389979, A389981.

Sequence in context: A284808 A335774 A006625 * A114382 A120141 A240593

Adjacent sequences: A389977 A389978 A389979 * A389981 A389982 A389983

Keyword

nonn,base

Author

Marco Ripà, Oct 20 2025

Status

approved