A392231 Positive integers (not multiples of 10) whose constant congruence speed is smaller than the product of the constant congruence speeds of all their prime factors (see A373387 for the definition of "constant congruence speed").

14, 21, 25, 28, 35, 42, 49, 56, 63, 75, 77, 84, 86, 91, 98, 105, 112, 119, 125, 129, 133, 147, 154, 161, 172, 175, 189, 196, 202, 203, 214, 215, 217, 231, 238, 245, 252, 258, 259, 266, 273, 275, 287, 294, 298, 301, 302, 303, 308, 314, 315, 321, 322, 325, 329, 336, 343, 344, 364, 371, 375

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)).

Here, prime factors are considered with their multiplicity (e.g., the prime factors of 75 are 3, 5, and 5 and thus the product of their constant congruence speeds equals 1*2*2 = 4).

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.

This sequence belongs to a family of 11 related sequences comparing the constant congruence speed A373387(m) with various combinations of the speeds of its prime factors. See Crossrefs.

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..61.

Gabriele Di Pietro, Congruence speed smaller than the product of their factors, Zenodo, 2026.

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, 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

a(1) = 14 since the constant congruence speed of 2 is 1, the constant congruence speed of 7 is 2, the constant congruence speed of 2*7 is 1, and 1 is smaller than 1*2.

Prog

(Python) # See Links.

Crossrefs

Cf. A317905, A373387, A389432, A389979, A389980, A389981.

Sequence in context: A330137 A230095 A133185 * A013629 A234802 A162780

Adjacent sequences: A392228 A392229 A392230 * A392232 A392233 A392234

Keyword

nonn,base

Author

Gabriele Di Pietro and Marco Ripà, Jan 04 2026

Status

approved