A392233 Composite numbers (not multiples of 10) whose constant congruence speed equals the product of the constant congruence speeds of all their prime factors.

4, 6, 8, 9, 12, 16, 22, 27, 33, 34, 36, 38, 39, 44, 45, 46, 48, 52, 54, 58, 62, 64, 66, 69, 72, 78, 81, 85, 87, 88, 92, 94, 96, 102, 104, 106, 108, 111, 114, 115, 116, 117, 121, 122, 123, 128, 134, 136, 138, 141, 142, 144, 146, 148, 152, 153, 155, 156, 158, 159, 162, 164, 165, 166, 168

Offset

1,1

Comments

For the definition of "constant congruence speed", see A373387 and 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.

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..65.

Gabriele Di Pietro, Congruence speed equal 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(2) = 6 since the constant congruence speed of 6 is 1, the constant congruence speed of 2 is 1, the constant congruence speed of 3 is 1 and 1 is equal to 1*1.

Prog

(Python) # see Links

Crossrefs

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

Subsequence of A067251.

Sequence in context: A350706 A036310 A391434 * A046760 A115684 A373879

Adjacent sequences: A392230 A392231 A392232 * A392234 A392235 A392236

Keyword

nonn,base

Author

Gabriele Di Pietro and Marco Ripà, Jan 04 2026

Status

approved