A380525 Squarefree numbers k such that for all factorizations of k as x*y, the sum (x * y') + (x' * y) is carryless when the addition is done in the primorial base, A049345. Here n' stands for A003415(n), the arithmetic derivative of n.

1, 2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 23, 26, 29, 31, 37, 38, 41, 43, 47, 53, 59, 61, 62, 67, 70, 71, 73, 74, 79, 83, 86, 89, 97, 101, 103, 107, 109, 113, 122, 127, 131, 134, 137, 139, 146, 149, 151, 154, 157, 158, 163, 167, 173, 179, 181, 186, 190, 191, 193, 194, 195, 197, 199, 206, 211, 218, 223, 227, 229, 233

Offset: 1

Author

Antti Karttunen, Feb 04 2025

Keywords

• nonn

Comments

A380468 is a subsequence. This differs from it by containing also the terms 70, 154, 190, 195, 455, 574, 645, 1054, 1085, ...

Examples

			For n=70, there are four factorizations into two factors: 1*70, 2*35, 5*14, 7*10, and thus, applying the formula (x' * y) + (x * y') we obtain
  0*70 + 1*70' = A003415(70) = 59, and A049345(59) = 1421.
  1*35 + 2*35' = 35 + 2*12, i.e., 1021 + 400 in primorial base, (giving 1421)
  1*14 + 5*14' = 14 + 5*9, i.e., 210 + 1211 in primorial base,
  1*10 + 7*10' = 10 + 7*7, i.e., 120 + 1301 in primorial base,
and as all these sums are carryless, 70 is included in this sequence.
For n = 1518 = 2*3*11*23, we obtain eight factorizations into two factors:
x*y:    | 1*1518   2*759   3*506   6*253  11*138   22*69   23*66   33*46
--------+----------------------------------------------------------------
x' * y  |      0   34111   22410   60021    4300   41411    2100   30210 (in primorial base)
x  * y' |  66421   32310   44011    6400   62121   25010   64321   36211
--------+----------------------------------------------------------------
Sum     |  66421   66421   66421   66421   66421   66421   66421   66421 = A049345(A003415(1518)), and as all these sums are carryless, 1581 is included in this sequence.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences related to primorial base.

Crossrefs

Intersection of A005117 and A358673.

Cf. A003415, A049345, A380524 (characteristic function).

Subsequences: A380468, A380526 (terms with at least 5 prime factors).

Programs

PARI

is_A380525 = A380524;