cp's OEIS Frontend

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.

%I A380525 #13 Feb 09 2025 18:10:17
%S A380525 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,
%T A380525 70,71,73,74,79,83,86,89,97,101,103,107,109,113,122,127,131,134,137,
%U A380525 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
%N 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.
%C A380525 A380468 is a subsequence. This differs from it by containing also the terms 70, 154, 190, 195, 455, 574, 645, 1054, 1085, ...
%H A380525 Antti Karttunen, <a href="/A380525/b380525.txt">Table of n, a(n) for n = 1..20000</a>
%H A380525 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>.
%e A380525 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
%e A380525   0*70 + 1*70' = A003415(70) = 59, and A049345(59) = 1421.
%e A380525   1*35 + 2*35' = 35 + 2*12, i.e., 1021 + 400 in primorial base, (giving 1421)
%e A380525   1*14 + 5*14' = 14 + 5*9, i.e., 210 + 1211 in primorial base,
%e A380525   1*10 + 7*10' = 10 + 7*7, i.e., 120 + 1301 in primorial base,
%e A380525 and as all these sums are carryless, 70 is included in this sequence.
%e A380525 For n = 1518 = 2*3*11*23, we obtain eight factorizations into two factors:
%e A380525 x*y:    | 1*1518   2*759   3*506   6*253  11*138   22*69   23*66   33*46
%e A380525 --------+----------------------------------------------------------------
%e A380525 x' * y  |      0   34111   22410   60021    4300   41411    2100   30210 (in primorial base)
%e A380525 x  * y' |  66421   32310   44011    6400   62121   25010   64321   36211
%e A380525 --------+----------------------------------------------------------------
%e A380525 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.
%o A380525 (PARI) is_A380525 = A380524;
%Y A380525 Intersection of A005117 and A358673.
%Y A380525 Cf. A003415, A049345, A380524 (characteristic function).
%Y A380525 Subsequences: A380468, A380526 (terms with at least 5 prime factors).
%K A380525 nonn
%O A380525 1,2
%A A380525 _Antti Karttunen_, Feb 04 2025