A394760 - OEIS

A394760

a(1) = 1, a(2) = 2; with s(n) the partial sum of a(i), i <= n, for n > 2, a(n) = least k != a(i), such that rad(k) = rad(s(n-1)), where rad = A007947.

%I #10 Apr 13 2026 19:21:33

%S 1,2,3,6,12,18,42,84,126,168,462,924,1386,1848,2772,7854,15708,23562,

%T 31416,47124,54978,62832,70686,322014,644028,966042,1288056,1932084,

%U 2254098,2576112,2898126,3542154,4186182,20930910,41861820,62792730,83723640,104654550,125585460

%N a(1) = 1, a(2) = 2; with s(n) the partial sum of a(i), i <= n, for n > 2, a(n) = least k != a(i), such that rad(k) = rad(s(n-1)), where rad = A007947.

%C If rad(s(n-1)) > rad(s(n-2)), then a(n) = rad(s(n-1)).

%C Let r = rad(s(n-1)). a(n) = r implies a(n+1) = 2*r for n >= 3.

%C Let R(r) = {m : rad(m) | r} sorted such that R(r,i) denotes the i-th smallest term.

%C The sequence can be interpreted as an irregular table T(i,j) = R(r,j)*r, where row i shares the same squarefree kernel r. Since T(i,j) == 0 (mod r), we may express the sequence of partial sums also as an irregular table S(i,j) = m*r that continues so long as rad(m) | r.

%C When rad(m) does not divide r, we then have S(i+1,1) = m'*r' where m' = m*r/r' and r' > r for r' = rad(S(i+1,1)).

%C Since the squarefree kernels r are nondecreasing as n increases, this sequence is not a permutation of natural numbers and only a finite number of smallest terms that have the same squarefree kernel appear in the sequence. For example, the intersection of A033845 and this sequence is {6, 12, 18}. Furthermore, not all squarefree numbers appear in this sequence.

%C Intersection of this sequence and A001694 is {1}.

%C Intersection of this sequence and the primes is {2, 3}.

%C The only primorials (in A002110) in this sequence appear to be {1, 2, 6}.

%C The behavior of this sequence regarding its partial sums is similar to that of A392975 regarding its partial sums.

%H Michael De Vlieger, <a href="/A394760/b394760.txt">Table of n, a(n) for n = 1..10000</a>

%e Table of n, a(n), s(n) for n = 1..16:

%e n a(n) = r * m s(n) rad(s(n)) = r

%e ----------------------------------------------

%e 1 1 = 1 * 1 1 1

%e 2 2 = 2 * 1 3 3

%e 3 3 = 3 * 1 6 2 * 3

%e 4 6 = 6 * 1 12 2 * 3

%e 5 12 = 6 * 2 24 2 * 3

%e 6 18 = 6 * 3 42 2 * 3 * 7

%e 7 42 = 42 * 1 84 2 * 3 * 7

%e 8 84 = 42 * 2 168 2 * 3 * 7

%e 9 126 = 42 * 3 294 2 * 3 * 7

%e 10 168 = 42 * 4 462 2 * 3 * 7 * 11

%e 11 462 = 462 * 1 924 2 * 3 * 7 * 11

%e 12 924 = 462 * 2 1848 2 * 3 * 7 * 11

%e 13 1386 = 462 * 3 3234 2 * 3 * 7 * 11

%e 14 1848 = 462 * 4 5082 2 * 3 * 7 * 11

%e 15 2772 = 462 * 6 7854 2 * 3 * 7 * 11 * 17

%e 16 7854 = 7854 * 1 15708 2 * 3 * 7 * 11 * 17

%e This sequence as an irregular table T(i,j) where row i has the same squarefree kernel:

%e i\j | 1 2 3 4 5 6 7 8

%e ----+--------------------------------------------------------

%e 0 | 1;

%e 1 | 2;

%e 2 | 3;

%e 3 | 6, 12, 18;

%e 4 | 42, 84, 126, 168;

%e 5 | 462, 924, 1386, 1848, 2772;

%e 6 | 7854, 15708, 23562, 31416, 47124, 54978, 62832, 70686;

%e ...

%e In this table, T(i,1) is squarefree and all the rest of the terms are nonsquarefree.

%t Block[{a, c, f, k, r, s, u, nn}, nn = 40; c[_] := False; u[_] := 1; f[x_] := Times @@ FactorInteger[x][[;; , 1]]; a = Range[1, 2]; Map[Set[c[#], True] &, a]; s = Total[a]; a~Join~Reap[Do[r = f[s]; While[Set[k, u[r]*r]; Nand[! c[k], Divisible[r, f@ u[r] ] ], u[r]++]; Sow[k]; s += k; Set[c[k], True], {n, Length[a], nn}] ][[-1, 1]] ]

%Y Cf. A005117, A007947, A392975.

%K nonn,easy

%O 1,2

%A _Michael De Vlieger_ and _James C. McMahon_, Mar 31 2026