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A352343 Numbers k such that k and k+1 are both lazy-Pell-Niven numbers (A352342).

%I A352343 #7 Mar 14 2022 02:41:50
%S A352343 1,24,63,209,216,459,560,584,656,729,999,1110,1269,1728,1859,1989,
%T A352343 2100,2196,2197,2255,2650,2651,2820,3443,3497,4080,4563,5291,5784,
%U A352343 5785,5837,5928,6252,6383,7344,7657,7812,8150,8203,8459,8670,8749,9251,9295,9372,9464,9840,9884
%N A352343 Numbers k such that k and k+1 are both lazy-Pell-Niven numbers (A352342).
%C A352343 Numbers k such that A352340(k) | k and A352340(k+1) | k+1.
%H A352343 Amiram Eldar, <a href="/A352343/b352343.txt">Table of n, a(n) for n = 1..10000</a>
%e A352343 24 is a term since 24 and 25 are both lazy-Pell-Niven numbers: the maximal Pell representation of 24, A352339(24) = 1210, has the sum of digits A352340(24) = 1+2+1+0 = 4 and 24 is divisible by 4, and the maximal Pell representation of 25, A352339(25) = 1211, has the sum of digits A352340(25) = 1+2+1+1 = 5 and 25 is divisible by 5.
%t A352343 pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; pellp[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; IntegerDigits[Total[3^(s - 1)], 3]]; lazyPellNivenQ[n_] := Module[{v = pellp[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] > 0 && v[[i + 1]] == 0 && v[[i + 2]] < 2, v[[i ;; i + 2]] += {-1, 2, 1}; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; Divisible[n, Plus @@ v[[i[[1, 1]] ;; -1]]]]; Select[Range[10^4], lazyPellNivenQ[#] && lazyPellNivenQ[#+1] &]
%Y A352343 Cf. A352339, A352340.
%Y A352343 Subsequence of A352342.
%Y A352343 Subsequences: A352344, A352345.
%Y A352343 Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090, A352108, A352321.
%K A352343 nonn,base
%O A352343 1,2
%A A352343 _Amiram Eldar_, Mar 12 2022