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A352322 Starts of runs of 3 consecutive Pell-Niven numbers (A352320).

%I A352322 #9 Mar 19 2022 04:16:41
%S A352322 4,28,110,168,984,1024,3123,3514,5740,6783,6923,8584,12664,16744,
%T A352322 18160,19670,23190,23470,24030,34503,34643,36304,40384,45880,47390,
%U A352322 50910,51190,51750,57607,61640,68104,73600,78403,78630,78910,79470,86674,89360,95824,101320
%N A352322 Starts of runs of 3 consecutive Pell-Niven numbers (A352320).
%C A352322 Conjecture: There are no runs of 4 consecutive Pell-Niven numbers (checked up to 2*10^8).
%H A352322 Amiram Eldar, <a href="/A352322/b352322.txt">Table of n, a(n) for n = 1..10000</a>
%e A352322 4 is a term since 4, 5 and 6 are all Pell-Niven numbers: the minimal Pell representation of 4, A317204(20) = 20, has the sum of digits 2+0 = 2 and 4 is divisible by 2, the minimal Pell representation of 5, A317204(5) = 100, has the sum of digits 1+0+0 = 1 and 5 is divisible by 1, and the minimal Pell representation of 6, A317204(6) = 101, has the sum of digits 1+0+1 = 2 and 6 is divisible by 2.
%t A352322 pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; pellNivenQ[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]; Divisible[n, Plus @@ IntegerDigits[Total[3^(s - 1)], 3]]]; seq[count_, nConsec_] := Module[{pn = pellNivenQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ pn, c++; AppendTo[s, k - nConsec]]; pn = Join[Rest[pn], {pellNivenQ[k]}]; k++]; s]; seq[30, 3]
%Y A352322 Cf. A000129, A265744, A317204.
%Y A352322 A182190 \ {0} is a subsequence.
%Y A352322 Subsequence of A352320 and A352321.
%Y A352322 Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716, A351721, A352091, A352109.
%K A352322 nonn,base
%O A352322 1,1
%A A352322 _Amiram Eldar_, Mar 12 2022