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%I #4 Mar 31 2023 07:01:12
%S 82004,84524,158235,516704,2921535,5801984,10846016,12374144,12603824,
%T 18738224,24252074,24887655,25691984,32409530,33696975,35356544,
%U 36149295,41078114,42541190,43485584
%N Numbers k such that k and k+1 are both primitive Zumkeller numbers (A180332).
%e 82004 is a term since 82004 and 82005 are both primitive Zumkeller numbers.
%t q[n_, d_, s1_, m1_] := Module[{s = s1, m = m1}, If[m == 0, False, While[d[[m]] > n, s -= d[[m]]; m--]; d[[m]] == n || If[s > n, q[n - d[[m]], d, s - d[[m]], m - 1] || q[n, d, s - d[[m]], m - 1], n == s]]];
%t (* after _M. F. Hasler_'s pari code at A006037 *)
%t zumQ[n_] := Module[{d = Most[Divisors[n]], m, s}, m = Length[d]; s = Total[d]; If[OddQ[s + n], False, q[(s + n)/2, d, s, m]]];
%t primZumQ[n_] := zumQ[n] && AllTrue[Most[Divisors[n]], ! zumQ[#] &];
%t seq[kmax_] := Module[{s = {}, zq1 = False, zq2}, Do[zq2 = primZumQ[k]; If[zq1 && zq2, AppendTo[s, k - 1]]; zq1 = zq2, {k, 2, kmax}]; s]; seq[3*10^6]
%o (PARI)
%o is1(n,d,s,m) = {m||return; while(d[m]>n, s-=d[m]; m--||return); d[m]==n || if(n<s, is1(n-d[m], d, s-d[m], m-1) || is1(n, d, s-d[m], m-1), n==s);} \\ after _M. F. Hasler_ at A006037
%o isZum(n) = {my(d = divisors(n)[^-1], s = vecsum(d), m = #d); if((s+n)%2, return(0), is1((s+n)/2, d, s, m)); }
%o isPrimZum(n) = {if(!isZum(n), return(0)); fordiv(n, d, if(d < n && isZum(d), return(0))); 1;}
%o lista(kmax) = {my(is1 = 0, is2); for(k = 2, kmax, is2 = isPrimZum(k); if(is1 && is2, print1(k-1, ", ")); is1 = is2);}
%Y Subsequence of A180332 and A328327.
%Y Similar sequences: A283418, A330872, A334882.
%Y Cf. A006037, A083207.
%K nonn,more
%O 1,1
%A _Amiram Eldar_, Mar 31 2023
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