A361934 Numbers k such that k and k+1 are both primitive Zumkeller numbers (A180332).

82004, 84524, 158235, 516704, 2921535, 5801984, 10846016, 12374144, 12603824, 18738224, 24252074, 24887655, 25691984, 32409530, 33696975, 35356544, 36149295, 41078114, 42541190, 43485584

Offset

1,1

Links

Table of n, a(n) for n=1..20.

Example

82004 is a term since 82004 and 82005 are both primitive Zumkeller numbers.

Mathematica

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]]];
(* after M. F. Hasler's pari code at A006037 *)
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]]];
primZumQ[n_] := zumQ[n] && AllTrue[Most[Divisors[n]], ! zumQ[#] &];
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]

Prog

(PARI)
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
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)); }
isPrimZum(n) = {if(!isZum(n), return(0)); fordiv(n, d, if(d < n && isZum(d), return(0))); 1; }
lista(kmax) = {my(is1 = 0, is2); for(k = 2, kmax, is2 = isPrimZum(k); if(is1 && is2, print1(k-1, ", ")); is1 = is2); }

Crossrefs

Subsequence of A180332 and A328327.

Similar sequences: A283418, A330872, A334882.

Cf. A006037, A083207.

Sequence in context: A138044 A098187 A146025 * A283418 A330872 A253957

Adjacent sequences: A361931 A361932 A361933 * A361935 A361936 A361937

Keyword

nonn,more

Author

Amiram Eldar, Mar 31 2023

Status

approved