A381071 Numbers k such that the sum of the proper divisors of k that have the same binary weight as k is larger than k, and no subset of these divisors sums to k.

1050, 3150, 4284, 4410, 5148, 6292, 6790, 7176, 8890, 10764, 17850, 18648, 19000, 19530, 32886, 33072, 33150, 35088, 35530, 35720, 35770, 38850, 41360, 43164, 45084, 49368, 49764, 50456, 50730, 52884, 54280, 54340, 58410, 58696, 59010, 59408, 63492, 66010, 68376

Offset

1,1

Comments

Analogous to weird numbers (A006037), as A380846 is analogous to perfect numbers (A000396).

Links

Amiram Eldar, Table of n, a(n) for n = 1..1724

Mathematica

divs[n_] := Module[{hw = DigitCount[n, 2, 1]}, Select[Divisors[n], DigitCount[#, 2, 1] == hw &]];
weirdQ[n_, d_, s1_, m1_] :=  weirdQ[n, d, s1, m1] = Module[{s = s1, m = m1}, If[m == 0, False, While[m > 0 && d[[m]] > n, s -= d[[m]]; m--]; If[m == 0, True, d[[m]] < n && If[s > n, weirdQ[n - d[[m]], d, s - d[[m]], m - 1] && weirdQ[n, d, s - d[[m]], m - 1], s < n && m < Length[d] - 1]]]];
q[n_] := Module[{d = divs[n], s, m}, s = Total[d] - n; m = Length[d] - 1; weirdQ[n, d, s, m]]; Select[Range[70000], q] (* based on a Pari code by M. F. Hasler at A006037 *)

Prog

(PARI) divs(n) = {my(h = hammingweight(n)); select(x -> hammingweight(x)==h, divisors(n)); }
is(n, d = divs(n), s = vecsum(d)-n, m = #d-1) = {if(m == 0, return(0)); while(m > 0 && d[m] > n, s -= d[m]; m--); if(m==0, return(1)); (d[m] < n &&
if(s > n, is(n-d[m], d, s-d[m], m-1) && is(n, d, s-d[m], m-1), s < n && m < #d-1)); } \\ based on a code by M. F. Hasler at A006037

Crossrefs

Cf. A000396, A380845, A380846.

Subsequence of A380929.

A381072 is a subsequence.

Similar sequences: A006037, A064114, A292986, A306984, A321146, A327948, A339939, A348525, A348631, A349285, A364862.

Sequence in context: A015064 A090057 A287173 * A204753 A252391 A020389

Adjacent sequences: A381068 A381069 A381070 * A381072 A381073 A381074

Keyword

nonn,base

Author

Amiram Eldar, Feb 13 2025

Status

approved