A348631 -id:A348631 - cp's OEIS frontend

A349285 (1+e)-weird numbers: (1+e)-abundant numbers k such that no subset of the aliquot (1+e)-divisors of k sums to k.

70, 836, 4030, 5830, 10430, 10570, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17570, 17990, 18410, 18830, 18970, 19390, 19670, 19810, 20510, 21490, 21770, 21910, 22190, 23170, 23590, 24290

Offset: 1

Amiram Eldar, Nov 13 2021

nonn

The (1+e)-abundant numbers are numbers k such that A051378(k) > 2*k (union of A333928 and A349284).
Is there any number besides 836 which is in this sequence but not in A348631? - R. J. Mathar, Nov 16 2021
The next term after 836 that is not in A348631 is a(89) = 45356. - Amiram Eldar, Nov 21 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A049603, A051378, A333928, A349284.

Similar sequences: A006037, A064114, A292986, A306984, A321146, A327948, A348525.

divQ[n_, m_] := (n > 0 && (m == 0 || Divisible[n, m])); oeDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[divQ, {f[[;; , 2]], IntegerExponent[d, f[[;; , 1]]]}]]; oeDivs[1] = {1}; oeDivs[n_] := Module[{d = Divisors[n]}, Select[d, oeDivQ[n, #] &]]; oesigma[1] = 1; oesigma[n_] := Total@oeDivs[n]; oeAbundantQ[n_] := oesigma[n] > 2*n; oeWeirdQ[n_] := oeAbundantQ[n] && Module[{d = Most[oeDivs[n]]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 0]; Select[Range[12000], oeWeirdQ]

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.

Original entry on oeis.org

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

Amiram Eldar, Feb 13 2025

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

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

Cf. A000396, A380845, A380846.
Subsequence of A380929.
A381072 is a subsequence.
Similar sequences: A006037, A064114, A292986, A306984, A321146, A327948, A339939, A348525, A348631, A349285, A364862.

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 *)

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

cp's OEIS Frontend

A349285 (1+e)-weird numbers: (1+e)-abundant numbers k such that no subset of the aliquot (1+e)-divisors of k sums to k.

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