A064114 -id:A064114 - cp's OEIS frontend

A348631 Nonexponential weird numbers: nonexponential abundant numbers (A348604) that are not equal to the sum of any subset of their nonexponential divisors.

70, 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, Oct 26 2021

nonn

			70 is a term since the sum of its nonexponential divisors, {1, 2, 5, 7, 10, 14, 35}, is 74 > 70, and no subset of these divisors sums to 70.

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

Cf. A160135, A348604.

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

dQ[n_, m_] := (n > 0 && m > 0 && Divisible[n, m]); expDivQ[n_, d_] := Module[{ft = FactorInteger[n]}, And @@ MapThread[dQ, {ft[[;; , 2]], IntegerExponent[d, ft[[;; , 1]]]}]]; neDivs[1] = {}; neDivs[n_] := Module[{d = Divisors[n]}, Select[d, ! expDivQ[n, #] &]]; nesigma[n_] := Total@neDivs[n]; neAbundantQ[n_] := nesigma[n] > n; neWeirdQ[n_] := neAbundantQ[n] && Module[{d = neDivs[n]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 0]; Select[Range[6000], neWeirdQ]

A328563 Nonsquarefree unitary weird numbers that are also weird numbers.

Original entry on oeis.org

1554070, 1596070, 1725430, 1859830, 1952230, 2095030, 2242870, 2293270, 2553670, 2607430, 2716630, 2772070, 3116470, 3481030, 3607030, 3670870, 3800230, 3998470, 4065670, 4410070, 4623430, 4841830, 5065270, 5140870, 5371030, 5527270, 5606230, 6009430, 6597430

Offset: 1

Amiram Eldar, Oct 19 2019

nonn

The nonsquarefree unitary weird numbers that are not weird numbers are listed in A328562.

Amiram Eldar, Table of n, a(n) for n = 1..15716 (terms below 10^10)

Subsequence of A064114.
Intersection of A006037 and A292705.
Cf. A328562.

weirdQ[n_, d_, s1_, m1_] := weirdQ[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, weirdQ[n - d[[m]], d, s - d[[m]], m - 1] && weirdQ[n, d, s - d[[m]], m - 1], s < n && m < Length[d] - 1]]];
aQ[n_] := ! SquareFreeQ[n] && Module[{d = Divisors[n]}, s = Total@d - n; m = Length[d] - 1; Total@Select[d, GCD[#, n/#] == 1 &] > 2 n && weirdQ[n, d, s, m]]; Select[Range[10^7], aQ]
(* after M. F. Hasler's pari code at A006037 *)

A339939 Coreful weird numbers: numbers k that are coreful abundant (A308053) but no subset of their aliquot coreful divisors sums to k.

Original entry on oeis.org

4900, 14700, 53900, 63700, 83300, 93100, 112700, 142100, 151900, 161700, 181300, 191100, 200900, 210700, 230300, 249900, 259700, 279300, 289100, 298900, 328300, 338100, 347900, 349448, 357700, 387100, 406700, 426300, 436100, 455700, 475300, 494900, 504700, 524300

Offset: 1

Amiram Eldar, Dec 23 2020

nonn

First differs from A321146 at n = 24.

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).

			4900 is a term since the sum of its aliquot coreful divisors, {70, 140, 350, 490, 700, 980, 2450}, is 5180 > 4900, and no subset of these divisors sums to 4900.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Subsequence of A308053.
Cf. A007947, A057723.
Similar sequences: A006037, A064114, A292986, A306984, A321146, A327948.

corDiv[n_] := Module[{rad = Times @@ FactorInteger [n][[;;,1]]}, rad * Divisors[n/rad]]; corWeirdQ[n_] := Module[{d = Most@corDiv[n], x}, Plus @@ d > n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 0]; Select[Range[10^5], corWeirdQ]

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.

Original entry on oeis.org

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

A342399 Unitary pseudoperfect numbers k such that no subset of the nontrivial unitary divisors {d|k : 1 < d < k, gcd(d, k/d) = 1} adds up to k.

Original entry on oeis.org

3510, 3770, 5670, 5810, 6790, 7630, 7910, 9590, 9730, 544310, 740870, 2070970, 4017310, 4095190, 5368510, 5569690, 5762330, 5838770, 5855290, 5856130, 5887630, 5902470, 5985770, 6006070, 6039530, 6075370, 6083630, 6181210, 6259610, 6471290, 7038710, 7065730, 7285390

Offset: 1

Amiram Eldar, Mar 10 2021

nonn

Numbers that are the sum of a proper subset of their aliquot unitary divisors but are not the sum of any subset of their nontrivial unitary divisors.
The unitary perfect numbers (A002827) which are a subset of the unitary pseudoperfect numbers (A293188) are excluded from this sequence since otherwise they would all be trivial terms: if k is a unitary perfect number then the sum of the divisors {d|k : 1 < d < k, gcd(d, k/d) = 1} is k-1, so any subset of them has a sum smaller than k.
The unitary pseudoperfect numbers are thus a disjoint union of the unitary perfect numbers, this sequence and A342398.
The unitary abundant numbers (A034683) are a disjoint union of the unitary weird numbers (A064114), this sequence and A342398.

			3510 is a term since it is a unitary pseudoperfect number, 3510 = 1 + 2 + 5 + 13 + 27 + 54 + 65 + 130 + 135 + 270 + 351 + 702 + 1755, and the set of nontrivial unitary divisors of 3510, {d|3510 : 1 < d < 3510, gcd(d, 3510/d) = 1} = {2, 5, 10, 13, 26, 27, 54, 65, 130, 135, 270, 351, 702, 1755}, has no subset that adds up to 3510.

Amiram Eldar, Table of n, a(n) for n = 1..206

The unitary version of A339343.
Subsequence of A034683 and A293188.
Cf. A002827, A064114, A342398.

q[n_] := Module[{d = Most @ Select[Divisors[n], CoprimeQ[#, n/#] &], x}, Plus @@ d > n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0 && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, 2, Length[d]}], {x, 0, n}], n] == 0]; Select[Range[10^4], q]

cp's OEIS Frontend

A348631 Nonexponential weird numbers: nonexponential abundant numbers (A348604) that are not equal to the sum of any subset of their nonexponential divisors.

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A328563 Nonsquarefree unitary weird numbers that are also weird numbers.

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A339939 Coreful weird numbers: numbers k that are coreful abundant (A308053) but no subset of their aliquot coreful divisors sums to k.

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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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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.

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PARI

A342399 Unitary pseudoperfect numbers k such that no subset of the nontrivial unitary divisors {d|k : 1 < d < k, gcd(d, k/d) = 1} adds up to k.

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