cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 16 results. Next

A328562 Unitary weird numbers (A064114) that are not weird numbers (A006037).

Original entry on oeis.org

5390, 11830, 17010, 20230, 25270, 37030, 51030, 58870, 67270, 93170, 95830, 117670, 129430, 153090, 153790, 154630, 196630, 243670, 260470, 314230, 343910, 352870, 373030, 436870, 459270, 480130, 482230, 554470, 658630, 714070, 742630, 801430, 831670, 851690, 893830
Offset: 1

Views

Author

Amiram Eldar, Oct 19 2019

Keywords

Comments

All the terms are nonsquarefree, since unitary weird numbers that are squarefree are necessarily also weird.
Nonsquarefree unitary weird numbers that are also weird numbers are listed in A328563.

Links

Crossrefs

Cf. A005117, A006037, A013929, A064114, A292705, A328563.

Programs

  • Mathematica
    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]]];
wQ[n_] := Module[{d = Divisors[n]}, s = Total@d - n; m = Length[d] - 1; weirdQ[n, d, s, m]];
uQ[n_] := Module[{d = Select[Divisors[n], GCD[#, n/#] == 1 &]}, s = Total@d - n; m = Length[d] - 1; weirdQ[n, d, s, m]];
aQ[n_] := uQ[n] && ! wQ[n]; Select[Range[10^6], aQ]
(* after M. F. Hasler's pari code at A006037 *)

A363295 Numbers k such that k and k+20 are consecutive unitary weird numbers (A064114).

Original entry on oeis.org

34121990, 34428290, 34766810, 34936070, 38014970, 38152010, 39506090, 39844610, 41400170, 42044990, 42552770, 42584990, 43769810, 46276490, 46308710, 47155010, 47324270, 47461310, 49016870, 49153910, 49323170, 49661690, 49863170, 50540210, 51015770, 51354290, 53079110
Offset: 1

Views

Author

Amiram Eldar, May 26 2023

Keywords

Comments

Conjecture: All the unitary weird numbers are of the form 20*k + 10, and thus 20 is the least gap between consecutive unitary weird numbers (verified for the 16326260 unitary weird numbers below 10^10).

Examples

			34121990 is a term since 34121990 = A064114(53235) and 34122010 = 34121990 + 20 = A064114(53236) are consecutive unitary weird numbers.

Links

Crossrefs

Cf. A064114, A306952, A363296.

A363296 Unitary weird numbers (A064114) with a record gap to the next unitary weird number.

Original entry on oeis.org

70, 5830, 2197790, 902388130, 2013240110
Offset: 1

Views

Author

Amiram Eldar, May 26 2023

Keywords

Comments

The record gap values are 3960, 4600, 5040, 5440, 7560, ... .
a(6) > 10^10, if it exists.

Examples

			70 is in the sequence since it is the first unitary weird number and the next unitary weird number after it is 4030 = 70 + 3960. The next gap which is larger than 3960 is 4600 and it occurs at 5830 which is followed by 10430 = 5830 + 4600.

Crossrefs

Cf. A064114, A306953, A363295.

A326808 Unitary weird numbers (A064114) that are divisible by 3.

Original entry on oeis.org

17010, 51030, 153090, 459270, 1377810, 4133430, 6004530, 6106590, 6140610, 6242670, 6344730, 6446790, 6514830, 6616890, 6752970, 6821010, 6957090, 7127190, 7161210, 7331310, 7365330, 7467390, 7535430, 7637490, 7773570, 7841610, 7875630, 7943670, 8147790, 8283870
Offset: 1

Views

Author

Amiram Eldar, Oct 19 2019

Keywords

Comments

While there are no known weird numbers (A006037) that are a multiple of 3, such unitary weird numbers exist, but they are relatively rare: out of the first 10^6 unitary weird numbers only 5834 are divisible by 3.

Links

Crossrefs

Cf. A006037, A064114.

A363297 Unitary weird numbers (A064114) with more unitary divisors than any smaller weird number.

Original entry on oeis.org

70, 4030, 4199030, 5702250610
Offset: 1

Views

Author

Amiram Eldar, May 26 2023

Keywords

Comments

The corresponding numbers of unitary divisors are 8, 16, 32, 64, ...
a(5) > 10^10, if it exists.

Examples

			The first 2 unitary weird numbers, 70 and 4030, have an increasing number of unitary divisors, 8 and 16. The least unitary weird number with more than 16 unitary divisors is the 6091st unitary weird number, 4199030, which has 32 unitary divisors.

Crossrefs

Cf. A034444, A064114, A335008.

A293188 Unitary pseudoperfect numbers: numbers that equal to the sum of a subset of their aliquot unitary divisors.

Original entry on oeis.org

6, 30, 42, 60, 66, 78, 90, 102, 114, 138, 150, 174, 186, 210, 222, 246, 258, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690, 714, 726, 750, 762, 770, 780, 786, 798, 822
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2017

Keywords

Comments

Supersequence of A002827.
The nonsquarefree terms are 60, 90, 150, 294, 420, 630, 660, 726, 750, 780, 840, ...

Examples

			150 is in the sequence since its unitary aliquot divisors are 1, 2, 3, 6, 25, 50, 75 and 150 = 25 + 50 + 75.

Links

Crossrefs

Cf. A002827, A005835, A034683, A064114.

Programs

  • Mathematica
    udiv[n_]:=Block[{d=Divisors[n]},Select[d,GCD[#,n/#]==1&]]; a={};n=0;While[Length[a]<100,n++;d=Most[udiv[n]];c = SeriesCoefficient[ Series[ Product[1+x^d[[i]],{i,Length[d]} ],{x,0,n}], n]; If[c>0,AppendTo[a,n]]];a (* after T. D. Noe at A005835 *)

A292986 Bi-unitary weird numbers: bi-unitary abundant numbers (A292982) that are not bi-unitary pseudoperfect (A292985).

Original entry on oeis.org

70, 4030, 5390, 5830, 7192, 7400, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 11830, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670, 19810
Offset: 1

Views

Author

Amiram Eldar, Sep 27 2017

Keywords

Comments

Analogous to weird numbers (A006037) with bi-unitary sigma (A188999) instead of sigma (A000203).

Links

Crossrefs

Cf. A006037, A064114, A188999, A292982, A292985.

Programs

  • Mathematica
    f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdiv[m_] := Select[Divisors[m], Last@Intersection[f@#, f[m/#]] == 1 &]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; bAbundantQ[n_] := bsigma[n] > 2 n; a = {}; n = 0; While[Length[a] < 5, n++; If[!bAbundantQ[n], Continue[]]; d = Most[bdiv[n]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c <= 0, AppendTo[a, n]]]; a (* after T. D. Noe at A005835 and Michael De Vlieger at A188999 *)

A306984 Infinitary weird numbers: infinitary abundant numbers (A129656) that are not infinitary pseudoperfect numbers (A306983).

Original entry on oeis.org

70, 4030, 5390, 5830, 7192, 7400, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 11830, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17920, 17990, 18410, 18830, 18970, 19390, 19670
Offset: 1

Views

Author

Amiram Eldar, Mar 18 2019

Keywords

Comments

Differs from bi-unitary weird numbers from n >= 32 (a(32) = 17920 is not bi-unitary weird).

Links

Crossrefs

Cf. A006037, A049417, A064114, A126168, A129656, A292986, A321146, A306983.

Programs

  • Mathematica
    idivs[x_] := If[x == 1, 1, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[ x ] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] ;s = {}; Do[d = Most[idivs[n]]; If[Total[d]

A327948 Nonunitary weird numbers: numbers that are nonunitary abundant but not nonunitary pseudoperfect.

Original entry on oeis.org

280, 3344, 16120, 23320, 28768, 31648, 37088, 41720, 42280, 43168, 43960, 45640, 46760, 48440, 50120, 50680, 53480, 54040, 55160, 55720, 59080, 62440, 63560, 64120, 65240, 66920, 67480, 69088, 70280, 71960, 73640, 75320, 75880, 77560, 78680, 79240, 82040
Offset: 1

Views

Author

Amiram Eldar, Sep 30 2019

Keywords

Links

Crossrefs

The nonunitary version of A006037.
Cf. A064597, A327945.
Cf. A064114, A292986, A306984, A321146.

Programs

  • Mathematica
    nudiv[n_] := Module[{d = Divisors[n]}, Select[d, GCD[#, n/#] > 1 &]]; s = {}; Do[d = nudiv[n]; If[Total[d] <= n, Continue[]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c == 0, AppendTo[s, n]], {n, 1, 10^5}]; s

A348525 Noninfinitary weird numbers: noninfinitary abundant numbers (A348274) that are not equal to the sum of any subset of their noninfinitary divisors.

Original entry on oeis.org

3344, 12636, 88900, 95900, 109900, 116900, 121100, 181424, 271472, 365552, 476272, 504016, 975568, 1016048, 1354288, 1375504, 1407824, 1552304, 1628528, 1641904, 1862608, 1882672, 1902736, 1909424, 1929488, 1962928, 1982992, 2003056, 2009744, 2029808, 2049872
Offset: 1

Views

Author

Amiram Eldar, Oct 21 2021

Keywords

Examples

			3344 is a term since the sum of its noninfinitary divisors, {2, 4, 8, 22, 38, 44, 76, 88, 152, 418, 836, 1672}, is 3360 > 3344, and no subset of these divisors sums to 3344.

Links

Crossrefs

Cf. A348271, A348274, A348341.
Similar sequences: A006037, A064114, A292986, A306984, A321146, A327948.

Programs

  • Mathematica
    q[n_] := !IntegerQ@ Log2@ DivisorSigma[0, n]; nidiv[1] = {}; nidiv[n_] := Complement[Divisors[n], Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; s = {}; Do[If[! q[n], Continue[]]; d = nidiv[n]; If[Total[d] <= n, Continue[]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c == 0, AppendTo[s, n]], {n, 1, 13000}]; s

Extensions

More terms from Amiram Eldar, Mar 25 2023
Showing 1-10 of 16 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.