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 1-8 of 8 results.

A335936 Infinitary weird numbers (A306984) whose number of divisors is not a power of 2.

Original entry on oeis.org

5390, 7400, 11830, 17920, 20230, 25270, 37030, 43750, 58870, 67270, 95830, 117670, 129430, 154630, 168070, 196630, 243670, 260470, 314230, 352870, 373030, 436870, 459270, 482230, 554470, 658630, 714070, 742630, 801430, 831670, 893830, 1024870, 1129030, 1201270
Offset: 1

Views

Author

Amiram Eldar, Jun 30 2020

Keywords

Comments

Weird numbers (A006037) whose number of divisors is a power of 2 (A036537) are also infinitary weird numbers (A306983), since all of their divisors are infinitary.

Links

Crossrefs

Intersection of A162643 and A306984.
Cf. A006037, A036537, A077609, A335935.

Programs

  • Mathematica
    fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; infabQ[n_] := isigma[n] > 2*n; idivs[x_] := If[x == 1, 1, Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[x] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; infwQ[n_] := infabQ[n] && Module[{d = Most @ idivs[n]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 0]; pow2Q[n_] := n == 2^IntegerExponent[n, 2]; seq = {}; Do[If[!pow2Q[DivisorSigma[0, n]] && infwQ[n], AppendTo[sm n]], {n, 1, 10^5}]; s

Extensions

More terms from Amiram Eldar, Mar 25 2023

A306983 Infinitary pseudoperfect numbers: numbers n equal to the sum of a subset of their proper infinitary divisors.

Original entry on oeis.org

6, 24, 30, 40, 42, 54, 56, 60, 66, 72, 78, 88, 90, 96, 102, 104, 114, 120, 138, 150, 168, 174, 186, 210, 216, 222, 246, 258, 264, 270, 280, 282, 294, 312, 318, 330, 354, 360, 366, 378, 384, 390, 402, 408, 420, 426, 438, 440, 456, 462, 474, 480, 486, 498, 504
Offset: 1

Views

Author

Amiram Eldar, Mar 18 2019

Keywords

Comments

Subsequence of A005835.

Links

Crossrefs

Cf. A005835, A007357, A049417, A126168, A129656, A293188, A292985, A318100, A306984.

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]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c > 0, AppendTo[s, n]], {n, 2, 1000}]; s

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

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

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Oct 26 2021

Keywords

Examples

			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.

Links

Crossrefs

Cf. A160135, A348604.
Similar sequences: A006037, A064114, A292986, A306984, A321146, A327948, A348525.

Programs

  • Mathematica
    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]

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

Views

Author

Amiram Eldar, Dec 23 2020

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Amiram Eldar, Nov 13 2021

Keywords

Comments

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

Links

Crossrefs

Cf. A049603, A051378, A333928, A349284.
Similar sequences: A006037, A064114, A292986, A306984, A321146, A327948, A348525.

Programs

  • Mathematica
    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

Views

Author

Amiram Eldar, Feb 13 2025

Keywords

Comments

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

Links

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.

Programs

  • 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 *)
  • 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
Showing 1-8 of 8 results.

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

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