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 1-3 of 3 results.

A265726 Primitive weird numbers whose abundance is a record.

Original entry on oeis.org

70, 836, 7192, 9272, 73616, 243892, 338572, 1188256, 1901728, 3963968, 28279232, 36228736, 91322752, 141659096, 263144192, 351295232, 664373504, 2113834496, 5522263024, 6933503488, 19179527168, 22755515392, 31574500724, 98620009472, 135895635968
Offset: 1

Views

Author

Douglas E. Iannucci and Robert G. Wilson v, Dec 14 2015

Keywords

Comments

Although the abundance A(n) = sigma(n) - 2n is increasing, the (relative) abundancy sigma(n)/n is decreasing, except at indices {3, 6, 8, 15, 16, 19, 24 ...}. No term has larger abundancy than 2 + 2/35, that of a(1). - M. F. Hasler, Nov 14 2018

Examples

			a(1) = 70 since it is the first term in A002975; its abundance is 4.
a(2) is 836 since its abundance, 8, exceeds that of a(1); 4.
a(3) is 7192 = A002975(5) since its abundance, 16, exceeds that of a(2) and that of A002975(1..4).

Links

Crossrefs

Cf. A002975, A258250, A258333, A258374, A258375, A258401, A258882, A258883, A258884, A258885, A265727, A265728.

Programs

  • Mathematica
    (* copy the terms from A002975, assign them equal to 'lst' and then *) f[n_] := DivisorSigma[1, n] - 2n; k = 1; lsu = {}; mx = 0; While[k < 647, ds = f@ lst[[k]]; If[ds > mx, mx = ds; AppendTo[lsu, lst[[k]]]]; k++]; lsu

A265728 Least primitive weird number, pwn, (A002975) whose abundance is divisible by the n-th prime (A000040), or 0 if no such pwn exists.

Original entry on oeis.org

70, 232374697216, 73616, 9272, 243892, 343876, 4128448, 519712, 1901728, 338572, 5568448, 6621632, 272240768, 4960448, 7470272, 1673087984, 146279296, 5440192, 91322752, 8134208, 35442304, 286717696, 54962343424, 110232704, 6460864, 2812606976, 44473216, 141659096, 33736064, 58668928, 9537494528, 37499776, 292335872, 795730688, 530110208, 18657360896, 16995175424, 664373504, 266311424, 23049995264, 15152370176, 17124699136, 64015565312, 52059008
Offset: 1

Views

Author

Douglas E. Iannucci and Robert G. Wilson v, Dec 14 2015

Keywords

Comments

No odd weird number exists below 10^21. The search is done on the volunteer computing project yoyo@home. - Wenjie Fang, Feb 23 2014

Examples

			a(1) = 70 since it is the least pwn whose abundance, 4, is divisible by the first prime, 2.
a(2) = 0 since there is no known odd pwn and if there were, there is no reason why the abundance would be == 0 (mod 3).
a(3) = 73616 since it is the first pwn whose abundance, 80, is divisible by the third prime, 5.

Links

Crossrefs

Cf. A002975, A258250, A258333, A258374, A258375, A258401, A258882, A258883, A258884, A258885, A265726, A265727.

Programs

  • Mathematica
    (* copy the terms from A002975, assign them equal to 'lst' and then *) f[n_] := Select[lst, Mod[ DivisorSigma[1, #] - 2#, Prime@ n] == 0 &][[1]]; Array[f, 30]

A319735 Primitive weird numbers (pwn; A002975) congruent to 2 mod 4.

Original entry on oeis.org

70, 4030, 5830, 4199030, 1550860550, 66072609790
Offset: 1

Views

Author

M. F. Hasler and Robert G. Wilson v, Sep 26 2018

Keywords

Comments

Primitive weird numbers divisible by 2 but not by 4.
10805836895078390 = 2 * 5 * 11 * 89 * 167 * 829 * 7972687 is a term.

Examples

			a(1) is 70 = 2 * 5 * 7 with abundance of 4;
a(2) is 4030 = 2 * 5 * 13 * 31 with abundance of 4;
a(3) is 5830 = 2 * 5 * 11 * 53 with abundance of 4;
a(4) is 4199030 = 2 * 5 * 11 * 59 * 647 with abundance of 20;
a(5) is 1550860550 = 2 * 5^2 * 29 * 37 * 137 * 211 with abundance of 20;
a(6) is 66072609790 = 2 * 5 * 11 * 127^2 * 167 * 223 with abundance of 4; etc.
From _M. F. Hasler_, Nov 28 2018: (Start)
The larger terms are in other sequences related to PWN with many prime factors. We have the following relations:
   a(3) = 70 = A258882(1) = A258374(3) = A258250(1) = A002975(1).
   a(3) = 4030 = A258883(1) = A258374(4) = A258401(1) = A258250(3) = A002975(3).
   a(3) = 5830 = A258883(2) = A258401(2) = A258250(4) = A002975(4).
   a(4) = 4199030 = A258884(1) = A258374(5) = A258401(11) = A265727(15).
   a(5) = 1550860550 = A258885(1) = A273815(1) = A258374(6).
   a(6) = 66072609790 = A258885(3) = A273815(3). (End)

References

  • Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, Maurizio Parton. Primitive weird numbers having more than three distinct prime factors. Rivista di Matematica della Università degli studi di Parma, 2016, 7(1), pp. 153-163. (hal-01684543)

Links

Crossrefs

Cf. A002975, A258375, A258401, A258882, A258883, A258884, A258885.

Programs

  • Mathematica
    (* import the b-file in A002975 and assign it to lst *);
Select[lst, IntegerExponent[#, 2] == 1 &]
Showing 1-3 of 3 results.

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