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

70, 4030, 5830, 4199030, 1550860550, 66072609790

Offset

1,1

Comments

Primitive weird numbers divisible by 2 but not by 4.

10805836895078390 = 2 * 5 * 11 * 89 * 167 * 829 * 7972687 is a term.

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

Table of n, a(n) for n=1..6.

G. Amato, M. Hasler, G. Melfi and M. Parton, Primitive weird numbers having more than three distinct prime factors, Riv. Mat. Univ. Parma, Vol. 7, No. 1 (2016) 153-163.

Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, Maurizio Parton, Primitive abundant and weird numbers with many prime factors, arXiv:1802.07178 [math.NT], 2018.

Stan Benkoski, Problem E2308, Amer. Math. Monthly, 79 (1972) 774.

S. J. Benkoski and P. Erdos, On weird and pseudoperfect numbers, Math. Comp., 28 (1974), pp. 617-623. Alternate link; 1975 corrigendum.

R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991.

Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.

Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Volume 147, February 2015, Pages 508-514.

Eric Weisstein's World of Mathematics, Weird Number.

Wikipedia, Weird number

Example

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)

Mathematica

(* import the b-file in A002975 and assign it to lst *);
Select[lst, IntegerExponent[#, 2] == 1 &]

Crossrefs

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

Sequence in context: A354281 A348631 A354283 * A258374 A363297 A146349

Adjacent sequences: A319732 A319733 A319734 * A319736 A319737 A319738

Keyword

nonn,more

Author

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

Status

approved