A299401
Number of primitive weird numbers (PWN) of the form 2^n*p*q*r, where p,q,r are odd primes.
Original entry on oeis.org
2, 7, 12, 18, 41, 130
Offset: 1
In the sequel, p,q,r denote arbitrary odd primes.
The a(1) = 2 PWN of the form 2*p*q*r are A258883(1..2): 4030 = 2*5*13*31 and 5830 = 2*5*11*53.
The a(2) = 7 PWN of the form 2^2*p*q*r are 45356, 91388, 243892, 254012, 338572, 343876 and 388076, with (p,q,r) = (17, 23, 29), (11, 31, 67), (11, 23, 241), (11, 23, 251), (13, 17, 383), (13, 17, 389) and (13, 17, 439).
The a(3) = 12 PWN of the form 2^3*p*q*r range from 1713592 to 173482552.
The a(4) = 18 PWN of the form 2^4*p*q*r range from 15126992 to 6587973136.
The a(5) = 41 PWN of the form 2^5*p*q*r range from 569494624 to 297512429728.
-
A299401(n,k=3,m=2^n,P=3,cnt=0,s)={if(k>1,forprime(p=P,,(s=sigma(m*p,-1))<2||next;p>P&&s*(1+1/p)^(k-1)<2&&break;/*printf("%d",[k,p]);*/cnt+=A299401(n,k-1,m*p,p)),s=sigma(m);my(p=1\(2*m/s-1)+1,d);while(PA005835(m*p,d=divisors(m*p),s+(s-m)*p,#d-1)&&cnt++));cnt}
A319735
Primitive weird numbers (pwn; A002975) congruent to 2 mod 4.
Original entry on oeis.org
70, 4030, 5830, 4199030, 1550860550, 66072609790
Offset: 1
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)
- 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)
- 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
A322524
Primitive weird numbers (pwn; A002975) divisible by 4 but not 8.
Original entry on oeis.org
836, 45356, 91388, 243892, 254012, 338572, 343876, 388076, 29465852, 120888092, 259858324, 260378492, 410832532, 775397948, 785187524, 903217276, 989226964, 1609445332, 2358115084, 3254323124, 3381352084, 3381872252, 3781448788, 3782267372, 5056717796, 5065605532
Offset: 1
a(1) = 836 = 2^2 * 11 * 19;
a(2) = 45356 = 2^2 * 17 * 23 * 29;
a(3) = 91388 = 2^2 * 11 * 31 * 67; etc.
Comments