A299401 Number of primitive weird numbers (PWN) of the form 2^n*p*q*r, where p,q,r are odd primes.

2, 7, 12, 18, 41, 130

Offset

1,1

Comments

The analog of A258333 for three odd factors.

Note that this sequence counts PWN with nonsquarefree odd part, which are excluded from A258883, see also A273815.

Links

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

Example

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.

Prog

(PARI) 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(P<p=precprime(p-1), /*print1([p]); */is_A005835(m*p, d=divisors(m*p), s+(s-m)*p, #d-1)&&cnt++)); cnt}

Crossrefs

Cf. A258883, A002975.

Sequence in context: A131190 A099353 A297432 * A188039 A133459 A023669

Adjacent sequences: A299398 A299399 A299400 * A299402 A299403 A299404

Keyword

nonn,more,hard

Author

M. F. Hasler, Feb 18 2018

Status

approved