A258882 Primitive weird numbers of the form 2^k*p*q with k > 0 and where p < q are odd primes.

70, 836, 7192, 7912, 9272, 10792, 17272, 73616, 83312, 113072, 519712, 539744, 555616, 682592, 786208, 1188256, 1229152, 1901728, 2081824, 2189024, 3963968, 4128448, 4145216, 4486208, 4559552, 4632896, 4960448, 5440192, 5568448, 6460864, 6621632, 7354304, 7470272, 8000704, 8134208

Offset: 1

Author

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

Keywords

• nonn

Comments

The number of terms < 10^n: 0, 1, 2, 5, 9, 15, 35, 61, 114, 204, 380, 696, 1703, 3548, 6726, 13137, ....

If 2^k*p*q is a weird number, it is necessarily primitive, and 2^(k+1) < p < 2^(k+2)-2 < q < 2^(2k+1).

No odd weird numbers are known and any even weird number must have at least 3 distinct prime factors, since all numbers of the form 2^k*p^m are deficient or pseudoperfect or perfect (iff m = 1 and p = 2^(k+1)-1 is a Mersenne prime). Sequence A258333 lists the number of terms in this sequence for given k. - M. F. Hasler, Jul 11 2016

Kravitz has shown that 2^k*p*q is a primitive weird number when the primes p and q satisfy p = (2^(k+1)*q-q-1)/(q+1-2^(k+1)). Many terms in this sequence are of this form, e.g., a(n) with n = 1, 2, 3, 4, 6, 7, 9, 10, 15, 23, 26, 38, 45, 75, 94, 144, 157, 187, 287, 327, 368, 370, 459, 607, 657, 658, .... Sequences A242025, A242998, ... are related to the special case where q is a Mersenne prime (A000668). - M. F. Hasler, Jul 12 2016

Weird numbers of the form 2^k*p*q are always primitive, so this condition could be omitted in the definition of this sequence. - M. F. Hasler, Jul 13 2016

About 35 years after Kravitz's work, the topic of weird numbers has regained interest after a CWU press release about students who used Kravitz's formula to find a large PWN of this form. See A242025 and A320875. - M. F. Hasler, Nov 20 2018

Examples

			a(1) = A002975(1) = 70 = 2*5*7.
a(2) = A002975(2) = 836 = 2^2*11*19.
A002975(3) = 4030 = 2*5*13*31 is not in this sequence since it is not of the required form.
The same is true for A002975(4) = 5830.
a(3) = A002975(5) = 7192 = 2^3*29*31, etc.
A002975(179) = 2319548096 = 2^6 * 137^2 * 1931 is the first term of A002975 with only two odd prime divisors, but not of the required form. - _M. F. Hasler_, Nov 20 2018
		

References

• S. Kravitz, A search for large weird numbers. J. Recreational Math. 9 (1976), 82-85 (1977). Zbl 0365.10003

Links

• Douglas E. Iannucci and Robert G. Wilson v, Table of n, a(n) for n = 1..15384, updated Dec 06 2015; corrected by _M. F. Hasler_, Jul 16 2016
• R. Bagula et al., A very big weird number, Number Theory group on LinkedIn (web.archive.org snapshot; page no longer available). Dec. 2013
• Central Washington University, CWU Math Students Break World Record for Largest Weird Number [alternate article]
• 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, Vol. 147, Feb 2015, pp 508-514.
• Eric Weisstein's World of Mathematics, Weird Number.
• Wikipedia, Weird number

Crossrefs

Cf. A002975, A258401 (PWN not of this form), A258374, A258375, A258883, A258884, A258885.

Cf. A242025, A242993, A242998, A242999, A243003 (related to the subsequence with q = (2^k*p-p-1)/(p+1-2^k) and p a Mersenne prime in A000668).

Cf. A320875 (more general case of Karavitz' formula).

Programs

Mathematica

(* copy the terms from A002975, assign them equal to 'lst' and then *) fQ[n_] := Block[{m = n}, While[ Mod[m, 2] == 0, m /= 2]; PrimeOmega@ m == 2]; Select[lst, fQ]

PARI

select(t->factor(t)[, 2][^1]=[1, 1]~, A002975) \\ Assuming that A002975 is defined as set or vector. - M. F. Hasler, Jul 11 2016

Formula

A258882 union A258401 is A002975.

Extensions

Edited by M. F. Hasler, Jul 11 2016, Nov 20 2018