OFFSET
3,1
COMMENTS
A proper subsequence of A002975.
Conjecture: a(n) = the smallest primitive weird number of the form 2^(n-2)*p*q where p*q is minimal.
Is it known that a(n) always exists? - Charles R Greathouse IV, Jun 11 2015
No, it is not even unconditionally proved that there are infinitely many primitive weird numbers. In view of this, the above formula a(n) = 2^(n-2)*p*q and the asymptotic formula a(n) ~ 2^(3n-2) are only conjectures. - M. F. Hasler, Jul 08 2016
The conjectured a(n) ~ 2^(3n-2) follows from the conjecture that a(n) = 2^(n-2)*p*q (cf. A258882) where q is the least prime larger than 2M = 2^n-2 such that 2^(n-2)*q*precprime((Mq-1)/(q-M)) is weird. I also conjecture that for all n > 7, q = nextprime(2^n-2). - M. F. Hasler, Jul 13 2016
LINKS
M. F. Hasler, Table of n, a(n) for n = 3..30 (first 16 terms from Robert G. Wilson v)
Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.
FORMULA
It appears that a(n) ~ 2^(3n-2). [Corrected by M. F. Hasler, Jul 13 2016]
EXAMPLE
a(3) = 2^1 * 5 * 7 = 70
a(4) = 2^2 * 11 * 19 = 836
a(5) = 2^3 * 29 * 31 = 7192
a(6) = 2^4 * 43 * 107 = 73616
a(7) = 2^5 * 109 * 149 = 519712
a(8) = 2^6 * 241 * 257 = 3963968
a(9) = 2^7 * 499 * 521 = 33277312
a(10) = 2^8 * 997 * 1031 = 263144192
a(11) = 2^9 * 2011 * 2053 = 2113834496
a(12) = 2^10 * 4049 * 4099 = 16995175424
a(13) = 2^11 * 8101 * 8191 = 135895635968
a(14) = 2^12 * 16273 * 16411 = 1093862207488
a(15) = 2^13 * 32603 * 32771 = 8752602423296
a(16) = 2^14 * 65287 * 65537 = 70102452125696
a(17) = 2^15 * 130729 * 131071 = 561472495910912
a(18) = 2^16 * 261637 * 262147 = 4494940873621504
a(19) = 2^17 * 523571 * 524287 = 35979456528646144
a(20) = 2^18 * 1047559 * 1048583 = 287952747599495168
a(21) = 2^19 * 2095721 * 2097169 = 2304288287017664512
a(22) = 2^20 * 4192267 * 4194319 = 18437851191624859648
a(23) = 2^21 * 8385719 * 8388617 = 147523287039340445696
a(24) = 2^22 * 16773149 * 16777259 = 1180308456157336305664
a(25) = 2^23 * 33548689 * 33554467 = 9443126304886073851904
a(26) = 2^24 * 67100681 * 67108879 = 75548667373415913488384
a(27) = 2^25 * 134206169 * 134217757 = 604410983292363190829056
a(28) = 2^26 * 268419077 * 268435459 = 4835408274665227893604352
a(29) = 2^27 * 536847791 * 536870923 = 38683960976635781347016704
a(30) = 2^28 * 1073709061 * 1073741827 = 309475567394195954395512832
MATHEMATICA
(* copy the terms from A002975, assign them equal to 'lst' and then *) Table[ Min@ Select[ lst, PrimeOmega@# == n &], {n, 3, 12}]
PROG
(PARI) a(n)=for(k=1, #A=A002975, bigomega(A[k])==n&&return(A[k])) \\ This assumes A002975 is defined as a set or vector with enough terms. A002975 could be replaced by A258882 (for which much larger terms are known) if we assume that all terms are in that sequence. - M. F. Hasler, Jul 08 2016
(PARI) A258375(n)={ forprime(q=2^n-1, , my(p=precprime((2^(n-1)-1)*(q+1)\(q-2^(n-1)+1)), P); is_A006037(2^(n-2)*p*q) || next; while( is_A006037(2^(n-2)*q*P=precprime(p-1)), p=P); return(2^(n-2)*p*q))} \\ This assumes that all terms are of the form 2^k*p*q. It seems to give correct results at least up to n=30. - M. F. Hasler, Jul 13 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, May 28 2015
EXTENSIONS
a(17) - a(20) from Robert G. Wilson v, Jun 14 2015
a(17) and a(19) corrected, and new terms a(21) - a(30), from M. F. Hasler, Jul 13 2016
STATUS
approved