A258375 - OEIS

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A258375

Least primitive weird number with n prime divisors, counting multiplicity.

70, 836, 7192, 73616, 519712, 3963968, 33277312, 263144192, 2113834496, 16995175424, 135895635968, 1093862207488, 8752602423296, 70102452125696, 561472495910912, 4494940873621504, 35979456528646144, 287952747599495168 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`3,1`
	COMMENTS	`A proper subsequence of A002975.` `Conjecture: a(n) = the smallest primitive weird number of the form 2^(n-2)pq where pq 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)pq 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)pq (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^kpq, 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)pq) \|\| next; while( is_A006037(2^(n-2)qP=precprime(p-1)), p=P); return(2^(n-2)pq))} \\ This assumes that all terms are of the form 2^kp*q. It seems to give correct results at least up to n=30. - M. F. Hasler, Jul 13 2016`
	CROSSREFS	`Cf. A006037, A002975, A258374, A258882.` `Sequence in context: A335008 A258882 A265726 * A306953 A302573 A251933` `Adjacent sequences: A258372 A258373 A258374 * A258376 A258377 A258378`
	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`

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Last modified April 15 20:40 EDT 2021. Contains 342977 sequences. (Running on oeis4.)