A002975 -id:A002975 - cp's OEIS frontend

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

Offset: 3

Robert G. Wilson v, May 28 2015

nonn

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

			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

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.

Cf. A006037, A002975, A258374, A258882.

(* copy the terms from A002975, assign them equal to 'lst' and then *) Table[ Min@ Select[ lst, PrimeOmega@# == n &], {n, 3, 12}]

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

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

It appears that a(n) ~ 2^(3n-2). [Corrected by M. F. Hasler, Jul 13 2016]

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

A258374 Least primitive weird number with n prime divisors, not counting multiplicity.

Original entry on oeis.org

70, 4030, 4199030, 1550860550

Offset: 3

Robert G. Wilson v, May 28 2015

A proper subset of A002975.
So far all terms are == 2 (mod 4).
From M. F. Hasler, Jul 18 2016: (Start)
a(3) = 2 * 5 * 7 = A002975(1) = A258882(1),
a(4) = 2 * 5 * 13 * 31 = A258401(1) = A258883(1),
a(5) = 2 * 5 * 11 * 59 * 647 = A258884(1),
a(6) = 2 * 5^2 * 29 * 37 * 137 * 211 = A258885(1) = A273815(1). (End)
a(7) <= 4 * 13 * 17 * 449 * 24809 * 228259243 * 11449243661 ≈ 2.6e28, a(8) <= 4 * 13 * 17 * 449 * 24809 * 223842061 * 1123622795959 * 16039588627050434791 ≈ 3.97 e49. - M. F. Hasler, Aug 02 2016
a(7) <= 2 * 5 * 11 * 89 * 167 * 829 * 7972687 ≈ 1.1e16. - M. F. Hasler, Feb 18 2018

Cf. A002975, A258375.

(* copy the terms from A002975, assign them equal to 'lst' and then *) Table[ Min@ Select[ lst, PrimeNu@# == n &], {n, 3, 6}]

A242998 Number of integers k such that R = (2^k*Q - Q - 1)/(Q + 1 - 2^k) is a prime number, when Q = A000668(n) is the n-th Mersenne prime.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

M. F. Hasler, Aug 17 2014

Related to the search of large primitive weird numbers: Kravitz has shown that 2^(k-1)*Q*R is a primitive weird number (cf. A002975) when Q > 2^k and R = (2^k*Q - Q - 1)/(Q + 1 - 2^k) both are prime. Here we count such primes for the special case where Q = 2^p - 1 is a Mersenne prime, p=A000043(n). For such Q one has R = 2^k - 1 + (2^k - 2)/(2^(p-k) - 1).
See A242025 for the resulting primes R, which however are there not listed in order of the p's.
This sequence gives the row lengths for the table A243003 whose rows hold the k-values leading to prime R, for a given Mersenne prime.

			For given p=A000043(n), the following k's yield a prime R:
p : k's (and resulting primes R, Q=2^p-1 and/or weird W=2^(k-1)*Q*R)
2 : -
3 : 2 (R=5, Q=7, W=70)
5 : 4 (R=29, Q=31, W=7192)
7 : 4 (R=17, Q=127, W=17272), 5 (R=41, Q=127, W=83312)
13 : 11 (R=2729, Q=8191, W=22889716736)
17 : 13 (R=8737, Q=131071, W=4690605371392)
19 : 16 (R=74897, W=1286718208049152), 17 (R=174761, W=6004730783793152)
31 : 16 (R=65537, W=2^15*(2^31-1)*R), 29 (R=715827881, W=2^28*(2^31-1)*R)
61 : 57 (R=153722867280912929, W=2^56*(2^61-1)*R)
89 : 78 (R=302379100949042568368129, W=2^77*(2^89-1)*R)
107 through 86243 : none.
107 through 3021377: none. _Robert Price_, Sep 05 2019
The present sequence lists the number of k's in each line.

S. Kravitz, A search for large weird numbers. J. Recreational Math. 9(1976), 82-85 (1977). Zbl 0365.10003
E. Weisstein, Weird numbers, on MathWorld - a Wolfram web ressource.

Cf. A258882 (PWN of the form 2^k*p*q), A000043 (Mersenne prime exponents), A000668.
Cf. A242025 (the primes R).
Row lengths of A242999 (values of p) and A243003 (values of k), cf. A242993 for the smallest possible k.
See also A320875 for more general solutions to R = (MQ-1)/(Q-M) = prime.

A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,
   1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,
   21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433,
   1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,
   24036583, 25964951, 30402457, 32582657, 37156667, 42643801,
   43112609};
lst = {};
For[i = 1, i <= 28, i++,
  p = A000043[[i]];
  kc = 0;
  For[k = 1, k < p, k++,
   r = 2^k - 1 + (2^k - 2)/(2^(p - k) - 1);
   If[! IntegerQ[r], Continue[]];
   If[PrimeQ[r], kc++]];
  AppendTo[lst, kc]];
lst (* Robert Price, Sep 05 2019 *)

A242998(n,p=A000043[n])={sum(k=p\2+1, p-1, Mod(2, 2^(p-k)-1)^k==2 && ispseudoprime(2^k-1+(2^k-2)/(2^(p-k)-1)))}

Typo in definition corrected by Jens Kruse Andersen, Aug 27 2014

a(29)-a(37) from Robert Price, Sep 05 2019

A273815 Primitive weird numbers (pwn) with nonsquarefree odd part.

Original entry on oeis.org

1550860550, 2319548096, 66072609790, 114141404156, 232374697216

Offset: 1

M. F. Hasler, Jul 08 2016

Equivalently, primitive weird numbers (A002975) with at least one odd prime factor with multiplicity > 1. A subsequence of A258401.
Although some of them have only few prime divisors, these primitive weird numbers are not in the sequences A258882, A258883, A258884 defined to list "pwn of the form 2^k p*...*r with primes p<... squarefree). Sequence A258885 (pwn with 6 prime divisors) does not have this restriction. - M. F. Hasler, Jul 26 2016
a(6) <= 2^10*2081^2*129083 = 572417848896512, which is also in the sequence. - M. F. Hasler, Feb 15 2018

			a(1) = 1550860550 = 2 * 5^2 * 29 * 37 * 137 * 211 = A258885(1): the smallest pwn with 6 (distinct) prime divisors.
a(2) = 2319548096 = 2^6 * 137^2 * 1931 = A258401(45), but not in A258882 nor A258883, cf. comment.
a(3) = 66072609790 = 2 * 5 * 11 * 127^2 * 167 * 223 = A258885(3).
a(4) = 114141404156 = 2^2 * 13^2 * 19 * 383 * 23203 = A258401(123), but not in A258884, cf. comment.
a(5) = 232374697216 = 2^8 * 797^2 * 1429 = A258401(143), but not in A258882 nor A258883, cf. comment.

Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, Maurizio Parton, Primitive abundant and weird numbers with many prime factors, Journal of Number Theory vol. 201 (2019) pp. 436-459. DOI: 10.1016/j.jnt.2019.02.027. (Preprint: arXiv:1802.07178.)

Cf. A002975, A258401, A258882, A258883, A258884, A258885.

select(t->vecmax(factor(t)[,2][^1])>1, A002975) \\ Assuming that A002975 is defined as vector holding enough terms of that sequence

A242025 Primes of the form R = 2^k-1+(2^k-2)/(2^(p-k)-1), where p are Mersenne prime exponents listed in A000043.

Original entry on oeis.org

5, 17, 29, 41, 2729, 8737, 65537, 74897, 174761, 715827881, 153722867280912929, 302379100949042568368129

Offset: 1

M. F. Hasler, Aug 17 2014

Related to the search for large primitive weird numbers: Kravitz has shown that 2^(k-1)*Q*R is a primitive weird number when Q > 2^k and R = (2^k*Q-Q-1)/(Q+1-2^k) both are prime, cf. subset A258882 of A002975. Here we consider such primes for the special case of Mersenne primes Q = 2^p-1, p in A000043. For such Q one has R = 2^k-1+(2^k-2)/(2^(p-k)-1), which must be an integer and prime number.
See A242998 for the number of exponents k leading to primes R, for given Q = A000668(n) = 2^p-1, p = A000043(n). But there is no one-to-one correspondence since the primes R are here listed according to their size (cf. example). The pairs (k,p) are given in A242999 and A243003.
Kravitz used his formula in 1976 to find the 53-digit PWN corresponding to a(11), cf. examples. In 2013, students of CWU used the same idea to find the next term in the series, corresponding to a(12), see examples. They found still larger PWN of the same form with other primes Q, see A320875. This renewed the interest in weird numbers and motivated several recent papers, cf. A002975. - M. F. Hasler, Nov 10 2018

			For given p = A000043(n), the following k yield a prime R and an associated (primitive) weird number W = 2^(k-1)*(2^p-1)*R in A258882 c A002975 c A006037:
For p = 2, no k yields a prime R = 2^k-1+(2^k-2)/(2^(p-k)-1).
For p = 3, k = 2 yields R = 5 and the (smallest) weird number W = 70 = A006037(1).
For p = 5, k = 4 yields R = 29 = a(3) and W = 7192 = A258882(3).
For p = 7, k = 4 yields R = 17 = a(2) and W = 17272 = A258882(7),
       and k = 5 yields R = 41 = a(4) and W = 83312 = A258882(9).
For p = 13, k = 11 yields R = 2729 = a(5) and W = 22889716736 = A258882(288)
For p = 17, k = 13 yields R = 8737 = a(6) and W = 4690605371392 = A258882(1203).
For p = 19, k = 16 yields R = 74897 = a(8), W = 1286718208049152 = A258882(7154),
        and k = 17 yields R = 174761 = a(9), W = 6004730783793152 = A258882(11466).
For p = 31, k = 16 yields R = 65537 = a(7) (smaller than both R's for p = 19),
        and k = 29 yields R = 715827881 = a(10).
For p = 61, only k = 57 yields a prime R = 153722867280912929 = a(11).
For p = 89, only k = 78 yields a prime R = 302379100949042568368129 = a(12).
For p = 107 through p = 86243, no k yields a prime R.
For p = 107 through p = 3021377, no k yields a prime R. - _Robert Price_, Sep 04 2019

M. F. Hasler, Table of n, a(n) for n = 1..12
CWU press release, CWU Math Students Break World Record for Largest Weird Number, Dec. 4, 2013.
S. Kravitz, A search for large weird numbers. J. Recreational Math. 9(1976), 82-85 (1977). Zbl 0365.10003

Cf. A258882 (PWN of the form 2^k*p*q).
Cf. A242993, A242998, A242999 and A243003: related to PWN with a prime factor R as defined here, with Q = A000668(n) = 2^A000043(n)-1.
Cf. A320875 for a more general pattern leading to more and larger PWN.

A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,
   1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,
   21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433,
   1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,
   24036583, 25964951, 30402457, 32582657, 37156667, 42643801,
   43112609};
lst = {};
For[i = 1, i <= Length[A000043], i++,
  p = A000043[[i]];
  For[k = 1, k < p, k++,
   r = 2^k - 1 + (2^k - 2)/(2^(p - k) - 1);
   If[! IntegerQ[r], Continue[]];
   If[PrimeQ[r], AppendTo[lst, r]]]];
Union[lst] (* Robert Price, Sep 04 2019 *)

A242999 Mersenne prime exponents p in A000043 such that R=2^k-1+(2^k-2)/(2^(p-k)-1) is prime for some k < p, listed with multiplicity (number of k's), see A243003 for the k-values.

Original entry on oeis.org

3, 5, 7, 7, 13, 17, 19, 19, 31, 31, 61, 89

Offset: 1

M. F. Hasler, Aug 17 2014

Related to the search for large primitive weird numbers: Kravitz has shown that 2^(k-1)*Q*R is a primitive weird number (cf A002975) when Q > 2^k and R = (2^k*Q-Q-1)/(Q+1-2^k) both are prime. In the special case of Mersenne primes Q = 2^p-1, p = A000043(n), considered here, one has R = 2^k-1+(2^k-2)/(2^(p-k)-1).
This sequence lists the p-values. See sequence A243003 for the k-values and A242998(n) for the number of possible k-values for a given p = A000043(n), i.e., the number of times this p appears here.
The next term, a(13), is larger than 80000 (if it exists).

			For given p = A000043(n), the following k's yield a prime R:
p : k's
2 : -
3 : 2
5 : 4
7 : 4, 5
13 : 11
17 : 13
19 : 16, 17
31 : 16, 29
61 : 57
89 : 78
107 through 86243 : none.
107 through 3021377: none. - _Robert Price_, Sep 04 2019
Thus the pairs (p,k) are (3,2), (5,4), (7,4), (7,5), (13,11), ... and the present sequence lists the first component of these pairs, sequence A243003 lists the second component.

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

A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,
   1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,
   21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433,
   1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,
   24036583, 25964951, 30402457, 32582657, 37156667, 42643801,
   43112609};
lst = {};
For[i = 1, i <= 10, i++,
  p = A000043[[i]];
  For[k = 1, k < p, k++,
   r = 2^k - 1 + (2^k - 2)/(2^(p - k) - 1);
   If[! IntegerQ[r], Continue[]];
   If[PrimeQ[r], AppendTo[lst, p]]]];
lst (* Robert Price, Sep 04 2019 *)

forprime(p=1,,ispseudoprime(2^p-1)||next;for(k=p\2+1,p-1,(k-1)%(p-k)==0 && isprime(2^k-1+(2^k-2)/(2^(p-k)-1))&&print1(p","))) \\ M. F. Hasler, Jul 19 2016

A243003 Pairs (p,k) such that p is in A000043 and R=2^k-1+(2^k-2)/(2^(p-k)-1) is prime: this sequence lists the k-values, see A242999 for the p-values. (Ordered by p, then k.)

Original entry on oeis.org

2, 4, 4, 5, 11, 13, 16, 17, 16, 29, 57, 78

Offset: 1

M. F. Hasler, Aug 17 2014

Related to the search for large primitive weird numbers: Kravitz has shown that 2^(k-1)*Q*R is a primitive weird number (cf A002975) when Q > 2^k and R = (2^k*Q-Q-1)/(Q+1-2^k) both are prime. Here we consider the special case where Q = 2^p-1 is a Mersenne prime, p = A000043(n). For such Q one has R = 2^k-1+(2^k-2)/(2^(p-k)-1). [First R formula corrected by Jens Kruse Andersen, Aug 18 2014]
This sequence lists the k-values, see sequence A242999 for the p-values and A242998(n) for the number of possible k-values for given p = A000043(n) resp. Q = A000668(n).
This sequence can also be considered as a table whose n-th row holds the possible k-values for the n-th Mersenne prime Q = A000668(n); sequence A242998 gives the row lengths of the table (which are mostly 0).

			For given p=A000043(n), the following k's yield a prime R:
p : k's
2 : -
3 : 2
5 : 4
7 : 4, 5
13 : 11
17 : 13
19 : 16, 17
31 : 16, 29
61 : 57
89 : 78
107 through 86243 : none.
107 through 3021377: none. - _Robert Price_, Sep 04 2019
Thus the pairs (p,k) are (3,2), (5,4), (7,4), (7,5), (13,11), ... and the present sequence lists the second component of these pairs, the first components are listed in A242999.

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

Cf. A258882 (PWN of the form 2^k*p*q).

Cf. A242993 (least k), A242998 (number of solution for given p in A000043), A242999 (values of p), A242025 list of all primes R.

A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,
   1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,
   21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433,
   1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,
   24036583, 25964951, 30402457, 32582657, 37156667, 42643801,
   43112609};
lst = {};
For[i = 1, i <= 10, i++,
  p = A000043[[i]];
  For[k = 1, k < p, k++,
   r = 2^k - 1 + (2^k - 2)/(2^(p - k) - 1);
   If[! IntegerQ[r], Continue[]];
   If[PrimeQ[r], AppendTo[lst, k]]]];
lst (* Robert Price, Sep 04 2019 *)

forprime(p=1,, ispseudoprime(2^p-1)||next; for(k=p\2+1, p-1, (k-1)%(p-k)==0 && isprime(2^k-1+(2^k-2)/(2^(p-k)-1))&&print1(k", "))) \\ M. F. Hasler, Jul 19 2016

One must have p/2 < k < p and (p-k) | (k-1).

A262228 Deficiency sequence: a(0) = 1, a(n) is the smallest prime p > a(n-1) such that the product of a(i), 1 <= i < n, is deficient (belongs to A005100).

Original entry on oeis.org

1, 2, 5, 11, 59, 653, 84761, 2763189059, 377406001499268899, 2638619515495963542360422694651593, 135435890329895562961039215198033899386421965445591860752412324961

Offset: 0

Chayim Lowen, Sep 15 2015

nonn

The product of the first n+1 terms is the smallest deficient multiple of the product of the first n terms.
The product of any finite number of distinct terms of this sequence is deficient.
a(n) for n > 0 is the lexicographically earliest sequence of primes P, such that the asymptotic density of the squarefree numbers (A005117) which are not divisible by any prime in P is 3/Pi^2 (A104141), i.e., half the asymptotic density of all the squarefree numbers. - Amiram Eldar, Nov 30 2020

			a(3) = 11 because A001065(2*5*7) = A001065(70) = 74 > 70, and A001065(2*5*11) = A001065(110) = 106 < 110.
From _M. F. Hasler_, Dec 14 2017: (Start)
Let Q(x) = 1/(2x/sigma(x) - 1), P(n) = Product( a(k), k<n): P(0) = 1 (empty product). Then:
Q(P(0)) = 1, a(0) = nextprime(1) = 2 = P(1).
Q(P(1)) = 3, a(1) = 5. (2*3 is perfect, P(2) = 2*5 is deficient.)
Q(P(2)) = 9, a(2) = 11. (2*5*7 is weird, P(3) = 2*5*11 is deficient.)
Q(P(3)) = 54, a(3) = 59. (P(3)*53 is weird, P(4) = 2*5*11*59 is deficient.)
Q(P(4)) = 648, a(4) = 653. (P(4)*647 is weird, P(5) = 2*5*11*59*653 is deficient.)
Q(P(5)) = 84758.4, a(5) = 84761. (P(5)*84751 is abundant and semiperfect: sum of all proper divisors except {1, 2, 11, 22, 55, 59, 590}; P(6) = 2*5*11*59*653*84761 is deficient.) (End)

Amiram Eldar, Table of n, a(n) for n = 0..14

Cf. A001065, A005100, A005117, A104141, A151800 (nextprime).

Cf. A002975 (primitive weird numbers), A000203 (sigma), A295001 (same definition but a(0) = 4).

a[0]=1; a[n_] := a[n] = NextPrime[1/(2*Product[a[i],{i,1,n-1}]/Product[a[i]+1,{i,1,n-1}]-1)]; Array[a, 11, 0] (* Amiram Eldar, Jun 10 2019 *)

lista(nn) = {print1(p=1, ", "); vp = [p]; for (n=2, nn, np = nextprime(1+floor(1/(2*prod(i=2, n-1, vp[i]/(vp[i]+1))-1))); vp = concat(vp, np); print1(np, ", "););} \\ Michel Marcus, Oct 16 2015

a=List(); m=1; for(n=0, 13, listput(a, p=nextprime(1\(2/sigma(m,-1)-1)+1)); p>default(primelimit)&&addprimes(p); m*=p); a \\ M. F. Hasler, Dec 14 2017

a(n) = A151800(floor(1/(2*(Product_{i=2..n-1} a(i)/(a(i)+1))-1))), where A151800 is the "next larger prime" function.
Lim_{n->infinity} A001065(Product_{i=0..n} a(i))/(Product_{i=0..n} a(i)) = 1. [Corrected by M. F. Hasler, Dec 04 2017]
Conjecture: log(a(n)) ~ e^(an+b) where a and b are approximately 0.6 and -1.6 respectively.

A242993 Least k such that R = (2^k*Q-Q-1)/(Q+1-2^k) is prime, where Q = A000668(n) is the n-th Mersenne prime, or 0 if no such k exists.

Original entry on oeis.org

0, 2, 4, 4, 11, 13, 16, 16, 57, 78, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

M. F. Hasler, Aug 17 2014

Kravitz has shown that 2^(k-1)*Q*R is a primitive weird number (cf. A002975) when Q > 2^k =: M+1 and R = (M*Q-1)/(Q-M) = M + (M^2-1)/(Q-M) both are prime. R cannot be an integer unless Q < M(M+1) which yields k > p/2 for Mersenne primes Q = 2^p-1. [Edited by M. F. Hasler, Nov 11 2018]

Sequence A242025 lists all primes R obtained in that way. Sequence A242998 gives the number of (k,R) for each Q in A000668. Sequence A242998 lists the primes p which give rise to a solution, with multiplicity, and A243003 lists the corresponding values of k. See the "main entry" A242025 for more information. - M. F. Hasler, Nov 11 2018

			For n = 2, Q = A000668(2) = 7, k = 2 yields the prime R = (2^k*Q-Q-1)/(Q+1-2^k) = 20/4 = 5 and the (smallest possible) weird number 2^(k-1)*Q*R = 2*7*5 = 70.
For n = 9, Q = A000668(9) = 2^61-1, k = 57 yields the prime R = 2^57-1 + (2^57-2)/(2^4-1) and the 53-digit primitive weird number 2^56*Q*R = 25541592347764814106588251084767772206406532903993344.
For n = 10, Q = A000668(10) = 2^89-1, k = 78 yields the prime R = 2^78-1 + (2^78-2)/(2^11-1) and the 74-digit primitive weird number 2^77*Q*R = 28283363272427014026275183563912621451964887156507346985599492888375328768.

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

Cf. A242025 (primes R obtained in that way), A242998 (number of such k values for given n), A242999 (p=A000043(n) listed A242998(n) times), A243003 (all values of k).

Cf. A258882 (weird numbers of the form 2^k*p*q).

A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,
   1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,
   21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433,
   1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,
   24036583, 25964951, 30402457, 32582657, 37156667, 42643801,
   43112609};
lst = {};
For[i = 1, i <= 25, i++,
  p = A000043[[i]];
  kc = 0;
  For[k = 1, k < p, k++,
   r = 2^k - 1 + (2^k - 2)/(2^(p - k) - 1);
   If[! IntegerQ[r], Continue[]];
   If[PrimeQ[r], kc = k; Break[]]];
  AppendTo[lst, kc]];
lst (* Robert Price, Sep 05 2019 *)

a(n)={p=A000043[n]; for(k=p\2+1,p-1, Mod(2,2^(p-k)-1)^k==2 && ispseudoprime(2^k-1+(2^k-2)/(2^(p-k)-1)) && return(k))}

Definition corrected by Jens Kruse Andersen, Aug 18 2014

a(28)-a(37) from Robert Price, Sep 05 2019

A295369 Number of squarefree primitive abundant numbers (A071395) with n prime factors.

Original entry on oeis.org

0, 0, 1, 18, 610, 216054, 12566567699

Offset: 1

Gianluca Amato, Feb 12 2018

Here primitive abundant number means an abundant number all of whose proper divisors are deficient numbers (A071395). The alternative definition (an abundant number having no abundant proper divisor, see A091191) would yield an infinite count for a(3): since 2*3 = 6 is perfect, all numbers of the kind 2*3*p with p > 3 would be primitive abundant.
See A287590 for the number of squarefree ODD primitive abundant numbers with n prime factors.
The actual numbers are listed in A298973. - M. F. Hasler, Feb 16 2018

			For n=3, the only squarefree primitive abundant number (SFPAN) is 2*5*7 = 70, which is also a primitive weird number, see A002975.
For n=4, the 18 SFPAN range from 2*5*11*13 = 1430 to 2*5*11*53 = 5830.
For n=5, the 610 SFPAN range from 3*5*7*11*13 = 15015 to 2*5*11*59*647 = 4199030.

Gianluca Amato, Primitive Weirds and Abundant Numbers, GitHub.
Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, and Maurizio Parton, Primitive abundant and weird numbers with many prime factors, arXiv:1802.07178 [math.NT], 2018.

Cf. A071395 (primitive abundant numbers), A287590 (counts of odd SFPAN), A298973, A249242 (using A091191).

A295369(n, p=1, m=1, sigmam=1) = {
  my(centerm = sigmam/(2*m-sigmam), s=0);
  if (n==1,
    if (centerm > p, primepi(ceil(centerm)-1) - primepi(p), 0),
    p = max(floor(centerm),p); while (0A295369(n-1, p=nextprime(p+1), m*p, sigmam*(p+1)), s+=c); s
  )
}

def A295369(n, p=1, m=1, sigmam=1):
  centerm = sigmam/(2*m-sigmam)
  if n==1:
    return prime_pi(ceil(centerm)-1) - prime_pi(p) if centerm > p else 0
  else:
    p = max(floor(centerm), p)
    s = 0
    while True:
       p = next_prime(p)
       c = A295369(n-1, p, m*p, sigmam*(p+1))
       if c <= 0: return s
       s+=c

cp's OEIS Frontend

A258375 Least primitive weird number with n prime divisors, counting multiplicity.

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A258374 Least primitive weird number with n prime divisors, not counting multiplicity.

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A242998 Number of integers k such that R = (2^k*Q - Q - 1)/(Q + 1 - 2^k) is a prime number, when Q = A000668(n) is the n-th Mersenne prime.

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A273815 Primitive weird numbers (pwn) with nonsquarefree odd part.

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A242025 Primes of the form R = 2^k-1+(2^k-2)/(2^(p-k)-1), where p are Mersenne prime exponents listed in A000043.

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A242999 Mersenne prime exponents p in A000043 such that R=2^k-1+(2^k-2)/(2^(p-k)-1) is prime for some k < p, listed with multiplicity (number of k's), see A243003 for the k-values.

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A243003 Pairs (p,k) such that p is in A000043 and R=2^k-1+(2^k-2)/(2^(p-k)-1) is prime: this sequence lists the k-values, see A242999 for the p-values. (Ordered by p, then k.)

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A262228 Deficiency sequence: a(0) = 1, a(n) is the smallest prime p > a(n-1) such that the product of a(i), 1 <= i < n, is deficient (belongs to A005100).

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