A320875 Least d > 0 such that both Q = M + 2d and R = M + (M^2-1)/(Q-M) are prime, where M = 2^n - 1 = A000225(n), or 0 if there is no such d.

0, 1, 2, 1, 3, 8, 2, 0, 6, 4, 66, 16, 20, 0, 6, 1, 2720, 0, 32, 0, 164, 8, 0, 524288, 153, 3573184, 2097152, 7354396, 19436, 4517888, 672, 0, 174080, 0, 262146, 1984, 48, 0, 4194296, 79, 30720, 128, 1825866, 4188889, 194396, 27227248, 0, 16384, 723, 0, 265227072, 22771712, 13982720, 134217728, 59885796, 587144, 19436, 0, 17179869152, 8388608

Offset: 1

Author

M. F. Hasler, Nov 11 2018

Keywords

• nonn

Comments

It is easy to see that R can't be an integer unless M < Q < M^2 + M.

Nonzero terms yield primitive weird numbers (PWN) 2^(n-1)*Q*R, cf. A258882.

This idea was used by S. Kravitz in 1976 and 35 years later by students of CWU to find the largest known PWN, cf. links and A242025, A242993, A242998, A242999, A243003. The 226 digits mentioned in the news article correspond not to a PWN but to the prime R for a(381) = 5456. The corresponding prime Q = M(381) + 2*5456 is the 54th prime after M(381), and only the third one for which R is an integer. The 127 digit PWN they found earlier corresponds to a non-minimal solution d = 34008 for n = 109. (It is a matter of seconds to find many much larger solutions, see examples.) This news led to renewed interest in this topic and a series of recent research papers, see references in A258882 and A002975.

Sequences A242025, A242993, A242998, A242999, A243003 consider PWN of the form 2^(k-1)*Q*R(k,Q) where the prime Q is fixed to be a Mersenne prime A000668, and k is varied to find a prime R.

Zero terms do not mean that there aren't PWN of the form 2^(n-1)*p*q with M+1 = 2^n < p < 2M < q < M(M+1). For example, a(8) = 0, but there are A258333(8) = 53 weird numbers with such (p,q). However, the two primes never satisfy the relation (p-M)(q-M) = M^2-1 which is considered here for (Q,R). - M. F. Hasler, Nov 20 2018

Examples

			a(109) = 8436 yields a 62-digit prime R and a 127 digit PWN 2^108*Q*R.
a(381) = 5456 yields a 226-digit prime R and a 455 digit PWN 2^380*Q*R. (This and the preceding one are mentioned in the News articles, cf LINKS.)
a(391) = 16386 leads to a 231-digit prime R and a 466-digit PWN 2^390*Q*R.
a(409) = 12360 leads to a 242-digit prime R and a 488-digit PWN 2^408*Q*R.
a(421) = 1661 leads to a 250-digit prime R and a 504-digit PWN 2^420*Q*R.
a(430) = 10304 leads to a 255-digit prime R and a 514-digit PWN 2^429*Q*R.
a(441) = 36080 leads to a 261-digit prime R and a 526-digit PWN 2^440*Q*R.
a(505) = 20726 leads to a 300-digit prime R and a 604-digit PWN 2^504*Q*R.
		

Links

• Daily Record, CWU students find longest 'weird' number, and also Yakima Herald, CWU math students calculate what no mathematician has before (backup on web.archive.org, page no longer available), both from Dec. 5, 2013
• S. Kravitz, A search for large weird numbers. J. Recreational Math. 9(1976), 82-85 (1977). Zbl 0365.10003

Crossrefs

Cf. A258882, subsequence of A002975.

Cf. A242025, A242993, A242998, A242999, A243003 (all related to the case Q = 2^p-1 in A000668, p in A000043).

Programs

PARI

a(n)={my(M=2^n-1,S=M^2-1); fordiv(S+!S,D, ispseudoprime(M+D)&& ispseudoprime(M+S/D)&& return(D/2))} \\ Much faster than the variant below, but requires increasingly more stack space (allocatemem()) for larger n.

PARI

A320875(n,L=0)={my(M=2^n-1,S=M^2-1); forprime(Q=M+1,if(L,L,M<