Jeppe Stig Nielsen - cp's OEIS frontend

A385564 Prime powers k such that lcm(1, 2, 3, ..., k)-1 is prime.

3, 4, 5, 7, 8, 19, 23, 29, 32, 47, 61, 97, 181, 233, 307, 401, 887, 1021, 1087, 1361, 1481, 2053, 2293, 5407, 5857, 11059, 14281, 27277, 27803, 36497, 44987, 53017

Offset: 1

Jeppe Stig Nielsen, Jul 03 2025

nonn

The prime associated with each a(n) is A057824(n).

a(33) > 10^5. Up to 10^5, contains 4, 8, 32 not in subsequence A154524. - Michael S. Branicky, Jul 04 2025

			k=32 is a prime power, so point at which A003418 attains a new value, namely lcm(1, 2, 3, ..., 32) = 144403552893600, and by subtracting one we get 144403552893599 which is a prime number, so 32 is a member of the sequence.

Intersection of A057825 and A000961.
Supersequence of A154524.
Cf. A057824, A051453, A003418.

Select[Range[6000],PrimePowerQ[#]&&PrimeQ[Fold[LCM,Range[#]]-1]&] (* James C. McMahon, Jul 09 2025 *)

L=1;for(k=2,6000,!isprimepower(k,&p)&&next();L*=p;ispseudoprime(L-1)&&print1(k,", "))

a(31)-a(32) from Michael S. Branicky, Jul 03 2025

A377838 Let p = prime(n), then a(n) is the least prime q < p such that p * q# + 1 is prime.

Original entry on oeis.org

2, 2, 3, 2, 3, 3, 5, 2, 2, 13, 3, 2, 5, 3, 2, 7, 3, 5, 5, 3, 5, 2, 2, 11, 3, 3, 3, 5, 2, 13, 2, 3, 7, 13, 3, 7, 7, 5, 2, 2, 3, 2, 5, 13, 11, 29, 5, 19, 5, 2, 2, 3, 2, 3, 3, 13, 3, 3, 2, 3, 2, 13, 3, 3, 5, 3, 5, 3, 7, 7, 2, 3, 3, 11, 5, 67, 3, 7, 17, 2, 7, 2, 7

Offset: 2

Jeppe Stig Nielsen, Nov 09 2024

nonn

The notation q# means A034386(q).

Buhler, Crandall and Penk conjecture a(n) exists for all n > 1.

			For a(122), consider 673, the 122nd prime. Search for primes of form 673*2*3*5*7*...*q + 1. The first such prime appears at q=509 (and 509 is less than 673). Therefore a(122) = 509.

Jeppe Stig Nielsen, Table of n, a(n) for n = 2..10001
J. P. Buhler, R. E. Crandall and M. A. Penk, Primes of the form n! ± 1 and 2 · 3 · 5 ··· p ± 1, Math. Comp. 38 (1982), 639-643.

Cf. A034386.

a(n)=p=prime(n);m=p;forprime(q=2,p-1,m*=q;ispseudoprime(m+1)&&return(q));error("none")

A370305 Numbers k such that the distance from exp(k) to the closest average of two consecutive primes is less than 1.

Original entry on oeis.org

1, 3, 16, 61, 74, 91, 113, 1441, 1566, 2170, 2499

Offset: 1

Jeppe Stig Nielsen, Feb 14 2024

Explicitly, abs( e^k - (prevprime(e^k)+nextprime(e^k))/2 ) < 1.
For k>1, the formula (prevprime(e^k)+nextprime(e^k))/2 either gives floor(e^k), for k = 61, 74, 2170, ..., or gives ceiling(e^k), for k = 3, 16, 91, 113, 1441, 1566, 2499, ... This partitions {a(n)}\{1} into two subsequences each of which can be conjectured to have relative density 1/2.
In cases k = 16, 61, 113, 2499, ... the distance is actually less than 0.5. Then the formula (prevprime(e^k)+nextprime(e^k))/2 yields round(e^k), the nearest integer to e^k.

			For k=16, e^16 is about 8886110.52. The next prime is 8886113, and the previous prime is 8886109, and their average 8886111 is at a distance of about 0.48 away from e^16.

Cf. A037028, A040016, A050808, A059303, A074496.

default(realprecision,2000);for(k=1,+oo,r=exp(k);abs(r-(precprime(r)+nextprime(r))/2)<1&&print1(k,", "))

A364076 Numbers k such that (12^k - 1)^2 - 2 is prime.

Original entry on oeis.org

3, 29, 51, 7824, 15456, 22614, 28312, 47014, 68835

Offset: 1

Jeppe Stig Nielsen, Jul 03 2023

Such primes are sometimes called Carol primes of base 12.

Prime-Wiki, Carol-Kynea prime.
Eric Weisstein's World of Mathematics, Near-Square Prime.

Cf. A091515 (b=2), A100901 (b=6), A100903 (b=10), A100905 (b=14), A364078 (b=18), A364080 (b=20), A100907 (b=22).

Cf. A364077

Select[Range[1500],PrimeQ[(12^#-1)^2-2]&] (* James C. McMahon, Jan 04 2024 *)

for(k=1,1200,ispseudoprime((12^k-1)^2-2)&&print1(k,", "))

A364077 Numbers k such that (12^k + 1)^2 - 2 is prime.

Original entry on oeis.org

1, 2, 8, 60, 513, 1047, 7021, 7506, 78858

Offset: 1

Jeppe Stig Nielsen, Jul 03 2023

Such primes are sometimes called Kynea primes of base 12.

Prime-Wiki, Carol-Kynea prime.
Eric Weisstein's World of Mathematics, Near-Square Prime.

Cf. A091513 (b=2), A100902 (b=6), A100904 (b=10), A100906 (b=14), A364079 (b=18), A364081 (b=20), A100908 (b=22).

Cf. A364076.

for(k=1,1200,ispseudoprime((12^k+1)^2-2)&&print1(k,", "))

A364078 Numbers k such that (18^k - 1)^2 - 2 is prime.

Original entry on oeis.org

2, 8, 30, 98, 110, 185, 912, 2514, 4074, 10208, 15123, 19395, 69354

Offset: 1

Jeppe Stig Nielsen, Jul 03 2023

Such primes are sometimes called Carol primes of base 18.

Prime-Wiki, Carol-Kynea prime.
Eric Weisstein's World of Mathematics, Near-Square Prime.

Cf. A091515 (b=2), A100901 (b=6), A100903 (b=10), A364076 (b=12), A100905 (b=14), A364080 (b=20), A100907 (b=22).

Cf. A364079.

for(k=1,1200,ispseudoprime((18^k-1)^2-2)&&print1(k,", "))

A364079 Numbers k such that (18^k + 1)^2 - 2 is prime.

Original entry on oeis.org

1, 10, 21, 25, 31, 1083, 40485, 82516

Offset: 1

Jeppe Stig Nielsen, Jul 03 2023

Such primes are sometimes called Kynea primes of base 18.

Prime-Wiki, Carol-Kynea prime.
Eric Weisstein's World of Mathematics, Near-Square Prime.

Cf. A091513 (b=2), A100902 (b=6), A100904 (b=10), A364077 (b=12), A100906 (b=14), A364081 (b=20), A100908 (b=22).

Cf. A364078.

for(k=1,1200,ispseudoprime((18^k+1)^2-2)&&print1(k,", "))

A364080 Numbers k such that (20^k - 1)^2 - 2 is prime.

Original entry on oeis.org

1, 2, 53, 183, 1281, 1300, 8041, 29936, 72820

Offset: 1

Jeppe Stig Nielsen, Jul 03 2023

Such primes are sometimes called Carol primes of base 20.

Prime-Wiki, Carol-Kynea prime.
Eric Weisstein's World of Mathematics, Near-Square Prime.

Cf. A091515 (b=2), A100901 (b=6), A100903 (b=10), A364076 (b=12), A100905 (b=14), A364078 (b=18), A100907 (b=22).

Cf. A364081.

for(k=1,1200,ispseudoprime((20^k-1)^2-2)&&print1(k,", "))

A364081 Numbers k such that (20^k + 1)^2 - 2 is prime.

Original entry on oeis.org

1, 15, 44, 77, 141, 208, 304, 1169, 3359, 5050, 22431, 34935, 92990

Offset: 1

Jeppe Stig Nielsen, Jul 03 2023

Such primes are sometimes called Kynea primes of base 20.

Prime-Wiki, Carol-Kynea prime.
Eric Weisstein's World of Mathematics, Near-Square Prime.

Cf. A091513 (b=2), A100902 (b=6), A100904 (b=10), A364077 (b=12), A100906 (b=14), A364079 (b=18), A100908 (b=22).

Cf. A364080.

for(k=1,1200,ispseudoprime((20^k+1)^2-2)&&print1(k,", "))

A363215 Integers p > 1 such that 3^d == 1 (mod p) where d = A000265(p-1).

Original entry on oeis.org

2, 11, 13, 23, 47, 59, 71, 83, 107, 109, 121, 131, 167, 179, 181, 191, 227, 229, 239, 251, 263, 277, 286, 311, 313, 347, 359, 383, 419, 421, 431, 433, 443, 467, 479, 491, 503, 541, 563, 587, 599, 601, 647, 659, 683, 709, 719, 733, 743, 757, 827, 829, 839, 863

Offset: 1

Jeppe Stig Nielsen, May 21 2023

nonn

Inspired by an incorrect definition of strong pseudoprime to base 3.

As is obvious from the data, it fails to include all primes. Does include some composite numbers (pseudoprimes), namely 121, 286, 24046, 47197, 82513, ...

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..10000
Wikipedia, Strong pseudoprime

Cf. A000265, A005935, A020229, A130433.

is(p)=my(d=p-1);d/=2^valuation(d,2);Mod(3,p)^d==1

from itertools import count, islice
def inA363215(n): return pow(3,n-1>>(~(n-1)&n-2).bit_length(),n)==1
def A363215_gen(startvalue=2): # generator of terms >= startvalue
    return filter(inA363215,count(max(startvalue,2)))
A363215_list = list(islice(A363215_gen(),20)) # Chai Wah Wu, May 22 2023

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User: Jeppe Stig Nielsen

A385564 Prime powers k such that lcm(1, 2, 3, ..., k)-1 is prime.

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A377838 Let p = prime(n), then a(n) is the least prime q < p such that p * q# + 1 is prime.

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A370305 Numbers k such that the distance from exp(k) to the closest average of two consecutive primes is less than 1.

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A364076 Numbers k such that (12^k - 1)^2 - 2 is prime.

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A364077 Numbers k such that (12^k + 1)^2 - 2 is prime.

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A364078 Numbers k such that (18^k - 1)^2 - 2 is prime.

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A364079 Numbers k such that (18^k + 1)^2 - 2 is prime.

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A364080 Numbers k such that (20^k - 1)^2 - 2 is prime.

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A364081 Numbers k such that (20^k + 1)^2 - 2 is prime.

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