A005382 -id:A005382 - cp's OEIS frontend

A122781 Nonprimes n such that 4^n==4 (mod n).

1, 4, 6, 12, 15, 28, 66, 85, 91, 186, 276, 341, 435, 451, 532, 561, 645, 703, 946, 1068, 1105, 1247, 1271, 1387, 1581, 1695, 1729, 1891, 1905, 2044, 2046, 2047, 2071, 2465, 2701, 2821, 2926, 3133, 3277, 3367, 3683, 4033, 4369, 4371, 4681, 4795

Offset: 1

Farideh Firoozbakht, Sep 12 2006

If both numbers q and 2q-1 are prime, then q*(2q-1) is in the sequence. So, A005382(n)*(2*A005382(n)-1) = A129521(n) form a subsequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..500

Contains A020136, A001567, A006935 (except n=2), and A129521 as subsequences.

Cf. A005382.

for n from 1 to 5000 do if 4^n mod n = 4 mod n and not isprime(n) then print(n) fi od; # Gary Detlefs, May 14 2012

Select[Range[4800], ! PrimeQ[ # ] && Mod[4^#, # ] == Mod[4, # ] &]
Join[{1,4},Select[Range[5000],!PrimeQ[#]&&PowerMod[4,#,#]==4&]] (* Harvey P. Dale, Apr 09 2018 *)

A181715 Length of the complete Cunningham chain of the second kind starting with prime(n).

Original entry on oeis.org

3, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1

Offset: 1

M. F. Hasler, Nov 17 2010

nonn

Number of iterations x -> 2x-1 needed to get a composite number, when starting with prime(n).
Dickson's conjecture implies that, for every positive integer r, there exist infinitely many n such that a(n) = r. - Lorenzo Sauras Altuzarra, Feb 12 2021
a(n) is the least k such that 2^k * (prime(n)-1) + 1 is composite. Note that a(n) is well defined since 2^(p-1) * (p-1) + 1 is divisible by p for odd primes p. - Jianing Song, Nov 24 2021

			2 -> 3 -> 5 -> 9 = 3^2, so a(1) = 3 and a(2) = 2. - _Jonathan Sondow_, Oct 30 2015

T. D. Noe, Table of n, a(n) for n = 1..10000
G. Löh, Long chains of nearly doubled primes, Math. Comp., 53 (1989), 751-759.
Michael Penn, Romanian Mathematical Olympiad Problem, Youtube video, 2020.
Wikipedia, Cunningham chain

Cf. A000040, A005382, A005408, A005602, A005603, A181697, A263879, A285700, A285706, A057326, A057327, A057328, A057329, A057330, A064812.

Cf. A137288 (positions of terms > 1).

a := proc(n)
   local c, l:
   c, l := 0, ithprime(n):
   while isprime(l) do c, l := c+1, 2*l-1: od:
   c:
end: # Lorenzo Sauras Altuzarra, Feb 12 2021

Table[p = Prime[n]; cnt = 1; While[p = 2*p - 1; PrimeQ[p], cnt++]; cnt, {n, 100}] (* T. D. Noe, Jul 12 2012 *)
Table[-1 + Length@ NestWhileList[2 # - 1 &, Prime@ n, PrimeQ@ # &], {n, 98}] (* Michael De Vlieger, Apr 26 2017 *)

a(n)= n=prime(n); for(c=1,1e9, is/*pseudo*/prime(n=2*n-1) || return(c))

a(n) < prime(n) for n > 1; see Löh (1989), p. 751. - Jonathan Sondow, Oct 28 2015
max(a(n), A181697(n)) = A263879(n) for n > 2. - Jonathan Sondow, Oct 30 2015
a(n) = A285700(A000040(n)). - Antti Karttunen, Apr 26 2017

Escape clause added to definition by N. J. A. Sloane, Feb 19 2021

Escape clause deleted from definition by Jianing Song, Nov 24 2021

A038549 Least number with exactly n divisors that are at most its square root.

Original entry on oeis.org

1, 4, 12, 24, 36, 60, 192, 120, 180, 240, 576, 360, 1296, 900, 720, 840, 9216, 1260, 786432, 1680, 2880, 15360, 3600, 2520, 6480, 61440, 6300, 6720, 2359296, 5040, 3221225472, 7560, 46080, 983040, 25920, 10080, 206158430208, 32400, 184320

Offset: 1

Tom Verhoeff

nonn

Least number of identical objects that can be arranged in exactly n ways in a rectangle, modulo rotation.
Smallest number which has n distinct unordered factorizations of the form x*y. - Lekraj Beedassy, Jan 09 2008
Note that an upper bound on a(n) is 3*2^(n-1), which is attained at n = 4 and the odd primes in A005382 (primes p such that 2p-1 is also prime). - T. D. Noe, Jul 13 2013

Paul Tek, Table of n, a(n) for n = 1..1000
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

Cf. A038548 (records), A072671, A004778, A086921.

Cf. A227068 (similar, but with limit < sqrt).

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a038549 = (+ 1) . fromJust . (`elemIndex` a038548_list)
-- Reinhard Zumkeller, Dec 26 2012

nn = 18; t = Table[0, {nn}]; found = 0; n = 0; While[found < nn, n++; c = Length[Select[Divisors[n], # <= Sqrt[n] &]]; If[c > 0 && c <= nn && t[[c]] == 0, t[[c]] = n; found++]]; t (* T. D. Noe, Jul 10 2013 *)

a(n) = min(A005179(2n-1), A005179(2n)).

More terms from David W. Wilson.

A069142 Primes p such that p+2, 2p+1, and 2p+3 are also prime.

Original entry on oeis.org

5, 29, 659, 809, 2129, 2549, 3329, 3389, 5849, 6269, 10529, 33179, 41609, 44129, 53549, 55439, 57329, 63839, 65099, 70379, 70979, 72269, 74099, 74759, 78779, 80669, 81929, 87539, 93239, 102299, 115469, 124769, 133979, 136949, 156419

Offset: 1

Neil Fernandez, Apr 08 2002

Previous name: Lower prime in a twin pair that yields another.

a(n) gives the terms for A005382(i)-A005384(j)=2. - J. M. Bergot, Mar 12 2015

			659 and 661 form a prime twin pair. Their sum is 1320. 1320 is sandwiched between 1319 and 1321, which form another prime twin pair. So 659 is in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A014574.
Cf. A005384, A005382.
Cf. A066388.

[p: p in PrimesUpTo(160000) | IsPrime(p+2) and IsPrime(2*p+1) and IsPrime(2*p+3)]; // Vincenzo Librandi, Apr 09 2013

p = q = 1; Do[q = Prime[n]; If[p + 2 == q && PrimeQ[2p + 1] && PrimeQ[2p + 3], Print[p]]; p = q, {n, 1, 10^4}]
Select[Prime[Range[15000]], PrimeQ[# + 2] && PrimeQ[2 # + 1] && PrimeQ[2 # + 3]&] (* Vincenzo Librandi, Apr 09 2013 *)

forprime(p=1,10^5,if(isprime(p+2)&&isprime(2*p+1)&&isprime(2*p+3),print1(p,", "))) \\ Derek Orr, Mar 11 2015

a(n) = A066388(n)-1. - R. J. Mathar, Nov 02 2023

Edited and extended by Robert G. Wilson v, Apr 11 2002

A105610 Numbers k such that both p1=2k+3 and p2=4k+5 are primes.

Original entry on oeis.org

0, 2, 8, 14, 17, 38, 47, 68, 77, 98, 104, 113, 134, 152, 164, 167, 182, 188, 218, 248, 272, 287, 299, 302, 308, 329, 344, 362, 404, 413, 437, 467, 482, 497, 503, 533, 584, 617, 638, 647, 698, 713, 728, 764, 803, 812, 827, 878, 932, 1004, 1013, 1043, 1064, 1067

Offset: 1

Zak Seidov, Apr 15 2005

nonn

p1 in A005382, p2 in A005383.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A005382, A005383.

Equals A123998 minus 1.

Select[Range[0,1067], PrimeQ[2#+3]&&PrimeQ[4#+5]&] (* James C. McMahon, Jan 26 2024 *)

from sympy import isprime
print([ k for k in range(0,1068) if isprime(2*k+3) and isprime(4*k+5)])
# Karl-Heinz Hofmann, Jan 27 2024

A020137 Pseudoprimes to base 8.

Original entry on oeis.org

9, 21, 45, 63, 65, 105, 117, 133, 153, 231, 273, 341, 481, 511, 561, 585, 645, 651, 861, 949, 1001, 1105, 1281, 1365, 1387, 1417, 1541, 1649, 1661, 1729, 1785, 1905, 2047, 2169, 2465, 2501, 2701, 2821, 3145, 3171, 3201, 3277, 3605, 3641, 4005, 4033, 4097

Offset: 1

David W. Wilson

nonn

This sequence is a subsequence of the sequence A122785. In fact the terms are odd composite terms of A122785. Theorem: If both numbers q and 2q-1 are primes (q is in the sequence A005382) and n=q*(2q-1) then 8^(n-1)==1 (mod n) (n is in the sequence) iff q is of the form 12k+1. 2701,18721,49141,104653,226801,665281,721801,... is the related subsequence. This subsequence is also a subsequence of the sequence A122785. - Farideh Firoozbakht, Sep 15 2006
Composite numbers k such that 8^(k-1) == 1 (mod k). - Michel Lagneau, Feb 18 2012
If q and 3q-2 are odd primes, then q*(3q-2) is in the sequence. - Davide Rotondo, May 25 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..613 from R. J. Mathar, terms 614..1000 from T. D. Noe)
Index entries for sequences related to pseudoprimes

Cf. A001567 (pseudoprimes to base 2), A005382, A122783, A122785.

Select[Range[4100], ! PrimeQ[ # ] && PowerMod[8, (# - 1), # ] == 1 &] (* Farideh Firoozbakht, Sep 15 2006 *)

A068080 Integers n such that n + phi(n) is a prime.

Original entry on oeis.org

1, 2, 3, 7, 15, 19, 31, 33, 35, 37, 51, 65, 69, 77, 79, 85, 91, 95, 97, 133, 139, 141, 143, 145, 157, 159, 161, 177, 187, 199, 209, 211, 213, 215, 217, 229, 235, 247, 255, 267, 271, 299, 303, 307, 319, 331, 335, 337, 339, 341, 345, 365, 367, 371, 379, 391, 393

Offset: 1

Amarnath Murthy, Feb 17 2002

The subsequence of prime terms is given by A005382. - Michel Marcus, Aug 22 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A050530.

[n: n in [1..400] |IsPrime(n+EulerPhi(n))]; // Vincenzo Librandi, Dec 19 2015

Select[Range[400], PrimeQ[# + EulerPhi[#]] &] (* Carl Najafi, Aug 22 2011 *)

isok(n) = isprime(n+eulerphi(n)); \\ Michel Marcus, Aug 22 2015

Edited and extended by Robert G. Wilson v, Feb 18 2002

A246373 Primes p such that if 2p-1 = product_{k >= 1} A000040(k)^(c_k), then p <= product_{k >= 1} A000040(k-1)^(c_k).

Original entry on oeis.org

2, 3, 7, 19, 29, 31, 37, 47, 67, 71, 79, 89, 97, 101, 103, 107, 109, 127, 139, 151, 157, 181, 191, 197, 199, 211, 223, 227, 229, 241, 251, 269, 271, 277, 283, 307, 317, 331, 337, 349, 359, 367, 373, 379, 397, 409, 421, 433, 439, 457, 461, 467, 487, 499, 521, 541, 547, 569, 571, 577, 601

Offset: 1

Antti Karttunen, Aug 25 2014

nonn

Primes p such that A064216(p) >= p, or equally, A064989(2p-1) >= p.

All primes of A005382 are present here, because if 2p-1 is prime q, Bertrand's postulate guarantees (after cases 2 and 3 which are in A048674) that there exists at least one prime r larger than p and less than q = 2p-1, for which A064989(q) = r.

			2 is present, as 2*2 - 1 = 3 = p_2, and p_{2-1} = p_1 = 2 >= 2.
3 is present, as 2*3 - 1 = 5 = p_3, and p_{3-1} = p_2 = 3 >= 3.
5 is not present, as 2*5 - 1 = 9 = p_2 * p_2, and p_1 * p_1 = 4, with 4 < 5.
7 is present, as 2*7 - 1 = 13 = p_6, and p_5 = 11 >= 7.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Wikipedia, Bertrand's postulate

Intersection of A000040 and A246372.
Subsequence: A005382.
A246374 gives the primes not here.
Cf. A064216, A064989, A246281.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
n = 0; forprime(p=2,2^31, if((A064989((2*p)-1) >= p), n++; write("b246373.txt", n, " ", p); if(n > 9999, break)));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A246373 (MATCHING-POS 1 1 (lambda (n) (and (prime? n) (>= (A064216 n) n)))))

A105652 Numbers k such that p1=2k+3, p2=4k+5 and p3=6k+7 are all prime.

Original entry on oeis.org

0, 2, 17, 104, 134, 152, 164, 167, 299, 362, 584, 617, 647, 764, 827, 1109, 1139, 1277, 1517, 1529, 1532, 2129, 2222, 2399, 2474, 2612, 2789, 2924, 3074, 3179, 3344, 3419, 3482, 3809, 3839, 3842, 3932, 4007, 4082, 4094, 4142, 4259, 4262, 4322, 4469, 4544

Offset: 1

Zak Seidov, Apr 16 2005

nonn

Except for 0, all terms == 2 or 14 (mod 15). - Robert Israel, Jun 08 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005382, A005383, A105610, A105653 - A105657, A174734.

[n: n in [0..5000] | IsPrime(2*n+3) and IsPrime(4*n+5) and IsPrime(6*n+7)]; // Vincenzo Librandi, Nov 13 2010

select(k -> andmap(isprime, [2*k+3,4*k+5,6*k+7]), [0, seq(seq(15*i+j,j=[2,14]),i=0..1000)]); # Robert Israel, Jun 08 2018

a(n) = (A174734(n)-3)/2. - Robert Israel, Jun 08 2018

A120628 Primes such that their double is 1 away from a prime number.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 23, 29, 31, 37, 41, 53, 79, 83, 89, 97, 113, 131, 139, 157, 173, 179, 191, 199, 211, 229, 233, 239, 251, 271, 281, 293, 307, 331, 337, 359, 367, 379, 419, 431, 439, 443, 491, 499, 509, 547, 577, 593, 601, 607, 619, 641, 653, 659, 661, 683

Offset: 1

Cino Hilliard, Aug 17 2006

This sequence is a variation of the sequence in the reference. However this sequence should have an infinite number of terms.

			19 is a prime and 19*2 = 38 which is one away from 37 which is prime.
13 is not in the table because 13*2 = 26 is one away from 25 and 27 both not prime.

R. Crandall and C. Pomerance, Prime Numbers A Computational Perspective, Springer Verlag 2002, p. 49, exercise 1.18.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005382, A005384.

Select[Range[683], PrimeQ[#] && Or[PrimeQ[2 # - 1], PrimeQ[2 # + 1]] &]  (* Ant King, Dec 12 2010 *)
Select[Prime[Range[200]],AnyTrue[2#+{1,-1},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 26 2020 *)

primepm2(n,k) { local(x,p1,p2,f1,f2,r); if(k%2,r=2,r=1); for(x=1,n, p1=prime(x); p2=prime(x+1); if(isprime(p1*k+r)||isprime(p1*k-r), print1(p1",") ) ) }

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A122781 Nonprimes n such that 4^n==4 (mod n).

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A181715 Length of the complete Cunningham chain of the second kind starting with prime(n).

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A038549 Least number with exactly n divisors that are at most its square root.

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A069142 Primes p such that p+2, 2p+1, and 2p+3 are also prime.

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A105610 Numbers k such that both p1=2k+3 and p2=4k+5 are primes.

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A020137 Pseudoprimes to base 8.

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Mathematica

A068080 Integers n such that n + phi(n) is a prime.

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Magma

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