A216822 -id:A216822 - cp's OEIS frontend

A069051 Primes p such that p-1 divides 2^p-2.

2, 3, 7, 19, 43, 127, 163, 379, 487, 883, 1459, 2647, 3079, 3943, 5419, 9199, 11827, 14407, 16759, 18523, 24967, 26407, 37339, 39367, 42463, 71443, 77659, 95923, 99079, 113779, 117307, 143263, 174763, 175447, 184843, 265483, 304039, 308827

Offset: 1

Benoit Cloitre, Apr 03 2002

nonn

These are the prime values of n such that 2^n == 2 (mod n*(n-1)). - V. Raman, Sep 17 2012
These are the prime values p such that n^(2^(p-1)) is congruent to n or -n (mod p) for all n in Z/pZ, the commutative ring associated with each term. This results follows from Fermat's little theorem. - Philip A. Hoskins, Feb 08 2013
A prime p is in this sequence iff p-1 belongs to A014741. For p>2, this is equivalent to (p-1)/2 belonging to A014945. - Max Alekseyev, Aug 31 2016
From Thomas Ordowski, Nov 20 2018: (Start)
Conjecture: if n-1 divides 2^n-2, then (2^n-2)/(n-1) is squarefree.
Numbers n such that b^n == b (mod (n-1)*n) for every integer b are 2, 3, 7, and 43; i.e., only prime numbers of the form A014117(k) + 1. (End)
These are primes p such that p^2 divides b^(2^p-2) - 1 for every b coprime to p. - Thomas Ordowski, Jul 01 2024

Charles R Greathouse IV, Table of n, a(n) for n = 1..3314 (First 553 terms from V. Raman)
Mersenne Forum, Prime Conjecture

Cf. A216822, A217468, A211203, A211349, A330382.

a(n)-1 form subsequence of A014741; (a(n)-1)/2 for n>1 forms a subsequence of A014945.

Filtered([1..350000],p->IsPrime(p) and (2^p-2) mod (p-1)=0); # Muniru A Asiru, Dec 03 2018

[p : p in PrimesUpTo(310000) | IsZero((2^p-2) mod (p-1))]; // Vincenzo Librandi, Dec 03 2018

Select[Prime[Range[10000]], Mod[2^# - 2, # - 1] == 0 &] (* T. D. Noe, Sep 19 2012 *)
Join[{2,3},Select[Prime[Range[30000]],PowerMod[2,#,#-1]==2&]] (* Harvey P. Dale, Apr 17 2022 *)

isA069051(p)=Mod(2,p-1)^p==2 && isprime(p); \\ Charles R Greathouse IV, Sep 19 2012

from sympy import prime
for n in range(1,350000):
    if (2**prime(n)-2) % (prime(n)-1)==0:
        print(prime(n)) # Stefano Spezia, Dec 07 2018

a(1) added by Charles R Greathouse IV, Sep 19 2012

A217465 Composite integers k such that 2^k == 2 (mod k*(k+1)).

Original entry on oeis.org

561, 1905, 4033, 4681, 5461, 6601, 8481, 11305, 13741, 13981, 16705, 23377, 30121, 31417, 41041, 49141, 52633, 57421, 88357, 88561, 101101, 107185, 121465, 130561, 162193, 196021, 196093, 204001, 208465, 219781, 266305, 276013, 278545, 282133, 285541, 314821, 334153, 341497, 390937, 399001

Offset: 1

V. Raman, Oct 04 2012

nonn

Terms A019320(k) belongs to this sequence for k in A297415. - Max Alekseyev, Dec 29 2017

Amiram Eldar, Table of n, a(n) for n = 1..12766 (terms below 10^12; terms 1..100 from Harvey P. Dale)
Mersenne Forum, Prime Conjecture, 2012.

Subsequence of A216822.
Contains A303531 as a subsequence.
Cf. A019320, A217466, A217468, A297415, A298758.

Select[Range[400000],!PrimeQ[#]&&PowerMod[2,#,#(#+1)]==2&] (* Harvey P. Dale, Oct 12 2012 *)

for(n=1,10000,if((2^n)%(n*(n+1))==2&&isprime(n)==0,printf(n",")))

forcomposite(n=4,10^6, if(Mod(2,n*(n+1))^n==2, print1(n", "))) \\ Charles R Greathouse IV, Aug 29 2024

from sympy import isprime
A217465_list = [n for n in range(1,10**6) if pow(2,n,n*(n+1)) == 2 and not isprime(n)] # Chai Wah Wu, Mar 25 2021

A217468 Composite values of n such that 2^n == 2 (mod n*(n-1)).

Original entry on oeis.org

91625794219, 8796093022207, 1557609722332488343, 18216643597893471403

Offset: 1

V. Raman, Oct 04 2012

No other terms below 2^64.
The next term is <= 25790417485109157029391019 = A019320(258). - Emmanuel Vantieghem, Dec 01 2014
Terms A019320(k) belongs to this sequence for k in A297412. - Max Alekseyev, Dec 29 2017

			a(1) = 91625794219 = (2^38 - 2^19 + 1)/3 = A019320(114).
a(2) = 8796093022207 = 2^43 - 1 = A019320(43).

Mersenne Forum, Prime Conjecture

Cf. A069051, A216822, A219037, A217465.

Intersection of A001567 and { 2*A014945(k) + 1 }.

is_A217468(n)={Mod(2,(n-1)*n)^n==2 & !isprime(n)}  \\ - M. F. Hasler, Oct 07 2012

A217466 Primes p such that 2^p == 2 (mod p*(p+1)).

Original entry on oeis.org

5, 13, 29, 37, 61, 73, 157, 181, 193, 277, 313, 397, 421, 457, 541, 613, 661, 673, 733, 757, 877, 997, 1093, 1153, 1201, 1213, 1237, 1289, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1933, 1993, 2017, 2137, 2341, 2473, 2557, 2593, 2797, 2857, 2917, 3061

Offset: 1

V. Raman, Oct 04 2012

nonn

Primes in A216822.

V. Raman, Table of n, a(n) for n = 1..10000
Mersenne Forum, Prime Conjecture

Cf. A216822.

Select[Prime[Range[500]],PowerMod[2,#,#(#+1)]==2&] (* Harvey P. Dale, Mar 25 2019 *)

for(n=1,10000,if((2^n)%(n*(n+1))==2&&isprime(n),printf(n",")))

from sympy import primerange
A217466_list = [p for p in primerange(1,10**6) if pow(2,p,p*(p+1)) == 2] # Chai Wah Wu, Mar 25 2021

A219037 Numbers k such that k divides 2^k + 2 and (k-1) divides 2^k + 1.

Original entry on oeis.org

2, 6, 66, 73786976294838206466

Offset: 1

Max Alekseyev, Nov 10 2012

Also, numbers k such that 2^k == k-2 (mod k*(k-1)).
The sequence is infinite: if m is in this sequence, then so is 2^m + 2.
No other terms below 10^20.

W. Sierpinski, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #18.

Kin Y. Li et al., Solution to Problem 323, Mathematical Excalibur 14(2), 2009, p. 3.

Intersection of A006517 and A055685.

Cf. A217468, A216822, A171959.

Conjecture: a(n+1) = 2^a(n) + 2 for all n.

A303531 Numbers k such that both k and (k+1)/2 are Fermat pseudoprimes to base 2 (A001567).

Original entry on oeis.org

31417, 130561, 41541241, 208969201, 224074369, 392099401, 851934601, 5186365801, 33847795741, 65096562721, 91625968981, 104070261901, 111794455501, 165814673473, 180007280041, 184020124201, 276032114281, 408164260141, 620052300421, 1240013784241, 1244667503017

Offset: 1

Max Alekseyev, Apr 25 2018

nonn

Numbers (k+1)/2 are listed in A298758.

Amiram Eldar, Table of n, a(n) for n = 1..3031 (terms below 2^64)

Subsequence of each of A001567, A216822, and A217465.

Cf. A005382, A298758, A303447, A303448.

Select[Cases[Import["https://oeis.org/A001567/b001567.txt", "Table"], {, }][[;; , 2]], !PrimeQ[(#+1)/2] && PowerMod[2, (#-1)/2, (#+1)/2] == 1 &] (* Amiram Eldar, Nov 09 2023 *)

isF(n) = {Mod(2, n)^n==2 && !isprime(n) && n>1};
isok(n) = (n%2) && isF(n) && isF((n+1)/2); \\ Michel Marcus, Apr 26 2018

a(n) = 2*A298758(n) - 1.

A297414 Numbers k such that 2^m == 2 (mod m*(m+1)), where m = A019320(k).

Original entry on oeis.org

1, 4, 9, 12, 25, 36, 40, 52, 80, 92, 124, 150, 208, 306, 361, 630, 656, 1648, 1780, 2508, 3300, 3540, 5728, 6260, 6450, 7500, 10820, 12656, 14076, 14132, 18836, 20960, 23456, 24272, 35280, 43136

Offset: 1

Max Alekseyev, Dec 29 2017

nonn

Also, numbers k such that A019320(k) belongs to A216822.

Cf. A019320, A216822, A297413, A297415.

is_A297414(k) = my(m=polcyclo(k, 2)); Mod(2, m*(m+1))^m==2;

A375792 Numbers k such that 2^k == 2 (mod k-th triangular number) and not 2^k == 2 (mod k-th oblong number).

Original entry on oeis.org

3, 11, 131, 4091, 5851, 17291, 283051, 289771, 346963, 1008547, 1082971, 3424651, 3919771, 6464611, 6852691, 7298131, 7514851, 8733691, 12752251, 16740371, 17227891, 19895611, 27393211, 30281371, 33875323, 40528531, 45744931, 68174107, 81011971, 98940403

Offset: 1

Juri-Stepan Gerasimov, Aug 29 2024

nonn

Conjecture: all terms of the sequence are prime numbers A000040.

The conjecture is false: 45812984491 = 1777 * 25781083 is in the sequence. - Charles R Greathouse IV, Aug 29 2024

Cf. A000217 (triangular numbers), A002378 (oblong numbers), A216822 (n such that 2^n == 2 (mod n*(n+1))), A375793 (n such that 2^n == 2 (mod n*(n+1) div 2)), A217465.

[n: n in [1..10^5] | 2^n mod (n*(n+1) div 2) eq 2 and not 2^n mod (n*(n+1)) eq 2];

is(n)=my(m=n*(n+1)); Mod(2,m)^n==m/2+2 \\ Charles R Greathouse IV, Aug 29 2024

a(19)-a(30) from Charles R Greathouse IV, Aug 29 2024

A375793 Numbers m such that 2^m == 2 (mod m-th triangular number).

Original entry on oeis.org

1, 3, 5, 11, 13, 29, 37, 61, 73, 131, 157, 181, 193, 277, 313, 397, 421, 457, 541, 561, 613, 661, 673, 733, 757, 877, 997, 1093, 1153, 1201, 1213, 1237, 1289, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1905, 1933, 1993, 2017, 2137, 2341, 2473, 2557, 2593, 2797, 2857, 2917, 3061, 3217, 3253, 3313, 3389, 3457

Offset: 1

Juri-Stepan Gerasimov, Aug 29 2024

nonn

a(19) = 561 is the first composite term of the sequence.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Supersequence of A216822, A217465, A217466 and A375792.

Cf. A000217, A272934, A289382.

[1] cat [m: m in [2..3500] | Modexp(2, m, m*(m+1) div 2) eq 2];

t:= n-> n*(n+1)/2:
q:= m-> is(2&^m-2 mod t(m)=0):
select(q, [$1..3457])[];  # Alois P. Heinz, Sep 21 2024

Select[Range[3457],Mod[2^#-2,#(#+1)/2 ]==0&] (* James C. McMahon, Sep 23 2024 *)

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A069051 Primes p such that p-1 divides 2^p-2.

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A217465 Composite integers k such that 2^k == 2 (mod k*(k+1)).

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A217468 Composite values of n such that 2^n == 2 (mod n*(n-1)).

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A217466 Primes p such that 2^p == 2 (mod p*(p+1)).

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A219037 Numbers k such that k divides 2^k + 2 and (k-1) divides 2^k + 1.

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A303531 Numbers k such that both k and (k+1)/2 are Fermat pseudoprimes to base 2 (A001567).

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A297414 Numbers k such that 2^m == 2 (mod m*(m+1)), where m = A019320(k).

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A375792 Numbers k such that 2^k == 2 (mod k-th triangular number) and not 2^k == 2 (mod k-th oblong number).

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