A217466 -id:A217466 - cp's OEIS frontend

A216822 Numbers n such that 2^n == 2 (mod n*(n+1)).

1, 5, 13, 29, 37, 61, 73, 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

Offset: 1

V. Raman, Sep 17 2012

a(17) = 561 is the first composite number in the sequence. - Charles R Greathouse IV, Sep 19 2012

Intersection of { A015919(n) } and { A192109(n)-1 }. - Max Alekseyev, Apr 22 2013

V. Raman and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2000 terms from V. Raman)
Mersenne Forum, Prime Conjecture

Cf. A069051 (prime n such that 2^n == 2 (mod n*(n-1))).
Cf. A217466 (prime terms of the sequence).
Cf. A217465 (composite terms of the sequence)

Select[Range[1, 10000], Mod[2^# - 2, # (# + 1)] == 0 &] (* T. D. Noe, Sep 19 2012 *)
Join[{1},Select[Range[3000],PowerMod[2,#,#(#+1)]==2&]] (* Harvey P. Dale, Oct 05 2022 *)

is(n)=Mod(2,n*(n+1))^n==2; \\ Charles R Greathouse IV, Sep 19 2012

A216822_list = [n for n in range(1,10**6) if n == 1 or pow(2,n,n*(n+1)) == 2] # Chai Wah Wu, Mar 25 2021

a(1)=1 prepended by Max Alekseyev, Dec 29 2017

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

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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A216822 Numbers n such that 2^n == 2 (mod n*(n+1)).

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

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

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