A217465 -id:A217465 - cp's OEIS frontend

A297415 Numbers k such that A019320(k) is in A217465.

25, 36, 52, 92, 124, 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

Cf. A019320, A217465, A297412.

Set difference of A297414 and ({1} U A072226).

is_A297415(n) = my(m=polcyclo(n, 2)); (m>1) && Mod(2, m*(m+1))^m==2 && !ispseudoprime(m);

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

Original entry on oeis.org

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

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

A298758 Numbers k such that both k and 2k-1 are Poulet numbers (Fermat pseudoprimes to base 2).

Original entry on oeis.org

15709, 65281, 20770621, 104484601, 112037185, 196049701, 425967301, 2593182901, 16923897871, 32548281361, 45812984491, 52035130951, 55897227751, 82907336737, 90003640021, 92010062101, 138016057141, 204082130071, 310026150211, 620006892121, 622333751509

Offset: 1

Amiram Eldar, Jan 26 2018

nonn

2*a(n) - 1 = A303531(n) belongs to A217465. - Max Alekseyev, Apr 24 2018
Numbers k such that both k and 2k+1 are Poulet numbers are listed in A303447.
If p is a prime such that 2*p-1 is also a prime (A005382) and k = (2^(2*p-1)+1)/3 and 2*k-1 are both composites, then k is a term of this sequence (Rotkiewicz, 2000). - Amiram Eldar, Nov 09 2023

Amiram Eldar, Table of n, a(n) for n = 1..3031 (terms below 2^64)
Andrzej Rotkiewicz, Arithmetic progressions formed by pseudoprimes, Acta Mathematica et Informatica Universitatis Ostraviensis, Vol. 8, No. 1 (2000), pp. 61-74.

Subsequence of A001567.

Cf. A005382, A217465, A303447, A303448.

s = Import["b001567.txt", "Data"][[All, -1]]; n = Length[s];
aQ[n_] := ! PrimeQ[n] && PowerMod[2, (n - 1), n] == 1;
a = {}; Do[p = 2*s[[k]] - 1; If[aQ[p], AppendTo[a, s[[k]]]], {k, 1, n}]; a (* using the b-File from A001567 *)

isP(n) = (Mod(2, n)^n==2) && !isprime(n) && (n>1);
isok(n) = isP(n) && isP(2*n-1); \\ Michel Marcus, Mar 09 2018

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.

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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A297415 Numbers k such that A019320(k) is in A217465.

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

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

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A298758 Numbers k such that both k and 2k-1 are Poulet numbers (Fermat pseudoprimes to base 2).

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

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