New test

[CRGreathouse]

N.B. Charles = CRGreathouse
Many thanks! I didn't realize that you had completed the count. What remains to be done?

Or have I misunderstood, and there may be more pseudoprimes in that range?

[CRGreathouse]

The composites in the sequence are
2²⁹ - 1 = 536870911, (2⁴⁷ + 1)/3 = 46912496118443, (2⁵⁹ + 1)/3 = 192153584101141163, ...

If the requirement that the number be relatively prime to 3 is dropped, this adds the composite 19781763. As far as I can tell this is of no special form.

There are no other members below 2⁶⁴. The next known member of the sequence is (2^83+1)/3. 2¹⁶¹⁰³⁹ - 1 is the only known larger member not of the form (2^k + 1)/3.

[CRGreathouse]

Originally Posted by CRGreathouse

It looks like 2^161039-1 is another composite member of this sequence. (I've been testing Mersenne numbers to see if any others would be in.)

No more to exponent 190,000. I'm stopping the search (need cores for other work!).

[allasc]

Значит если для последовательности A175625 добавить условие

- (n-1)/2 не имеет делителей вида (2^k-1), где k - целое и 1<(2^k-1)<(n-1)/2

то мы пока для такого теста исключений незнаем.
Дополнительное условие совсем немного уменьшает последовательность A175625
перве элементы неудовлетворяющие этому условию 31 и 683

ага и тогда мы приходим к повторению последовательности A005385 например
вроде совпадают
хотя ... Необходимо взять большой интервал за проверку совпадения
или по другому, всели (n-1)/2 будут являтся простыми, если мы будем искать для них только делители вида 2^k-1
----

if for sequence A175625 add a condition

- (n-1)/2 has no kind dividers (2^k-1), where k - whole and 1 <(2^k-1) <(n-1)/2

That we while, for such test, exceptions do not know.
The additional condition almost does not reduce sequence A175625
The first elements unsatisfactory to this condition 31 and 683

Then we come to a repetition of the sequence A005385 for example :)
...It is necessary to take the big interval for coincidence check
Or on another, instal (n-1)/2 will be primary if we search for them only for kind dividers 2^k-1

[allasc]

извините
4931
9719
и т.д.
не удовлетворяют этому правилу
---
Excuse
4931
9719
Etc.
Do not satisfy to this rule

[allasc]

Сегодня нашол новое исключение, вида 2^k-1 для последовательности A175625

(2^2741333597-1)
делитель - 21930668777

незнаю, есть ли между (2^29-1) и (2^2741333597-1) другие исключения вида (2^k-1)

2741333597 - простое число
2741333597 - является элементом последовательности A054723

2741333597 - является элементом последовательности...
%I A175905
%S A175905 5,29,2045,40133,971837,5063357,7354397,16554917,17786525,42244637,
%T A175905 52717277,79704029
%N A175905 Numbers n such that n=4*(2*i+1)+1, 2^(n-2) = 1(mod (2*i+1))
%K A175905 nonn
%O A175905 1,1
%A A175905 Alzhekeyev Ascar M (allasc(AT)mail.ru), Oct 12 2010

(почемуто не добавили досих пор в OEIS мою последовательность, поэтому привожу заявку)

------------------

Today has found a new exception of type 2 ^ k-1 for the sequence A175625

(2 ^ 2741333597-1)
divider - 21930668777

Do not know if there is between (2^29-1) and (2 ^ 2741333597-1) other exceptions to the form (2 ^ k-1)

2741333597 - prime
2741333597 - is an element of the sequence A054723

2741333597 - is an element of the sequence ...
%I A175905
%S A175905 5,29,2045,40133,971837,5063357,7354397,16554917,17786525,42244637,
%T A175905 52717277,79704029
%N A175905 Numbers n such that n=4*(2*i+1)+1, 2^(n-2) = 1(mod (2*i+1))
%K A175905 nonn
%O A175905 1,1
%A A175905 Alzhekeyev Ascar M (allasc(AT)mail.ru), Oct 12 2010

(For some reason, have not added in my OEIS sequence, so bring a request)

[maxal]

Originally Posted by allasc

%I A175905
%S A175905 5,29,2045,40133,971837,5063357,7354397,16554917,17786525,42244637,
%T A175905 52717277,79704029
%N A175905 Numbers n such that n=4*(2*i+1)+1, 2^(n-2) = 1(mod (2*i+1))
%K A175905 nonn
%O A175905 1,1
%A A175905 Alzhekeyev Ascar M (allasc(AT)mail.ru), Oct 12 2010

Here are some more terms:
5, 29, 2045, 40133, 971837, 5063357, 7354397, 16554917, 17786525, 42244637, 52717277, 79704029, 84896957, 153424637, 262984997, 288644957, 328721213, 350252957, 353294757, 393411197, 498253253, 613578149, 634102757, 876046277, 994083077, 1393941437, 1993041317, 2545527197, 2553581477, 2741333597, 2757855557, 3158632037, 3221114717, 3523083077, 3581590853, 3599313629, 4077466877, 4079478405, 4660648709, 5061728765, 5857588037, 5981690717, 6155624477, 7367115197, 7823832293, 10542609029, 10704424733, 10950220037, 11509974533, 12116527997, 12122824925, 12424184957, 12675200189, 14000891933, 14864931677, 16585568477, 16754707517, 17067038213, 17089219037, 18328983077, 19069948829, 19148413437, 20682718757, 21112915677, 21396169797, 21488265797, 22221217757, 22452481157, 24716959877, 26440276997, 29459761157, 32338103717, 33149209277, 33459002117, 34040117477, 35528999717, 37982647397, 38215696829, 38373883877

[allasc]

немного измененый тест
slightly modified test

получилась немного урезанная версия последовательности A175625
is somewhat stripped down version of the sequence A175625

%I A175942
%S A175942 5,11,23,47,59,83,107,167,179,227,263,347,359,383,467,479,503,563,587,
%T A175942 683,719,839,863,887,983,1019,1187,1283,1307,1319,1367,1439,1487,1523,
%U A175942 1619,1823,1907,2027,2039,2063,2099,2207,2447,2459,2543,2579,2819,2879
%N A175942 Numbers n such that gcd(n, 2) = 1, 2^(2*n-1) = 2 (mod 3*n), and 2^(n-2) = 2 (mod 3*(n-1)/2)

composite members of the form (2^k + 1)/3 preserved
composite members of the form (2^k - 1) disappeared (536870911 absent)

[allasc]

another test

in oeis A186645

Consider the numbers satisfying the following condition
2^(n-1) = 1 (mod n)
n = 1 (mod B)
----------------------
value of B is obtained as follows
we first find the value of X
2^(n-1) = X (mod n*n)
it is clear that the Y value is an integer
Y=(X-1)/n
and finally we determine the value of B
Y=B(mod n)
-------------------------------------------------------

the first sequence value
7, 11, 13, 19, 31, 71, 127, 379, 491, 2047, 2633, 2659, 8191, 13249, 26893, 70687, 74597, 87211, 131071, 184511, 524287, 642581, 1897121, 2676301, 2703739, 8388607, 15456151, 52368101 ....

obvious that all composite numbers in this sequence belong pseudosimple base 2
sequence A001567
checked for all values of this sequence <=999986341201
and most interesting is that exceptions are only numbers Marsenne
2047=2^11-1; 8388607=2^23-1; 536870911=2^29-1; 37438953471=2^37-1
This sequence contains all elements of the sequence A065341 (Mersenne composites: 2^prime(n) - 1 is not a prime.)

Other (no Mersenne numbers) composites not found, their existence is under question.
Message in Russian http://dxdy.ru/post417173.html#p417173

[allasc]

modified the last test (a bit to improve performance) and got an interesting result

in oeis A186884

Consider the numbers satisfying the following condition
2^(n-1) = 1 (mod n)
n = 2^p (mod B)

where p >=0 integer and 2^p<n
----------------------
value of B is obtained as follows
we first find the value of X
2^(n-1) = X (mod n*n)
it is clear that the B value is an integer
B=(X-1)/n
-------------------------------------------------------

the first sequence value
5, 7, 11, 13, 17, 19, 31, 71, 127, 173, 199, 233, 251, 257, 379, 491, 613, 881, 2047, 2633, 2659, 3457, 5501, 5683, 8191, 11497, 13249, 15823, 16879, 18839, 22669, 24763, 25037, 26893, 30139, 45337, 48473, 56671, 58921, 65537, 70687, 74531, 74597, 77023, 79669, 87211, 92237, 102407, 131071, 133493, 181421, 184511, 237379, 250583, 254491, 281381....

obvious that all composite numbers in this sequence belong pseudosimple base 2
sequence A001567
checked for all values of this sequence <=999986341201
and most interesting is that exceptions are numbers Marsenne
2047=2^11-1; 8388607=2^23-1; 536870911=2^29-1; 37438953471=2^37-1
Also detected another composite number: 4294967297 = 2 ^ 32 + 1

This sequence contains all elements of the sequence A065341 (Mersenne composites: 2^prime(n) - 1 is not a prime.)
And composites of form 2^x+1.

In other words, all known composite numbers have the form (2 ^ x + 1) or (2 ^ x - 1).

Whether this statement is true for the all of numbers this sequence?

Message in Russian http://dxdy.ru/post418266.html#p418266

[allasc]

another test

in oeis A187849

Numbers n with property that 2^(n-1)=1(mod n) and 2^(m-1) = 1(mod m), where m=n*p-n+1; n=2^k*p+1; k - integer, p - odd integer

563, 1291, 1733, 1907, 2477, 2609, 2693, 2837, 3533, 3677, 4157, 4517, 5693, 12809, 15077, 19997, 25603, 28517, 29573, 29837, 31517, 32237, 32717, 34949, 37277, 43613, 43973, 44453, 50333, 52253.....

All composites in this sequence are 2-pseudoprimes, A001567. Checked all integers <= 999994510007533. Not found composite number.

For each element of n, p integer in most cases is also a prime number.

Maybe. If k = 1 and p is a composite number, then p = 0 (mod 3).

The first examples.

n = 1291 = 2 * 645 +1; p = 645 = 215 * 3

n = 25603 = 2 * 12801 +1; p = 12801 = 4267 * 3

n = 424843 = 2 * 212421 +1; p = 212421 = 70807 * 3

n = 579883 = 2 * 289941 +1; p = 289941 = 96647 * 3

n = 4325443 = 2 * 2162721 +1; p = 2162721 = 720907 * 3

etc.

http://dxdy.ru/post422789.html#p422789