xddd

[sweety439]

This is the smallest base b (b>=2) such that there is a unique prime with period length n, but gcd(Phi(n,b),n) > 1, i.e. Phi(n,b) is not prime, but Phi(n,b)/gcd(Phi(n,b),n) is prime.

If n is in https://oeis.org/A253235, then gcd(Phi(n,b),n) = 1 for all b, thus there cannot be such b, and if Schinzel's hypothesis H is true, then there are infinitely many such bases b for all n not in https://oeis.org/A253235

[sweety439]

List of primes of the form k*2^n+1 for -5000<=k<=5000 (k odd), -5000<=n<=5000 (n != 0)

we choose absolute value for negative numbers (i.e. k<0)
we choose numerator for noninteger rational numbers (i.e. n<0)

[sweety439]

a-file for A039951, see https://oeis.org/A039951/a039951_1.txt

[sweety439]

Originally Posted by sweety439

List of primes of the form k*2^n+1 for -5000<=k<=5000 (k odd), -5000<=n<=5000 (n != 0)

we choose absolute value for negative numbers (i.e. k<0)
we choose numerator for noninteger rational numbers (i.e. n<0)

OEIS sequences for the data:

Smallest |n|:

A033809 (k>0, n>0)
A046067 (k>0, n>=0)
A078680 (k>0, n>0, allow even k)
A040076 (k>0, n>=0, allow even k)
A108129 (k<0, n>0)
A046069 (k<0, n>=0)
A050412 (k<0, n>0, allow even k)
A040081 (k<0, n>=0, allow even k)
A067760 (k>0, n<0)
A252168 (k<0, n<0)
A096502 (k<0, n<0, 2^|n| > |k|)
A276417 (k<0, n<0, 2^|n| < |k|)

A078683 (k>0, n>0, corresponding primes)
A057025 (k>0, n>=0, corresponding primes)
A050921 (k>0, n>=0, allow even k, corresponding primes)
A052333 (k<0, n>0, corresponding primes)
A057026 (k<0, n>=0, corresponding primes)
A038699 (k<0, n>=0, allow even k, corresponding primes)
A123252 (k>0, n<0, corresponding primes)
A096822 (k<0, n<0, 2^|n| > |k|, corresponding primes)

[sweety439]

Also the n-values for specific k:

k=-15: A002237 (positive n), A059612 (negative n)
k=-13: A001773 (positive n), A096818 (negative n)
k=-11: A001772 (positive n), A096817 (negative n)
k=-9: A002236 (positive n), A059610 (negative n)
k=-7: A001771 (positive n), A059609 (negative n)
k=-5: A001770 (positive n), A059608 (negative n)
k=-3: A002235 (positive n), A050414 (negative n)
k=-1: A000043 (positive n), A000043 (negative n)
k=3: A002253 (positive n), A057732 (negative n)
k=5: A002254 (positive n), A059242 (negative n)
k=7: A032353 (positive n), A057195 (negative n)
k=9: A002256 (positive n), A057196 (negative n)
k=11: A002261 (positive n), A102633 (negative n)
k=13: A032356 (positive n), A102634 (negative n)
k=15: A002258 (positive n), A057197 (negative n)

[sweety439]

This is the dual problem of problem 49 of twin primes (see https://mersenneforum.org/forumdisplay.php?f=86, http://mersenneforum.org/showthread.php?t=10754, http://mersenneforum.org/showthread.php?t=6545)

For odd k < 237 which are divisible by 3, does there always exist n>1 such that 2^n+k and 2^n-k are both primes? (for k = 237, such prime pairs cannot exist, since the same as twin primes of the form k*2^n+1 and k*2^n-1)

status:

3,3
9,5
15,5
21,5
27,5
33,6
39,7
45,6
51,9
57,8
63,unknown
69,7
75,8
81,9
87,unknown
93,8
99,7
105,7
111,7
117,8
123,7
129,9
135,14
141,unknown
147,10
153,8
159,11
165,9
171,unknown
177,8
183,8
189,15
195,unknown
201,unknown
207,unknown
213,10
219,unknown
225,11
231,9

remaining k values are {63, 87, 141, 171, 195, 201, 207, 219}, compare with the original twin prime conjecture for k*2^n+1 and k*2^n-1, its remaining k values are {111, 123, 153, 159, 171, 183, 189, 219, 225}, the only two mixed-remaining k values are 171 and 219

[sweety439]

OEIS sequences of smallest Proth primes (k*b^n+1), smallest Riesel primes (k*b^n-1), smallest dual Proth primes (b^n+k), smallest dual Riesel primes (b^n-k):

k      smallest Proth primes (k*b^n+1)    smallest Riesel primes (k*b^n-1)    smallest dual Proth primes (b^n+k)    smallest dual Riesel primes (b^n-k)
2      A119624    A119591    A138066    A255707
b-1    A305531    A??????    A076845    A113516
b+1    A??????    A??????    A346149    A178250
n      A240234    A240235    A093324    A084746

(all entries in the list currently with no OEIS sequences (A??????) are Williams primes ((b+-1)*b^n+-1), for the data, see https://www.rieselprime.de/ziki/Williams_prime and http://harvey563.tripod.com/wills.txt, and there are also subsequences with prime indices in OEIS: A087139 and A122396)

[sweety439]

Smallest k such that n*k is palindromic in dozenal (all n and k are written in dozenal)

Next term (n=X1X1) is very large, such number

[chalsall]

Originally Posted by sweety439

Smallest k such that n*k is palindromic in dozenal (all n and k are written in dozenal)

When you write "dozenal", is this simply base 12? I seem to remember seeing that word before...

As far as I understand it, pure math is the same no matter what base (let alone dimensions). More than happy to be corrected on that.

Programmers usually actually work in base two (even if they don't realize it).

[Dr Sardonicus]

Originally Posted by sweety439

Smallest k such that n*k is palindromic in dozenal (all n and k are written in dozenal)

Next term (n=X1X1) is very large, such number

Assuming that n = X1X1 is a positive integer congruent to 1 (mod 12), then



is a base twelve palindromic number divisible by n. It may not be the smallest such.

Nil sapientiae odiosius obscuritate nimia

[sweety439]

Originally Posted by Dr Sardonicus

Assuming that n = X1X1 is a positive integer congruent to 1 (mod 12), then



is a base twelve palindromic number divisible by n. It may not be the smallest such.

Nil sapientiae odiosius obscuritate nimia

No, X1X1 (base 12) = 17545 (base 10), it is hard to find a positive multiple of 17545 which is palindromic number in base 12 (A029957)