A144487 -id:A144487 - cp's OEIS frontend

A068712 Primes of the form 5*2^k + 3.

13, 23, 43, 83, 163, 643, 1283, 10243, 20483, 1310723, 5242883, 335544323, 1342177283, 21474836483, 85899345923, 43980465111043, 87960930222083, 5629499534213123, 22517998136852483, 1441151880758558723

Offset: 1

Amarnath Murthy, Mar 05 2002

nonn

			1283 is a term as a concatenation of 2^7 and 3.

Vincenzo Librandi, Table of n, a(n) for n = 1..38

Cf. A058586, A068715, A144487, A211486.

[a: n in [1..100] | IsPrime(a) where a is 5*2^n + 3]; // Vincenzo Librandi, Dec 08 2011

Select[5*2^Range[60]+3, PrimeQ]  (* Harvey P. Dale, Feb 04 2011 *)

for(n=1,150, if(isprime(2^n*5+3)==1,print1(2^n*5+3,",")))

More terms from Benoit Cloitre, Mar 09 2002

A243429 Primes of the form 2^n + 39.

Original entry on oeis.org

41, 43, 47, 71, 103, 167, 1063, 2087, 8231, 131111, 536870951, 8589934631, 549755813927, 8796093022247, 154742504910672534362390567, 40564819207303340847894502572071, 162259276829213363391578010288167, 2722258935367507707706996859454145691687

Offset: 1

Vincenzo Librandi, Jun 05 2014

nonn

Associated n: 1, 2, 3, 5, 6, 7, 10, 11, 13, 17, 29, 33, 39, 43, 87, 105, 107, 131, 253, 329, ....

Vincenzo Librandi, Table of n, a(n) for n = 1..34

Cf. primes of the form 2^n+k: A092506 (k=1), A057733 (k=3), A123250 (k=5), A104066 (k=7), A104070 (k=9), A156940 (k=11), A104067 (k=13), A144487 (k=15), A156973 (k=17), A104068 (k=19), A156983 (k=21), A176922 (k=23), A104072 (k=25), A104071 (k=27), A156974 (k=29), A104069 (k=31), A176926 (k=33), A176927 (k=35), A176924 (k=37), this sequence (k=39), A176925 (k=41), A243430 (k=43), A243431 (k=45), A243432 (k=47), A104073 (k=49).

[a: n in [0..500] | IsPrime(a) where a is 2^n+39];

Select[Table[2^n + 39, {n, 0, 500}], PrimeQ]

A144670 Triangle read by rows where T(m,n)=2mn+m+n-7.

Original entry on oeis.org

-3, 0, 5, 3, 10, 17, 6, 15, 24, 33, 9, 20, 31, 42, 53, 12, 25, 38, 51, 64, 77, 15, 30, 45, 60, 75, 90, 105, 18, 35, 52, 69, 86, 103, 120, 137, 21, 40, 59, 78, 97, 116, 135, 154, 173, 24, 45, 66, 87, 108, 129, 150, 171, 192, 213, 27, 50, 73, 96, 119, 142, 165, 188, 211, 234, 257

Offset: 1

Vincenzo Librandi, Jan 28 2009

Numbers n such that, if 2^(s-1)=n then [A144487] is not prime.

Let p (prime number), n=(p^2-15)/2 mod(p).

			Triangle begins:
-3;
0, 5;
3, 10, 17;
6, 15, 24, 33;
9, 20, 31, 42, 53;
12, 25, 38, 51, 64, 77;
15, 30, 45, 60, 75, 90, 105;
18, 35, 52, 69, 86, 103, 120, 137;
21, 40, 59, 78, 97, 116, 135, 154, 173;
24, 45, 66, 87, 108, 129, 150, 171, 192, 213;
= = = = = = = =

Vincenzo Librandi, Rows n = 100, flattened

Cf. A057197, A144487.

[2*n*k + n + k -7: k in [1..n], n in [1..12]]; // Vincenzo Librandi, Oct 15 2012

t[n_,k_]:=2 n*k+n+k-7; Table[t[n, k], {n, 12}, {k, n}] // Flatten (* Vincenzo Librandi, Oct 15 2012 *)

A211486 Primes of the form 5+3*2^k.

Original entry on oeis.org

11, 17, 29, 53, 101, 197, 389, 773, 49157, 196613, 1572869, 12582917, 50331653, 402653189, 1610612741, 12884901893, 824633720837, 54043195528445957, 432345564227567621, 3458764513820540933, 226673591177742970257413, 59421121885698253195157962757

Offset: 1

Vincenzo Librandi, Apr 13 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..40

Cf. A057913 (n such that 3*2^n + 5 is prime).

Cf. A057197, A144487.

[ a: n in [0..250] | IsPrime(a) where a is 5+3*2^n ];

Mathematica
```
Select[5+2^Range[0,2000]*3,PrimeQ]
```

{for(n=0, 80, if(isprime(k=5+3*2^n), print1(k, ", ")))}

A237418 Primes of the form 4^k + 15.

Original entry on oeis.org

19, 31, 79, 271, 1039, 4111, 65551, 4194319, 67108879, 1073741839, 4294967311, 1099511627791, 4398046511119, 70368744177679, 4722366482869645213711, 75557863725914323419151

Offset: 1

Vincenzo Librandi, Feb 22 2014

Subsequence of A144487.

			65551 is a term because 4^8 + 15 = 65551 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..26

Cf. A000040, A144487, A253773 (corresponding k's).

[a: n in [1..70] | IsPrime(a) where a is 4^n+15];

Select[Table[4^n + 15, {n, 1, 200}],PrimeQ]

a(n) = 4^A253773(n) + 15. - Elmo R. Oliveira, Nov 28 2023

A193109 Least k such that 2^x + k produces primes for x=1..n and composite for x=n+1.

Original entry on oeis.org

0, 1, 9, 3, 225, 15, 65835, 1605, 19425, 2397347205, 153535525935

Offset: 1

Arkadiusz Wesolowski, Jul 21 2011

All terms except the first four are congruent to 15 mod 30.
a(10) was found in 2005 by T. D. Noe and a(11) was found in the same year by Don Reble.
Other known values: a(13) = 29503289812425.
a(12) > 10^13. - Tyler Busby, Feb 19 2023

Another version of A110096.

Cf. A008597, A057733, A092506, A104070, A144487.

Table[k = 0; While[i = 1; While[i <= n && PrimeQ[2^i + k], i++]; i <= n || PrimeQ[2^i + k], k++]; k, {n, 9}] (* T. D. Noe, Jul 21 2011 *)

is(k, n) = for(x=1, n, if(!isprime(k+2^x), return(0))); 1;
a(n) = {my(s=2); forprime(p=3, n, if(znorder(Mod(2, p))==(p-1), s*=p)); forstep(k=s*(n>1)/2, oo, s, if(is(k, n) && !isprime(k+2^(n+1)), return(k))); } \\ Jinyuan Wang, Jul 30 2020

A295196 Numbers n > 1 such that 2^(n-1) and (2n-m)2^(((n-1)/2) - floor(log_2(n))) are congruent to 1 (mod n) for at least one of m = 3, m = 7 and m = 15.

Original entry on oeis.org

7, 23, 31, 47, 71, 79, 263, 271, 1031, 1039, 2063, 4111, 32783, 65543, 65551, 262151, 1048583, 4194319, 8388623, 67108879, 268435463, 1073741831, 1073741839, 4294967311

Offset: 1

Jonas Kaiser, Nov 16 2017

This definition arises from the conjecture that pseudoprime numbers (A001567) occur only at certain distances m from the next smaller number of the form 2^n. So, if we know that a certain distance does not appear with pseudoprime numbers, we are able to calculate these numbers using Fermat's little theorem and we know that it has to be prime. To "plot" the distance of pseudoprime numbers to 2^n use m = A001567(n) - 2^floor(log_2(A001567(n))). So, the first values of m which do not have a "safe prime number distance" (values with "safe prime number distance" are those values for m which pseudoprime numbers never have) should be m = 1, 49, 81, 85, 129, 133, 273, 275, 289, 321, ....
Conjecture 1: There are no composite numbers in this sequence and perhaps infinitely many primes.
Conjecture 2: For m = 7 this definition generates A104066 and for m = 15 this definition generates A144487 (A057197).
Conjecture 3: There are (infinitely many?) m for which this definition generates nothing but (infinitely many?) primes of the form p = 2^k + m.
It appears that this sequence is a subsequence of A139035.

Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv:1608.00862 [math.GM], 2016.

Cf. A057197, A104066, A139035, A144487, A244626, A294717, A293394.

twoDistableQ[n_] := MemberQ[Mod[(2n - {3, 7, 15}) PowerMod[2, (n - 1)/2 - Floor@ Log2@ n, n], n], 1]; p = 3; twoDistablesList = {}; While[p < 1000000000, If[twoDistableQ@ p, AppendTo[ twoDistablesList, p]]; p = NextPrime@ p]; twoDistablesList (* Robert G. Wilson v, Nov 17 2017 *)

a(n) = (n%2) && lift((Mod(2, n)^(n-1))==1) && (lift((Mod((2*n-3), n)*Mod(2, n)^(((n-1)/2)-floor(log(n)/log(2)))) == 1)||lift((Mod((2*n-7), n)*Mod(2, n)^(((n-1)/2)-floor(log(n)/log(2)))) == 1)||lift((Mod((2*n-15), n)*Mod(2, n)^(((n-1)/2)-floor(log(n)/log(2)))) == 1))

is(n)=if(Mod(2,n)^(n-1)!=1, return(0)); my(m=Mod(2,n)^(n\2-logint(n,2))); ((2*n-3)*m==1 || (2*n-7)*m==1 || (2*n-15)*m==1) && n>1 \\ Charles R Greathouse IV, Nov 17 2017

a(17)-a(24) from Charles R Greathouse IV, Nov 17 2017

cp's OEIS Frontend

A068712 Primes of the form 5*2^k + 3.

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A243429 Primes of the form 2^n + 39.

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A144670 Triangle read by rows where T(m,n)=2mn+m+n-7.

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A211486 Primes of the form 5+3*2^k.

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A237418 Primes of the form 4^k + 15.

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A193109 Least k such that 2^x + k produces primes for x=1..n and composite for x=n+1.

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A295196 Numbers n > 1 such that 2^(n-1) and (2*n-m)*2^(((n-1)/2) - floor(log_2(n))) are congruent to 1 (mod n) for at least one of m = 3, m = 7 and m = 15.

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A295196 Numbers n > 1 such that 2^(n-1) and (2n-m)2^(((n-1)/2) - floor(log_2(n))) are congruent to 1 (mod n) for at least one of m = 3, m = 7 and m = 15.