Pierre CAMI - cp's OEIS frontend

A329941 Least prime, p, such that 2p3^n - 1 and 2p3^n + 1 are twin primes.

2, 11, 2, 5, 43, 29, 53, 311, 113, 109, 367, 859, 647, 11, 2, 619, 13, 1051, 157, 2801, 3767, 5, 337, 1721, 3517, 41, 4013, 1879, 1873, 13649, 4637, 2909, 8387, 6521, 1453, 6599, 1277, 4801, 167, 1031, 11213, 4129, 4933, 199, 1427, 859, 9227, 5581, 863, 11959, 10453

Offset: 1

Pierre CAMI, Nov 24 2019

nonn

			2*2*3^1 - 1 = 11; 11 and 13 are twin primes so a(1)=2.
2*11*3^2 - 1 = 197; 197 and 199 are twin primes so a(2)=11 as no other prime p < 11 gives twin primes.

Cf. A130327.

Array[Block[{p = 2}, While[! AllTrue[2 p 3^# + {-1, 1}, PrimeQ], p = NextPrime@ p]; p] &, 51] (* Michael De Vlieger, Dec 24 2019 *)

a(n) = {my(p=2); while (!isprime(2*p*3^n - 1) || !isprime(2*p*3^n + 1), p = nextprime(p+1)); p;} \\ Michel Marcus, Nov 25 2019

A329920 Smallest k such that 6kA121940(n)-1 and 6kA121940(n)+1 are twin primes.

Original entry on oeis.org

1, 2, 2, 15, 36, 10, 13, 26, 30, 228, 24, 138, 520, 59, 110, 456, 700, 670, 146, 300, 390, 53, 2335, 340, 159, 340, 65, 475, 785, 1145, 759, 3557, 490, 169, 990, 1527, 704, 3379, 1426, 1927, 2397, 600, 1603, 4809, 9815, 58, 35, 364, 361, 123, 2197, 4054, 1867, 1827, 5048

Offset: 1

Pierre CAMI, Nov 24 2019

nonn

			A121940(1)=7, 6*1*7-1=41, 41 and 43 are twin primes so a(1)=1.
A121940(2)=91, 6*2*91-1=1091, 1091 and 1093 are twin primes so a(2)=2.

Cf. A001359, A002476, A006512, A121940, A173937, A329336, A329916.

lista(nn) = {my(pp = 1); forprime (p = 1, nn, if (Mod(p, 6) == +1, pp *= p; my(k=1); while (!isprime(6*k*pp-1) || !isprime(6*k*pp+1), k++); print1(k, ", ");););} \\ Michel Marcus, Nov 25 2019

A330304 Prime numbers P such that Q=2P-1, R=4Q+1, S=6R+1, T=8S-1, U=10T+1 and V=12U-1 are all prime numbers.

Original entry on oeis.org

2, 34057, 36847, 207997, 612967, 11035807, 14015167, 19251097, 19587577, 25602547, 26953957, 28060717, 29722177, 29808277, 32894437, 40874857, 41691607, 49713127, 53064877, 54539827, 69143017, 85320577, 101516137, 110327797, 110712247, 123088117, 131584417, 140028607, 150780517

Offset: 1

Pierre CAMI, Dec 13 2019

nonn

Subsequence of A005382.
a(1) = A005382(1), a(2) = A005382(505), a(3) = A005382(536), a(4) = A005382(2084), a(5) = A005382(5105); a(6) > A005382(10000).
P, Q, R, S, T, U, V are 7 primes in near-geometric progression (2, 4, 6, 8, 10, 12 plus or minus one) starting P = a(n).

			2*2-1=3, 4*3+1=13, 6*13+1=79, 8*79-1=631, 10*631+1=6311, 12*6311-1=75731, where 2, 3, 13, 79, 631, 6311 and 75731 are all prime numbers; so 2 is the first term.

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A005382.

forprime(P=2,130000000,my(Q=2*P-1,R=4*Q+1,S=6*R+1,T=8*S-1,U=10*T+1,V=12*U-1);if(isprime(Q)&&isprime(R)&&isprime(S)&&isprime(T)&&isprime(U)&&isprime(V),print1(P,", "))) \\ Hugo Pfoertner, Dec 17 2019

a(12) and a(15) corrected by Chai Wah Wu, Jan 17 2020

A329916 Smallest k such that 6kA057130(n)-1 and 6kA057130(n)+1 are twin primes.

Original entry on oeis.org

1, 2, 3, 23, 11, 18, 77, 46, 84, 76, 22, 30, 3, 107, 26, 198, 136, 23, 236, 284, 167, 269, 381, 405, 379, 374, 620, 481, 606, 505, 163, 1414, 348, 639, 1696, 1429, 850, 2050, 740, 117, 362, 35, 3961, 72, 1307, 1816, 9410, 5705, 972, 368, 5083, 4387, 3296, 6039

Offset: 1

Pierre CAMI, Nov 24 2019

nonn

A057130 gives the product of prime numbers (-1 mod 6) in the order of occurrence.

			A057130(1)=5, 6*1*5-1=29, and 29 and 31 are twin primes, so a(1)=1.
A057130(2)=55, 6*2*55-1=659, and 659 and 661 are twin primes, so a(2)=2.

Cf. A001359, A006512, A007528, A057130, A060256, A173937, A329736.

lista(nn) = {my(pp = 1); forprime (p = 1, nn, if (Mod(p, 6) == -1, pp *= p; my(k=1); while (!isprime(6*k*pp-1) || !isprime(6*k*pp+1), k++); print1(k, ", ");););} \\ Michel Marcus, Nov 25 2019

A329736 Smallest odd prime P such that P32^n - 1 and P32^n + 1 are twin primes.

Original entry on oeis.org

3, 5, 3, 5, 43, 11, 3, 19, 17, 5, 113, 59, 317, 331, 307, 241, 127, 829, 23, 149, 127, 11, 3023, 1091, 787, 971, 1523, 2741, 727, 1051, 227, 211, 727, 89, 1163, 71, 367, 1031, 577, 89, 1213, 1151, 3, 1021, 283, 2699, 4933, 59, 647, 709, 3083, 541, 1483, 2069

Offset: 1

Pierre CAMI, Nov 20 2019

nonn

			3*3*2^1 - 1 =  17,  17 and  19 are twin primes so a(1)=3.
5*3*2^2 - 1 =  59,  59 and  61 are twin primes so a(2)=5.
3*3*2^3 - 1 =  71,  71 and  73 are twin primes so a(3)=3.
5*3*2^4 - 1 = 119, 119 and 121 are twin primes so a(4)=5.

Daniel Starodubtsev, Table of n, a(n) for n = 1..1000

Cf. A001359, A006512, A013597, A051886, A173937, A210651.

Array[Block[{p = 3}, While[! AllTrue[3 p*2^# + {-1, 1}, PrimeQ], p = NextPrime@ p]; p] &, 54] (* Michael De Vlieger, Nov 21 2019 *)

for(n=1,54,my(m=3*2^n);forprime(k=3,oo,my(j=k*m);if(ispseudoprime(j-1)&&ispseudoprime(j+1),print1(k,", ");break))) \\ Hugo Pfoertner, Nov 21 2019

a(n) = my(p=3, q); while (!isprime(q=p*3*2^n - 1) || !isprime(q+2), p = nextprime(p+1)); p; \\ Michel Marcus, May 06 2020

A327638 a(0)=1, a(1)=5; for n > 2, a(n) is the smallest odd number j not divisible by 3 such that (3*j+1)/2^k = a(n-1) for some k.

Original entry on oeis.org

1, 5, 13, 17, 11, 7, 37, 49, 65, 43, 229, 305, 203, 541, 721, 961, 5125, 6833, 4555, 6073, 32389, 172741, 230321, 153547, 818917, 4367557, 5823409, 7764545, 5176363, 6901817, 18404845, 98159173, 523515589, 2792083141, 3722777521, 19854813445, 105892338373

Offset: 0

Pierre CAMI, Sep 20 2019

nonn

Also a reverse Collatz sequence with odd numbers. - Paul Conradi, Oct 23 2020

Odd numbers in A225570. - Michel Marcus, Oct 24 2020

			a(0)=1, a(1)=5 by definition, then a(2) = (2*5-1)/3 or (8*5-1)/3; as (2*5-1)/3 is divisible by 3, a(2) = (8*5-1)/3 = 13.

Michel Marcus, Table of n, a(n) for n = 0..295

Intersection of A005408 and A225570.

lista(nn) = {print1(1, ", ", 5); x = 5; for (n = 2, nn, if(x%3 == 1, x = (4*x-1)/3, x = (2*x-1)/3); if(x%3 == 0, x = 4*x + 1); print1(", ",  x)); } \\ Jinyuan Wang, Sep 21 2019

If a(n-1) == 1 (mod 3) then a(n) = (4*a(n-1)-1)/3 or (16*a(n-1)-1)/3, whichever value is not divisible by 3.

If a(n-1) == -1 (mod 3) then a(n) = (2*a(n-1)-1)/3 or a(n)=(8*a(n-1)-1)/3, whichever value is not divisible by 3.

More terms from Jinyuan Wang, Sep 21 2019

A327581 a(1) is the smallest prime p such that 6p^2-1 and 6p^2+1 are twin primes; for n > 1, a(n) is the smallest prime q > a(n-1) such that 6q^prime(n)-1 and 6q^prime(n)+1 are twin primes or 0 if no solution exists.

Original entry on oeis.org

5, 0, 2557, 51137, 52057, 55373, 88867, 95273, 179947, 236653, 993647, 1010467, 1935533, 2031767, 2138803, 2849317, 8031343, 11696563, 11715133, 18125993, 22615493, 26766633, 26801393, 29963077, 39377893, 58282927, 70354657, 98988257, 119772847, 141442493, 145460123

Offset: 1

Pierre CAMI, Sep 17 2019

nonn

For prime(2) = 3 there is no solution such that 6*q^3-1 and 6*q^3+1 with q prime are twin primes. Because 7 divides 6*p^3-1 when p == 3, 5, 6 mod 7, 7 divides 6*p^3+1 when p == 1, 2, 4 mod 7. Therefore p can only be 7. But then 6*7^3-1 = 11^2*17 and 6*7^3+1 = 29*71 are not prime numbers, so a(2)=0.

Pierre CAMI, PFGW Script

findp(n, pmin) = {my(pmin = nextprime(pmin+1), q); forprime(p=pmin, , if (isprime(q=6*p^prime(n)-1) && isprime(q+2), return(p));); }
lista(nn) = {my(lasta = 2, newa); print1(findp(1, lasta), ", 0"); for (n=3, nn, newa = findp(n, lasta); print1(", ", newa); lasta = newa;); } \\ Michel Marcus, Sep 20 2019

A300507 Define the set of generalized Syracuse sequences starting with a positive odd integer 2n+1=x(1) then if x(i) is odd and prime set x(i+1)=2x(i)+1, if x(i) is odd not prime set x(i+1)=3*x(i)+1 and if x(i) is even then set x(i+1)=x(i)/2. a(n) is the index i when x(i) reaches 1 or 1163.

Original entry on oeis.org

4, 446, 444, 445, 448, 443, 236, 444, 441, 508, 8, 442, 511, 235, 506, 443, 514, 440, 509, 507, 233, 934, 445, 441, 512, 512, 438, 234, 937, 505, 889, 442, 515, 480, 241, 439, 239, 508, 510, 506, 892, 232, 10, 933, 503, 427, 444, 440, 461, 457, 478, 420, 509

Offset: 0

Pierre CAMI, Mar 07 2018

nonn

For n<40 the sequences reach 1, for n=40 the sequence reaches 1163 for x(889) and recover 1163 for x(889+931) a cycle of 961 values.

			For n=1 after 135 tripling(+1), 47 doubling(+1) and 263 halfing x(446)=1, so a(1)=446.

Cf. A006577, A294748, A297307, A300286.

f(x) = if (x % 2, if (isprime(x), 2*x+1, 3*x+1), x/2);
a(n) = {x = f(2*n+1); nb = 2; while (! ((x == 1) || (x == 1163)), x = f(x); nb++); nb;} \\ Michel Marcus, Mar 07 2018

A300286 Define a set of generalized Syracuse sequences starting with x(1)=2n+1 a positive odd integer, if x(i) is odd prime set x(i+1)=67x(i)+1, if x(i) is odd not prime set x(i+1)=3*x(i)+1 and if x(i) is even then set x(i+1)=x(i)/2. Then a(n) is the first index i > 1 at which x(i) reaches 1.

Original entry on oeis.org

4, 209166, 13, 207226, 207229, 384614, 384602, 32, 104820, 403030, 8, 30, 403033, 118516, 39365, 403070, 403036, 118323, 11641, 118425, 118514, 89369, 104824, 180241, 11644, 39371, 118321, 118294, 89372, 118423, 119595, 39372, 11647, 403093, 384607, 47436, 124886

Offset: 0

Pierre CAMI, Mar 02 2018

nonn

I define the generalized Syracuse sequences as follows:
Start with an odd positive number x(1)=2*k+1; then, for i >= 1, if x(i) is an odd prime set x(i+1)=p*x(i)+1 with p a prime, if x(i) is an odd nonprime set x(i+1)=3*x(i)+1, and if x(i) is even then set x(i+1)=x(i)/2.
If p=3 the sequences are the Syracuse sequences in which it does not matter whether odd x(i) is prime or not.
For all the prime numbers p other than 3, if x(i) is odd, the value of x(i+1) depends on whether x(i) is prime.
Among prime numbers p < 97, 67 is the only one for which x(i) reaches 1 for any k < 125 and for k=125, x(1)=251, x(8113)=887, x(8113+8099)=887 a cycle of 8099 values.
All the sequences for p < 423 eventually enter a loop (not tested above, but I conjecture that it is the case for any prime, although with different end cycles).

			For p=67 and k=2, we have x(1)=2*2+1=5, x(2)=67*5+1=336, x(3)=336/2=168, x(4)=168/2=84, x(5)=84/2=42, x(6)=42/2=21, x(7)=3*21+1=64, x(8)=64/2=32, x(9)=32/2=16, x(10)=16/2=8, x(11)=8/2=4, x(12)=4/2=2, x(13)=2/2=1; x(i) reaches 1 at i=13, so a(2)=13.

Pierre CAMI, PFGW Script

Cf. A006577, A294748, A297307.

f(n) = if (n % 2, if (isprime(n), 67*n+1, 3*n+1), n/2);
a(n) = {my(k = f(2*n+1), nb = 2); while (k != 1, k = f(k); nb++); nb;} \\ Michel Marcus, Mar 28 2018

A294748 Define one of the generalized Syracuse sequences starting with a positive odd integer 2k+1=x(1), then if x(i) is an odd prime set x(i+1)=2x(i)+1, if x(i) is odd not prime set x(i+1)=3x(i)+1, if x(i) is even then set x(i+1)=x(i)/2. This sequence gives the positive odd integers 2k+1=x(1) for sequences reaching x(i)=1.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 63, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 125, 127, 129, 131, 133, 135, 137, 141, 143, 145

Offset: 1

Pierre CAMI, Feb 18 2018

nonn

The sequence of positive odd integers not in this sequence begins {61, 81, 123, 139, ...}. When x(1) is any of these, the sequence x(i) enters a cycle of 931 values x(i) = x(i+931)=1163.

Pierre CAMI, PFGW Script

Cf. A006577, A297307, A300286.

f(n) = if (n % 2, if (isprime(n), 2*n+1, 3*n+1), n/2);
isok(n) = {if (n%2, while (1, n = f(n); if (n==1, return (1)); if (n==1163, return (0));););} \\ Michel Marcus, Mar 28 2018

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User: Pierre CAMI

A329941 Least prime, p, such that 2*p*3^n - 1 and 2*p*3^n + 1 are twin primes.

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A329920 Smallest k such that 6*k*A121940(n)-1 and 6*k*A121940(n)+1 are twin primes.

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A330304 Prime numbers P such that Q=2*P-1, R=4*Q+1, S=6*R+1, T=8*S-1, U=10*T+1 and V=12*U-1 are all prime numbers.

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A329916 Smallest k such that 6*k*A057130(n)-1 and 6*k*A057130(n)+1 are twin primes.

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A329736 Smallest odd prime P such that P*3*2^n - 1 and P*3*2^n + 1 are twin primes.

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A327638 a(0)=1, a(1)=5; for n > 2, a(n) is the smallest odd number j not divisible by 3 such that (3*j+1)/2^k = a(n-1) for some k.

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A327581 a(1) is the smallest prime p such that 6*p^2-1 and 6*p^2+1 are twin primes; for n > 1, a(n) is the smallest prime q > a(n-1) such that 6*q^prime(n)-1 and 6*q^prime(n)+1 are twin primes or 0 if no solution exists.

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A300507 Define the set of generalized Syracuse sequences starting with a positive odd integer 2*n+1=x(1) then if x(i) is odd and prime set x(i+1)=2*x(i)+1, if x(i) is odd not prime set x(i+1)=3*x(i)+1 and if x(i) is even then set x(i+1)=x(i)/2. a(n) is the index i when x(i) reaches 1 or 1163.

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A300286 Define a set of generalized Syracuse sequences starting with x(1)=2*n+1 a positive odd integer, if x(i) is odd prime set x(i+1)=67*x(i)+1, if x(i) is odd not prime set x(i+1)=3*x(i)+1 and if x(i) is even then set x(i+1)=x(i)/2. Then a(n) is the first index i > 1 at which x(i) reaches 1.

A329941 Least prime, p, such that 2p3^n - 1 and 2p3^n + 1 are twin primes.

A329920 Smallest k such that 6kA121940(n)-1 and 6kA121940(n)+1 are twin primes.

A330304 Prime numbers P such that Q=2P-1, R=4Q+1, S=6R+1, T=8S-1, U=10T+1 and V=12U-1 are all prime numbers.

A329916 Smallest k such that 6kA057130(n)-1 and 6kA057130(n)+1 are twin primes.

A329736 Smallest odd prime P such that P32^n - 1 and P32^n + 1 are twin primes.

A327581 a(1) is the smallest prime p such that 6p^2-1 and 6p^2+1 are twin primes; for n > 1, a(n) is the smallest prime q > a(n-1) such that 6q^prime(n)-1 and 6q^prime(n)+1 are twin primes or 0 if no solution exists.

A300507 Define the set of generalized Syracuse sequences starting with a positive odd integer 2n+1=x(1) then if x(i) is odd and prime set x(i+1)=2x(i)+1, if x(i) is odd not prime set x(i+1)=3*x(i)+1 and if x(i) is even then set x(i+1)=x(i)/2. a(n) is the index i when x(i) reaches 1 or 1163.

A300286 Define a set of generalized Syracuse sequences starting with x(1)=2n+1 a positive odd integer, if x(i) is odd prime set x(i+1)=67x(i)+1, if x(i) is odd not prime set x(i+1)=3*x(i)+1 and if x(i) is even then set x(i+1)=x(i)/2. Then a(n) is the first index i > 1 at which x(i) reaches 1.