A373124
Sum of indices of primes between powers of 2.
Original entry on oeis.org
1, 2, 7, 11, 45, 105, 325, 989, 3268, 10125, 33017, 111435, 369576, 1277044, 4362878, 15233325, 53647473, 189461874, 676856245, 2422723580, 8743378141, 31684991912, 115347765988, 421763257890, 1548503690949, 5702720842940, 21074884894536, 78123777847065
Offset: 0
Row-sums of the sequence of all positive integers as a triangle with row-lengths A036378:
1
2
3 4
5 6
7 8 9 10 11
12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27 28 29 30 31
32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
For indices of primes between powers of 2:
For primes between powers of 2:
For squarefree numbers between powers of 2:
Cf.
A000040,
A000120,
A014499,
A029837,
A029931,
A035100,
A069010,
A070939,
A112925,
A112926,
A211997.
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Table[Total[PrimePi/@Select[Range[2^(n-1)+1,2^n],PrimeQ]],{n,10}]
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ip(n) = primepi(1<A007053
t(n) = n*(n+1)/2; \\ A000217
a(n) = t(ip(n+1)) - t(ip(n)); \\ Michel Marcus, May 31 2024
A353738
Length of longest n-digit optimal prime ladder (base 2).
Original entry on oeis.org
0, 2, 2, 1, 5, 3, 7, 5, 15, 15, 19, 24, 39, 48, 35, 64, 57, 51, 59, 61, 67, 61, 61
Offset: 1
There are no 1-digit primes in base 2, so a(1) = 0.
The 2-digit optimal prime ladder 10 - 11 is tied for the longest amongst 2-digit primes in binary, so a(2) = 2.
The 3-digit optimal prime ladder 101 - 111 is tied for the longest amongst 3-digit primes in binary, so a(3) = 2.
The only 4-digit primes in binary, 1011 and 1101, are disconnected, so a(3) = 1.
The 5-digit optimal prime ladder 10001 - 10011 - 10111 - 11111 - 11101 is tied for the longest amongst 5-digit primes in binary, so a(5) = 5.
A367966
Smallest Sophie Germain prime >= 2^n.
Original entry on oeis.org
2, 2, 5, 11, 23, 41, 83, 131, 281, 593, 1031, 2063, 4211, 8243, 16421, 32771, 65633, 131321, 262193, 524351, 1048889, 2097629, 4194581, 8388953, 16777259, 33554771, 67108913, 134218433, 268435631, 536871311, 1073741891, 2147483693, 4294967681, 8589934631, 17179869659
Offset: 0
For n = 0, a(0) = 2 because 2 is prime, 2*(2) + 1 = 5 is prime, 2 >= 2^0 where 2^0 = 1, and 1 is not prime.
For n = 1, a(1) = 2 because 2 is prime, 2*(2) + 1 = 5 is prime, 2 >= 2^1 where 2^1 = 2.
For n = 2, a(2) = 5 because 5 is prime, 2*(5) + 1 = 11 is prime, 5 >= 2^2 where 2^2 = 4, and 4 is not prime.
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a:= proc(n) option remember; local p; for p from 2^n
while not andmap(isprime, [p, 2*p+1]) do od; p
end:
seq(a(n), n=0..44); # Alois P. Heinz, Dec 13 2023
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a={}; nmax=35; For[n=0, n<=nmax, n++, k=2^n; While[!PrimeQ[k] || !PrimeQ[2k+1], k++]; AppendTo[a,k]]; a (* Stefano Spezia, Dec 10 2023 *)
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a(n) = forprime(p=2^n, , if (isprime(2*p+1), return(p))); \\ Michel Marcus, Dec 12 2023
A340959
Table read by antidiagonals of the smallest prime >= n^k, n >= 1 and k >= 0.
Original entry on oeis.org
2, 2, 2, 2, 2, 2, 2, 3, 5, 2, 2, 5, 11, 11, 2, 2, 5, 17, 29, 17, 2, 2, 7, 29, 67, 83, 37, 2, 2, 7, 37, 127, 257, 251, 67, 2, 2, 11, 53, 223, 631, 1031, 733, 131, 2, 2, 11, 67, 347, 1297, 3137, 4099, 2203, 257, 2, 2, 11, 83, 521, 2411, 7789, 15629, 16411, 6563
Offset: 1
Table begins:
2, 2, 2, 2, 2, 2, ...
2, 2, 5, 11, 17, 37, ...
2, 3, 11, 29, 83, 251, ...
2, 5, 17, 67, 257, 1031, ...
2, 5, 29, 127, 631, 3137, ...
...;
yielding the triangle:
2;
2, 2;
2, 2, 2;
2, 3, 5, 2;
2, 5, 11, 11, 2;
2, 5, 17, 29, 17, 2;
...
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T[n_,k_]:=NextPrime[n^k-1];Flatten[Table[T[n-k,k],{n,11},{k,0,n-1}]] (* Stefano Spezia, Feb 01 2021 *)
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T(n,k) = nextprime(n^k); \\ Michel Marcus, Feb 01 2021
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