A014234 -id:A014234 - cp's OEIS frontend

A373124 Sum of indices of primes between powers of 2.

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

Gus Wiseman, May 31 2024

nonn

Sum of k such that 2^n+1 <= prime(k) <= 2^(n+1).

			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:
- sum A373124 (this sequence)
- length A036378
- min A372684 (except initial terms), delta A092131
- max A007053
For primes between powers of 2:
- sum A293697
- length A036378
- min A104080 or A014210
- max A014234, delta A013603
For squarefree numbers between powers of 2:
- sum A373123
- length A077643, run-lengths of A372475
- min A372683, delta A373125, indices A372540
- max A372889, delta A373126, indices A143658
Cf. A000040, A000120, A014499, A029837, A029931, A035100, A069010, A070939, A112925, A112926, A211997.

Table[Total[PrimePi/@Select[Range[2^(n-1)+1,2^n],PrimeQ]],{n,10}]

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

A378252 Least prime power > 2^n.

Original entry on oeis.org

2, 3, 5, 9, 17, 37, 67, 131, 257, 521, 1031, 2053, 4099, 8209, 16411, 32771, 65537, 131101, 262147, 524309, 1048583, 2097169, 4194319, 8388617, 16777259, 33554467, 67108879, 134217757, 268435459, 536870923, 1073741827, 2147483659, 4294967311, 8589934609

Offset: 0

Gus Wiseman, Nov 30 2024

nonn

Prime powers are listed by A246655.

Conjecture: All terms except 9 are prime. Hence this is the same as A014210 after 9. Confirmed up to n = 1000.

Subtracting 2^n appears to give A013597 except at term 3.
For prime we have A014210.
For previous we have A014234.
For perfect power we have A357751.
For squarefree we have A372683.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, diffs A375708 and A375735.
A031218 gives the greatest prime power <= n.
A244508 counts prime powers between powers of 2.
Prime powers between primes are counted by A080101 and A366833.
Cf. A007918, A037035, A053707, A059305, A065514, A069584, A151800, A304521, A377051, A377282, A377283, A377289, A377432, A378249.

Table[NestWhile[#+1&,2^n+1,!PrimePowerQ[#]&],{n,0,20}]

a(n) = my(x=2^n+1); while (!isprimepower(x), x++); x; \\ Michel Marcus, Dec 03 2024

from itertools import count
from sympy import primefactors
def A378252(n): return next(i for i in count(1+(1<Chai Wah Wu, Dec 02 2024

A092507 (Smallest prime >= 2^n) + (largest prime <= 2^n).

Original entry on oeis.org

2, 4, 8, 18, 30, 68, 128, 258, 508, 1030, 2052, 4092, 8192, 16400, 32792, 65520, 131058, 262172, 524286, 1048596, 2097156, 4194312, 8388620, 16777210, 33554472, 67108860, 134217738, 268435446, 536870858, 1073741832, 2147483616

Offset: 0

Jorge Coveiro, Apr 05 2004

nonn

For n=0 we just take a(0)=2, the least prime >= 2^0, as there is no prime <= 2^0. - Robert Israel, Nov 01 2018

Robert Israel, Table of n, a(n) for n = 0..3317

Cf. A014210, A014234, A058249.

[2, seq( (nextprime(2^x-1)+prevprime(2^x+1)),x=1..20)]; # Corrected by Robert Israel, Nov 01 2018

PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Table[PrevPrim[2^n+1] + NextPrim[2^n-1], {n, 31}] (* Robert G. Wilson v, Apr 14 2004 *)

a(n) = A014210(n) + A014234(n) for n >= 2. - Robert Israel, Nov 01 2018

More terms from Robert G. Wilson v, Apr 14 2004

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

Michael S. Branicky, May 09 2022

A prime ladder (in base b) starts with a prime, ends with a prime, and each step produces a new prime by changing exactly one base-b digit.
A shortest such construct between two given primes is optimal.
Analogous to a word ladder (see Wikipedia link).
Here, n-digit primes do not allow leading 0 digits.
If all n-digit primes are disconnected, a(n) = 1; if there are no n-digit primes, a(n) = 0.

			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.

Wikipedia, Word Ladder
Zach Wissner-Gross, Can You Build The Longest Ladder?, Riddler Classic, May 06 2022.

Cf. A000040, A004676, A104080, A014234, A145667, A353737.

a(n) is the number of vertices of a longest shortest path in the graph G = (V, E), where V = {n-digit base-2 primes} and E = {(v, w) | H_2(v, w) = 1}, where H_b is the Hamming distance in base b.

a(23) from Michael S. Branicky, May 21 2022

A365265 Numbers k for which sqrt(k/2) divides k and the symmetric representation of sigma(k) consists of a single part and its width at the diagonal equals 1.

Original entry on oeis.org

2, 8, 18, 32, 128, 162, 200, 392, 512, 882, 968, 1352, 1458, 2048, 2178, 3042, 3872, 5000, 5202, 5408, 6498, 8192, 9248, 9522, 11552, 13122, 15138, 16928, 17298, 19208, 26912, 30752, 32768, 36992, 43218, 43808, 46208, 53792, 58482, 59168, 67712, 70688

Offset: 1

Hartmut F. W. Hoft, Aug 29 2023

nonn

Every number a(n) has the form 2^(2*i + 1) * s^2, i>= 0 and s odd, the single middle divisor of a(n) is sqrt(a(n)/2), and sqrt(2*a(n)) - 1 = floor((sqrt(8*n + 1) - 1)/2) = A003056(a(n)).
The least number in the sequence with 3 odd prime divisors is a(126) = 1630818 = 2^1 * 3^2 * 7^2 * 43^2.
Conjecture: Let a(n) = 2^(2i+1) * s^2, i>=0 and s odd, be a number in the sequence.
(1) For any odd prime divisor p of s, number a(n) * p^2 is in the sequence.
(2) For any odd prime p not a divisor of s, number a(n) * p^2 is in the sequence if p satisfies sqrt(2*a(n)) < p < 2*a(n).

			a(5) = 128 = 2^7  has 2^3 as its single middle divisor, and its symmetric representation of sigma consists of one part of width 1.
a(10) = 882 = 2 * 3^2 * 7^2 has 3 * 7 as its single middle divisor, its symmetric representation of sigma is the smallest in this sequence of maximum width 3, consists of one part, and has width 1 at the diagonal.
A table of ranges for the single odd prime factor p for numbers k in the sequence having the form 2^(2i+1) * p^(2j), i>=0 and j>0, indexed by exponent 2i+1 of 2 in number k. The lower bound is A014210(i+1) and the upper bound is A014234(2(i+1)) = A104089(i+1):
---------------------
  2i+1  /---- p ----/
---------------------
  1       3  ..    3
  3       5  ..   13
  5      11  ..   61
  7      17  ..  251
  9      37  .. 1021
...

Intersection of A361903 and A361905.
Also subsequence of the following sequences: A001105, A071562, A238443 = A174973, A319796, A320137.
The powers of 2 with an odd index (A004171) form a subsequence.
Cf. A003056, A014210, A014234, A104089, A235791, A237048, A237270, A237271, A237593, A249223, A250068.

(* a2[ ] and its support functions are defined in A249223 *)
a365265Q[n_] := Module[{list=If[Divisible[n, Sqrt[n/2]], a2[n], {0}]}, Last[list]==1&&AllTrue[list, #>0&]]
a365265[{m_, n_}] := Select[Range[m, n], a365265Q]
a365265[{1,75000}]

A375235 Records of A112591.

Original entry on oeis.org

1, 6, 12, 28, 58, 126, 252, 506, 1012, 2042, 4082, 8190, 16366, 32742, 65518, 131056, 262114, 524280, 1048554, 2097146, 4194278, 8388594, 16777208, 33554390, 67108858, 134217716, 268435396, 536870852, 1073741814, 2147483614, 4294967284, 8589934580, 17179869158

Offset: 1

Bill McEachen, Aug 06 2024

nonn

Sequence closely parallel to A000295.

			The first term of A112591 = 1 is a record and is a(1). The next A112591 value > 1 is 6 which is a(2).

Cf. A000295, A014210 (primes where records occur), A014234, A112591.

a[n_] := BitXor @@ NextPrime[2^n, {-1, 1}]; a[1] = 1; Array[a, 33] (* Amiram Eldar, Aug 08 2024 *)

a(n)= if(n==1,1,bitxor(precprime(2^n), nextprime(2^n) ))

a(n) = previous_prime(2^n) XOR next_prime(2^n) = A112591(A014234(n)) for n > 1.

More terms from Amiram Eldar, Aug 06 2024

A128945 Numbers n such that the greatest prime < 2^n is a twin prime member.

Original entry on oeis.org

2, 3, 4, 5, 6, 10, 12, 16, 20, 149, 150, 476, 594, 788, 1574, 1664, 1691, 6117, 6242

Offset: 1

Cino Hilliard, Apr 28 2007

A014234(a(n)) is in A001097 (twin primes).

			For n=5, 31 is the greatest prime < 2^5 and is a member of the twin prime pair 29, 31.

Select[Range[1700],AnyTrue[NextPrime[2^#,-1]+{2,-2},PrimeQ]&] (* The program generates the first 17 terms of the sequence. *) (* Harvey P. Dale, Mar 26 2025 *)

g(n,b)=for(x=1,n,y=precprime(b^x);if(ispseudoprime(y-2),print1(x",")))

Edited by and terms a(15)-a(17) from Ray Chandler, May 12 2007

a(18), a(19) Donovan Johnson, Feb 21 2008

A334150 Primes p such that p AND q = 1, where q is the next prime after p and AND is the bitwise operation.

Original entry on oeis.org

3, 13, 61, 251, 4093, 32749, 65521, 8388593, 33554393, 1073741789, 137438953447, 9007199254740881, 36028797018963913, 144115188075855859, 147573952589676412909, 37778931862957161709471, 75557863725914323419121, 2417851639229258349412301, 4835703278458516698824647

Offset: 1

Michel Marcus, Apr 16 2020

Cf. A175330.
Subsequence of A014234 (largest prime <= 2^n).
Cf. A214415 (exponents of corresponding powers of 2).

s = {}; p = 2; Do[q = NextPrime[p]; If[BitAnd[p, q] == 1, AppendTo[s, p]]; p = q, {10^5}]; s (* Amiram Eldar, Apr 16 2020 *)
Select[ NextPrime[ 2^Range[82], -1], BitAnd[#, NextPrime@ #] == 1 &] (* Giovanni Resta, Apr 16 2020 *)

isok(p) = isprime(p) && (bitand(p, nextprime(p+1)) == 1);

a(9)-a(10) from Amiram Eldar, Apr 16 2020

a(11)-a(19) from Giovanni Resta, Apr 16 2020

cp's OEIS Frontend

A373124 Sum of indices of primes between powers of 2.

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A378252 Least prime power > 2^n.

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A092507 (Smallest prime >= 2^n) + (largest prime <= 2^n).

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A353738 Length of longest n-digit optimal prime ladder (base 2).

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A365265 Numbers k for which sqrt(k/2) divides k and the symmetric representation of sigma(k) consists of a single part and its width at the diagonal equals 1.

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A375235 Records of A112591.

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A128945 Numbers n such that the greatest prime < 2^n is a twin prime member.

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Mathematica

PARI

Extensions

A334150 Primes p such that p AND q = 1, where q is the next prime after p and AND is the bitwise operation.

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