A094076 -id:A094076 - cp's OEIS frontend

A103148 Numbers n such that 2^n+25229 is prime.

3, 5, 7, 9, 11, 19, 21, 29, 33, 41, 77, 81, 99, 101, 109, 119, 141, 145, 161, 163, 171, 183, 201, 209, 227, 241, 299, 303, 321, 367, 395, 413, 459, 501, 689, 777, 889, 989, 1317, 1839, 2027, 2197, 2571, 3041, 3143, 4541, 4701, 5265, 5463, 6449, 7061, 7289, 9291, 9663, 12287, 17441, 23033, 23297, 26543, 32939, 33543

Offset: 1

Lei Zhou, Feb 03 2005

nonn

This is the longest sequence that I have found of the class "2^n+/-(odd integer) is prime".

			2^3+25229 = 25237 is prime
2^11+25229 = 27277 is prime
2^2+25229 = 25233 = 3x13x647

Cf. A094076, A000043.

Do[If[PrimeQ[2^n + 25229], Print[n]], {n, 33600}]

is(n)=ispseudoprime(2^n+25229) \\ Charles R Greathouse IV, Jun 13 2017

A232068 Least k such that prime(n) + 2^(k+L) is a prime, where L is the length of binary representation of prime(n): L = A070939(A000040(n)). a(n)=-1 if no such k exists.

Original entry on oeis.org

0, 0, 1, 1, 0, 8, 1, 2, 0, 7, 0, 5, 0, 203, 1, 3, 2, 3, 0, 3, 3, 0, 2, 1, 0, 1, 2, 7, 0, 1, 1, 5, 2, 1, 8, 2, 0, 501, 3, 1, 8, 3, 0, 1, 2, 0, 0, 1, 22, 3, 1, 4, 5, 0, 2, 4, 11, 1, 6, 1, 2, 5, 0, 3, 0, 7, 1, 0, 1, 18, 8, 3, 13, 5, 6, 2, 3, 34, 1, 2, 3, 4, 19, 5, 6, 4, 1

Offset: 2

Alex Ratushnyak, Nov 17 2013

Prime(n) is in A065047 if and only if a(n)=0.

Cf. A000040, A070939, A065047, A094076.

A232084 Least k such that prime(n) + 2^(k+L) - 2^L is a prime, where L is the length of binary representation of prime(n): L = A070939(A000040(n)). a(n) = -1 if no such k exists.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 4, 4, 1, 2, 1, 2, 1, 2, 4, 2, 3, 4, 1, 2, 2, 1, 5, 4, 1, 2, 2, 4, 1, 6, 18, 20, 2, 4, 2, 3, 1, 4, 2, 2, 3, 6, 1, 12, 2, 1, 1, 96, 2, 4, 4, 2, 2, 1, 3, 3, 4, 6, 6, 4, 3, 6, 1, 4, 1, 2, 2, 1, 56, 2, 3, 8, 4, 4, 3, 4, 2, 4, 4, 3, 4, 4, 18, 20, 2, 8, 2, 2

Offset: 2

Alex Ratushnyak, Nov 17 2013

Least number of 1's that must be prepended to the binary representation of prime(n) such that the result is another prime.

Prime(n) is in A065047 if and only if a(n) = 1.

			a(6) = 1 because 13 in binary is 1101, and 29 (11101 in binary) is a prime.
a(7) = 2 because 17 in binary is 10001, and 113 (1110001 in binary) is a prime.
a(8) = 4 because 19 in binary is 10011, and 499 (111110011 in binary) is a prime.

Cf. A000040, A070939, A065047, A094076, A023758.

A307715 Decimal expansion of Sum_{t>0} log((t + 1)/t)^2.

Original entry on oeis.org

9, 7, 7, 1, 8, 9, 1, 8, 3, 2, 6, 8, 9, 3, 6, 5, 5, 4, 4, 5, 7, 8, 8, 5, 7, 4, 9, 4, 7, 6, 4, 3, 4, 7, 4, 8, 0, 7, 7, 3, 9, 2, 5, 0, 6, 4, 7, 4, 7, 2, 3, 9, 0, 1, 7, 7, 0, 2, 0, 9, 8, 9, 7, 5, 5, 3, 1, 8, 4, 4, 5, 2, 9, 3, 9, 2, 3, 9, 3, 3, 5, 6, 2, 9, 0, 1, 2, 3, 2, 1, 0, 7, 9, 7, 4, 3, 2, 0, 3, 3, 5, 9, 2, 3, 2

Offset: 0

Stefano Spezia, Apr 24 2019

This constant appears at several places in the literature:
1) In the asymptotic formula of the number of minimal covering systems with exactly n elements (see Theorem 1.1 in Balister, Bollobás, Morris, Sahasrabudhe and Tiba) and
2) in the maximal size of the iterated divisor function
(see Theorem 1 in Buttkewitz, Elsholtz, Ford and Schlage-Puchta) and
3) in the maximal order of the iterated r_2 function, which counts the number of representations as sums of 2 squares (see Theorems 2.1. and 2.3 in Elsholtz, M. Technau and N. Technau). - Modified by C. Elsholtz, Apr 15 2025

			0.9771891832689365544578857494764347480773925064747239017702...

P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe and M. Tiba, The structure and number of Erdős covering systems, arXiv:1904.04806 [math.CO], 2019. J. Eur. Math. Soc. 26, 75-109 (2024).
Y. Buttkewitz, C. Elsholtz, K. Ford, and J.-C. Schlage-Puchta, The maximal order of iterated multiplicative functions, arxiv:1108.1815 [math.NT], 2011. IMRN issue 17, 4051-4061 (2012).
C. Elsholtz, M. Technau, and N. Technau, The maximal order of iterated multiplicative functions, arxiv:1709.04799 [math.NT], 2017. Mathematika, 65 (4) 2019, 990-1009.
P. Erdős, Egy kongruenciarendszerekrol szóló problémáról, (On a problem concerning congruence-systems, in Hungarian), Mat. Lapok, 4 (1952), 122-128.

Cf. A001008, A002805, A010553, A080340, A094076, A275489, A294593, A296195, A335831.

First[RealDigits[NSum[(Log[(t + 1)/t])^2, {t, 1, Infinity}, NSumTerms -> 100, Method -> {"NIntegrate", "MaxRecursion" -> 10}, WorkingPrecision -> 100]]]

sumpos(t=1, log((t + 1)/t)^2) \\ Michel Marcus, Apr 26 2019

From Amiram Eldar, Jun 17 2023: (Start)
Equals 2 * Sum_{k>=1} H(k) * (zeta(k+1)-1) / (k+1), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
Equals -Sum_{k>=1} zeta'(2*k) / k. (End)

A361744 A(n,k) is the least m such that there are k primes in the set {prime(n) + 2^i | 1 <= i <= m}, or -1 if no such number exists; square array A(n,k), n > 1, k >= 1, read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 3, 3, 2, 4, 5, 4, 1, 6, 11, 6, 3, 2, 7, 47, 8, 5, 4, 1, 12, 53, 10, 7, 8, 13, 2, 15, 141, 16, 9, 20, 21, 6, 3, 16, 143, 18, 15, 38, 33, 30, 7, 1, 18, 191, 20, 23, 64, 81, 162, 39, 3, 4, 28, 273, 28, 29, 80, 129, 654, 79, 5, 12, 2

Offset: 2

Jean-Marc Rebert, Mar 22 2023

			p = prime(2) = 3, m=1, u = {p + 2^k | 1 <= k <= m} = {5} contains one prime, and no lesser m satisfies this, so A(2,1) = 1.
Square array A(n,k) n > 1 and k >= 1 begins:
 1,     2,     3,     4,     6,     7,    12,    15,    16,    18, ...
 1,     3,     5,    11,    47,    53,   141,   143,   191,   273, ...
 2,     4,     6,     8,    10,    16,    18,    20,    28,    30, ...
 1,     3,     5,     7,     9,    15,    23,    29,    31,    55, ...
 2,     4,     8,    20,    38,    64,    80,   292,  1132,  4108, ...
 1,    13,    21,    33,    81,   129,   285,   297,   769,  3381, ...
 2,     6,    30,   162,   654,   714,  1370,  1662,  1722,  2810, ...
 3,     7,    39,    79,   359,   451,  1031,  1039, 11311, 30227, ...
 1,     3,     5,     7,     9,    13,    15,    17,    23,    27, ...

Cf. A057732 (1st row), A094076 (1st column).
Cf. A361679.
Cf. A019434 (primes 2^n+1), A057732 (2^n+3), A059242 (2^n+5), A057195 (2^n+7), A057196(2^n+9), A102633 (2^n+11), A102634 (2^n+13), A057197 (2^n+15), A057200 (2^n+17), A057221 (2^n+19), A057201 (2^n+21), A057203 (2^n+23).
Cf. A205558 and A231232 (with 2*k instead of 2^k).

A(n, k)= {my(nb=0, p=prime(n), m=1); while (nb

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A103148 Numbers n such that 2^n+25229 is prime.

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A232068 Least k such that prime(n) + 2^(k+L) is a prime, where L is the length of binary representation of prime(n): L = A070939(A000040(n)). a(n)=-1 if no such k exists.

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A232084 Least k such that prime(n) + 2^(k+L) - 2^L is a prime, where L is the length of binary representation of prime(n): L = A070939(A000040(n)). a(n) = -1 if no such k exists.

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A307715 Decimal expansion of Sum_{t>0} log((t + 1)/t)^2.

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A361744 A(n,k) is the least m such that there are k primes in the set {prime(n) + 2^i | 1 <= i <= m}, or -1 if no such number exists; square array A(n,k), n > 1, k >= 1, read by antidiagonals.

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