Displaying 1-10 of 49 results found.
Numbers k such that 2k-1 is prime.
+0
85
2, 3, 4, 6, 7, 9, 10, 12, 15, 16, 19, 21, 22, 24, 27, 30, 31, 34, 36, 37, 40, 42, 45, 49, 51, 52, 54, 55, 57, 64, 66, 69, 70, 75, 76, 79, 82, 84, 87, 90, 91, 96, 97, 99, 100, 106, 112, 114, 115, 117, 120, 121, 126, 129, 132, 135, 136, 139, 141, 142, 147, 154, 156, 157
COMMENTS
The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004 Positions of prime numbers among odd numbers. - Zak Seidov, Mar 26 2013 Also, the integers remaining after removing every third integer following 2, and, recursively, removing every p-th integer following the next remaining entry (where p runs through the primes, beginning with 5). - Pete Klimek, Feb 10 2014 Also, numbers k such that k^2 = m^2 + p, for some integers m and every prime p > 2. Applicable m values are m = k - 1 (giving p = 2k - 1). Less obvious is: no solution exists if m equals any value in A047845, which is the complement of ( A006254 - 1). - Richard R. Forberg, Apr 26 2014 If you define a different type of multiplication (*) where x (*) y = x * y + (x - 1) * (y - 1), (which has the commutative property) then this is the set of primes that follows. - Jason Atwood, Jun 16 2019
PROG
(Python)
from sympy import prime
Numbers k such that 2*k + 3 is a prime.
+0
65
0, 1, 2, 4, 5, 7, 8, 10, 13, 14, 17, 19, 20, 22, 25, 28, 29, 32, 34, 35, 38, 40, 43, 47, 49, 50, 52, 53, 55, 62, 64, 67, 68, 73, 74, 77, 80, 82, 85, 88, 89, 94, 95, 97, 98, 104, 110, 112, 113, 115, 118, 119, 124, 127, 130, 133, 134, 137, 139, 140, 145, 152, 154, 155
COMMENTS
The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004 n is in the sequence iff none of the numbers (n-3k)/(2k+1), 1 <= k <= (n-1)/5, is positive integer. - Vladimir Shevelev, May 31 2009
MAPLE
select(t -> isprime(2*t+3), [$0..1000]); # Robert Israel, Feb 19 2015
MATHEMATICA
Select[Range[0, 200], PrimeQ[2#+3]&] (* Harvey P. Dale, Jun 10 2014 *)
PROG
(PARI) [k | k<-[0..99], isprime(2*k+3)] \\ for illustration
(Sage) [n for n in (0..200) if is_prime(2*n+3) ] # G. C. Greubel, May 21 2019 (GAP) Filtered([0..200], k-> IsPrime(2*k+3) ) # G. C. Greubel, May 21 2019
CROSSREFS
Numbers n such that 2n+k is prime: A005097 (k=1), this seq(k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19). - Jason Kimberley, Sep 07 2012
EXTENSIONS
Offset changed from 0 to 1 in 2008: some formulas here and elsewhere may need to be corrected.
a(n) = p*(p-1)/2 for p = prime(n).
+0
30
1, 3, 10, 21, 55, 78, 136, 171, 253, 406, 465, 666, 820, 903, 1081, 1378, 1711, 1830, 2211, 2485, 2628, 3081, 3403, 3916, 4656, 5050, 5253, 5671, 5886, 6328, 8001, 8515, 9316, 9591, 11026, 11325, 12246, 13203, 13861, 14878, 15931, 16290, 18145, 18528, 19306
COMMENTS
Whereas A034953 is the sequence of triangular numbers with prime indices, this is the sequence of triangular numbers with numbers one less than primes for indices. - Alonso del Arte, Aug 17 2014 a(n) is both the number of quadratic residues and the number of nonresidues modulo prime(n)^2 that are coprime to prime(n).
For k coprime to prime(n), k^a(n) == +-1 (mod prime(n)^2). (End)
FORMULA
a(n) = (phi(prime(n))^2 + phi(prime(n)))/2, where phi(n) is Euler's totient function, A000010. - Alonso del Arte, Aug 22 2014
MAPLE
a:= n-> (p-> p*(p-1)/2)(ithprime(n)):
PROG
(Magma) [ (k-1)*k/2 where k is NthPrime(n): n in [1..44] ]; // Klaus Brockhaus, Nov 18 2008 (PARI) { n=0; forprime (p=2, prime(1000), write("b008837.txt", n++, " ", p*(p - 1)/2) ) } \\ Harry J. Smith, Jul 25 2009
Length of period of continued fraction for sqrt(prime(n)).
+0
18
1, 2, 1, 4, 2, 5, 1, 6, 4, 5, 8, 1, 3, 10, 4, 5, 6, 11, 10, 8, 7, 4, 2, 5, 11, 1, 12, 6, 15, 9, 12, 6, 9, 18, 9, 20, 17, 18, 4, 5, 14, 21, 16, 13, 1, 20, 26, 4, 2, 5, 11, 12, 17, 14, 1, 12, 3, 24, 21, 13, 18, 5, 14, 16, 17, 11, 34, 19, 14, 7, 15, 4, 20, 5, 30, 8, 9, 21, 1, 21, 18, 37, 16
COMMENTS
The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004 Note that primes of the form n^2+1 ( A002496) have a continued fraction whose period length is 1; odd primes of the form n^2+2 ( A056899) have length 2; odd primes of the form n^2-2 ( A028871) have length 4. - T. D. Noe, Nov 03 2006 For an odd prime p, the length of the period is odd if p=1 (mod 4) or even if p=3 (mod 4). - T. D. Noe, May 22 2007
MAPLE
with(numtheory): for i from 1 to 150 do cfr := cfrac(ithprime(i)^(1/2), 'periodic', 'quotients'); printf(`%d, `, nops(cfr[2])) od:
MATHEMATICA
Table[p=Prime[n]; Length[Last[ContinuedFraction[Sqrt[p]]]], {n, 100}] (* T. D. Noe, May 22 2007 *) Length[ContinuedFraction[Sqrt[#]][[2]]]&/@Prime[Range[100]] (* Harvey P. Dale, Sep 28 2024 *)
CROSSREFS
Cf. A003285, A130272 (primes at which the period length sets a new record).
Product of three numbers: n-th prime, previous number, and following number.
+0
15
6, 24, 120, 336, 1320, 2184, 4896, 6840, 12144, 24360, 29760, 50616, 68880, 79464, 103776, 148824, 205320, 226920, 300696, 357840, 388944, 492960, 571704, 704880, 912576, 1030200, 1092624, 1224936, 1294920, 1442784, 2048256, 2247960, 2571216, 2685480, 3307800
COMMENTS
a(n) is the order of the matrix group SL(2,prime(n)). - Tom Edgar, Sep 28 2015
FORMULA
a(n) = prime(n)*(prime(n)^2-1). - Tom Edgar, Sep 28 2015 Product_{n>=1} (1 + 1/a(n)) = A065487. Product_{n>=1} (1 - 1/a(n)) = A065470. (End)
MATHEMATICA
Table[(Prime[n] + 1) Prime[n](Prime[n] - 1), {n, 1, 100}]
PROG
(PARI) a(n) = prime(n)*(prime(n)^2-1);
(Magma) [6] cat [NthPrime(n)*(NthPrime(n)^2-1): n in [2..40]]; // Vincenzo Librandi, Sep 29 2015
Product of a prime and the following number.
+0
14
6, 12, 30, 56, 132, 182, 306, 380, 552, 870, 992, 1406, 1722, 1892, 2256, 2862, 3540, 3782, 4556, 5112, 5402, 6320, 6972, 8010, 9506, 10302, 10712, 11556, 11990, 12882, 16256, 17292, 18906, 19460, 22350, 22952, 24806, 26732, 28056, 30102
COMMENTS
1/a(n) is the asymptotic density of numbers whose prime(n)-adic valuation is positive and even. - Amiram Eldar, Jan 23 2021
FORMULA
a(n) = prime(n)*(prime(n)+1).
Product_{n>=1} (1 + 1/a(n)) = zeta(2)/zeta(3) ( A306633). Product_{n>=1} (1 - 1/a(n)) = A065463. (End)
EXAMPLE
a(3)=30 because prime(3)=5 and prime(3)+1=6, hence 5*6 = 30.
MATHEMATICA
Table[(Prime[n] + 1) Prime[n], {n, 1, 100}] (* Artur Jasinski, Feb 06 2007 *)
2, 5, 13, 29, 47, 73, 107, 151, 197, 257, 317, 397, 467, 571, 659, 769, 883, 1019, 1151, 1291, 1453, 1607, 1783, 1987, 2153, 2371, 2593, 2791, 3037, 3307, 3541, 3797, 4073, 4357, 4657, 4973, 5303, 5641, 5939, 6301, 6679, 7019, 7477
COMMENTS
There are n distinct successive primes p (not appearing in the sequence) such that a(n) < p < a(n+1). - David James Sycamore, Jul 22 2018
FORMULA
a(n) is asymptotic to (n*(n+1)/2) * log(n*(n+1)/2) = (n*(n+1)/2) * (log(n)+log(n+1)-log(2)) ~ (n^2 + n)*(2 log n)/2 ~ (n^2 + n)*(log n). - Jonathan Vos Post, Mar 12 2006
PROG
(Haskell)
a011756 n = a011756_list !! (n-1)
a011756_list = map a000040 $ tail a000217_list
CROSSREFS
Primes in leading diagonal of triangle in A078721.
Half of product of three numbers: n-th prime, previous and following number.
+0
9
3, 12, 60, 168, 660, 1092, 2448, 3420, 6072, 12180, 14880, 25308, 34440, 39732, 51888, 74412, 102660, 113460, 150348, 178920, 194472, 246480, 285852, 352440, 456288, 515100, 546312, 612468, 647460, 721392, 1024128, 1123980, 1285608
COMMENTS
Except the first term, a(n) is the area of the integer-sided isosceles triangle ABC with AB=AC such that the altitude AH is of prime(n) length.
The couples (a(n), altitude) are (12,3), (60,5), (168,7), (660,11), (1092,13), ... and the sequence of the ratio a(n)/prime(n) is {4, 12, 24, 60, 84, 144, 180, ...} - see A084921. - Michel Lagneau, Oct 23 2013 a(n) is also equal to the number of reducible quadratic polynomials in the field of size prime(n). - James East, Apr 26 2024
MATHEMATICA
Table[(Prime[n] + 1) Prime[n](Prime[n] - 1)/2, {n, 1, 100}]
PROG
(Magma) [(NthPrime(n)+1)*NthPrime(n)*(NthPrime(n)-1)/2: n in [1..40]]; // Vincenzo Librandi, Apr 09 2017
1/3 of product of three numbers: the n-th prime, the previous number and the following number.
+0
9
2, 8, 40, 112, 440, 728, 1632, 2280, 4048, 8120, 9920, 16872, 22960, 26488, 34592, 49608, 68440, 75640, 100232, 119280, 129648, 164320, 190568, 234960, 304192, 343400, 364208, 408312, 431640, 480928, 682752, 749320, 857072, 895160, 1102600
COMMENTS
Number of irreducible monic cubic polynomials over GF(prime(n)). - Robert Israel, Jan 06 2015
MAPLE
seq((ithprime(n)^3 - ithprime(n))/3, n=1..100); # Robert Israel, Jan 06 2015
MATHEMATICA
Table[(Prime[n] + 1) Prime[n] (Prime[n] - 1)/3, {n, 100}]
Difference between the sum of next prime(n) natural numbers and the sum of next n primes.
+0
8
1, 4, 9, 10, 54, 71, 191, 236, 446, 1025, 1310, 2259, 3245, 3820, 5048, 7321, 10060, 11473, 15328, 18358, 20381, 25672, 30222, 36561, 46367, 53031, 58108, 65444, 70971, 78391, 104184, 116542, 133095, 142728, 169931, 181324, 203429, 226622
COMMENTS
Group the natural numbers with prime(n) elements in each group. (1,2),(3,4,5),(6,7,8,9,10),(11,12,13,14,15,16,17),... The sum corresponding the groups is 3,12,40,98,... Group the prime numbers such that the n-th group contains n primes. (2),(3,5),(7,11,13),(17,19,23,29),... The sum corresponding the groups is 2,8,31,88,... The required difference is 1,4,9,10,...
The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004
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