Displaying 1-10 of 49 results found.
3, 9, 24, 52, 118, 209, 362, 552, 828, 1263, 1759, 2462, 3323, 4269, 5397, 6828, 8598, 10489, 12767, 15323, 18024, 21184, 24670, 28675, 33428, 38579, 43935, 49713, 55708, 62149, 70277, 78923, 88376, 98106, 109281, 120757, 133160, 146526, 160554
EXAMPLE
a(3)=T(2)+T(3)+T(5)=3+6+15=24
MATHEMATICA
Accumulate[(#(#+1))/2&/@Prime[Range[50]]] (* Harvey P. Dale, Dec 01 2015 *)
PROG
(PARI) s=0; forprime(p=2, 300, s+=t(p); print1(s", "))
Numbers k such that 2k-1 is prime.
+10
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.
+10
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).
+10
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)).
+10
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.
+10
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.
+10
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.
+10
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.
+10
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}]
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