A034953 -id:A034953 - cp's OEIS frontend

A085739 Partial sums of A034953(n).

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

Offset: 1

Jon Perry, Jul 21 2003

nonn

			a(3)=T(2)+T(3)+T(5)=3+6+15=24

Cf. A034953.

Accumulate[(#(#+1))/2&/@Prime[Range[50]]] (* Harvey P. Dale, Dec 01 2015 *)

s=0; forprime(p=2,300,s+=t(p); print1(s","))

Corrected by T. D. Noe, Oct 25 2006

A006254 Numbers k such that 2k-1 is prime.

Original entry on oeis.org

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

Offset: 1

Marc LeBrun

a(n) is the inverse of 2 modulo prime(n) for n >= 2. - Jean-François Alcover, May 02 2017
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

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000040, A067076, A086801.
Equals A005097 + 1. A130291 is an essentially identical sequence.
Cf. A065091.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (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).
Numbers n such that 2n-k is prime: this seq(k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

[n: n in [0..1000] | IsPrime(2*n-1)]; // Vincenzo Librandi, Nov 18 2010

Rest@Prime@Range@70/2 + 1/2 (* Robert G. Wilson v, Jun 16 2006 *)
Select[Range[200], PrimeQ[2#-1]&] (* Harvey P. Dale, Apr 06 2014 *)

a(n)=prime(n+1)\2+1 \\ Charles R Greathouse IV, Mar 20 2013

from sympy import prime
def A006254(n): return prime(n+1)+1>>1 # Chai Wah Wu, Aug 02 2024

a(n) = (A000040(n+1) + 1)/2 = A067076(n-1) + 2 = A086801(n-1)/2 + 2.
a(n) = (1 + A065091(n))/2. - Omar E. Pol, Nov 10 2007
a(n) = sqrt((A065091^2 + 2*A065091+1)/4). - Eric Desbiaux, Jun 29 2009
a(n) = A111333(n+1). - Jonathan Sondow, Jan 20 2016

More terms from Erich Friedman

More terms from Omar E. Pol, Nov 10 2007

A067076 Numbers k such that 2*k + 3 is a prime.

Original entry on oeis.org

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

Offset: 1

David Williams, Aug 17 2002

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
Zeta(s) = Sum_{n>=1} 1/n^s = 1/1 - 2^(-s) * Product_{p=prime=(2*A067076)+3} 1/(1 - (2*A067076+3)^(-s)). - Eric Desbiaux, Dec 15 2009
This sequence is a subsequence of A047949. - Jason Kimberley, Aug 30 2012

Harry J. Smith, Table of n, a(n) for n = 1..1000
Mutsumi Suzuki, Vincenzo Librandi's method for sequential primes (Librandi's description in Italian).

Cf. A086801, A047949.
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
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Filtered([0..200], k-> IsPrime(2*k+3) ); # G. C. Greubel, May 21 2019

[n: n in [0..200]| IsPrime(2*n+3)]; // Vincenzo Librandi, Feb 23 2012

select(t -> isprime(2*t+3), [$0..1000]); # Robert Israel, Feb 19 2015

(Prime[Range[100]+1]-3)/2  (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008, modified by G. C. Greubel, May 21 2019 *)
Select[Range[0,200],PrimeQ[2#+3]&] (* Harvey P. Dale, Jun 10 2014 *)

[k | k<-[0..99], isprime(2*k+3)] \\ for illustration

A067076(n) = (prime(n+1)-3)/2 \\ M. F. Hasler, Feb 14 2024

[n for n in (0..200) if is_prime(2*n+3) ] # G. C. Greubel, May 21 2019

a(n) = A006254(n) - 2 = A086801(n+1)/2. [Corrected by M. F. Hasler, Feb 14 2024]
a(n) = A089253(n) - 4. - Giovanni Teofilatto, Dec 14 2003
Conjecture: a(n) = A008507(n) + n - 1 = A005097(n) - 1 = A102781(n+1) - 1. - R. J. Mathar, Jul 07 2009
a(n) = A179893(n) - A000040(n). - Odimar Fabeny, Aug 24 2010

Offset changed from 0 to 1 in 2008: some formulas here and elsewhere may need to be corrected.

A008837 a(n) = p*(p-1)/2 for p = prime(n).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Ken Levasseur

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
From Jianing Song, Apr 13 2019: (Start)
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)

Harry J. Smith, Table of n, a(n) for n = 1..1000

Half the terms of A036689.
Cf. A000217 (triangular numbers), A112456 (least triangular number divisible by n-th prime). - Klaus Brockhaus, Nov 18 2008
Cf. A000010, A000040, A072230, A034953, A271780.
Column 1 of A257253. (Row 1 of A257254).

[ (k-1)*k/2 where k is NthPrime(n): n in [1..44] ]; // Klaus Brockhaus, Nov 18 2008

a:= n-> (p-> p*(p-1)/2)(ithprime(n)):
seq(a(n), n=1..65);  # Alois P. Heinz, Apr 20 2022

Table[Prime[n] * (Prime[n] - 1)/2, {n, 22}] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Table[Binomial[Prime[n], 2], {n, 40}] (* Alonso del Arte, Aug 22 2014, based on the formula from Enrique Pérez Herrero *)
(#(#-1))/2&/@Prime[Range[50]] (* Harvey P. Dale, Oct 02 2019 *)

{ n=0; forprime (p=2, prime(1000), write("b008837.txt", n++, " ", p*(p - 1)/2) ) } \\ Harry J. Smith, Jul 25 2009

(define (A008837 n) (/ (A036689 n) 2)) ;; Antti Karttunen, May 01 2015

a(n) = binomial(prime(n), 2) = A000217(A000040(n)-1). - Enrique Pérez Herrero, Dec 10 2011
a(n) = (1/2)*A072230(A000040(n)). - L. Edson Jeffery, Apr 07 2012
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
a(n) = A036689(n)/2. - Antti Karttunen, May 01 2015
Product_{n>=2} (1 - 1/a(n)) = A271780. - Amiram Eldar, Nov 22 2022

Offset changed from 2 to 1 by Harry J. Smith, Jul 25 2009

A054269 Length of period of continued fraction for sqrt(prime(n)).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, May 05 2000

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

T. D. Noe, Table of n, a(n) for n = 1..10000
A. I. Gliga, On continued fractions of the square root of prime numbers

Cf. A003285, A130272 (primes at which the period length sets a new record).

with(numtheory): for i from 1 to 150 do cfr := cfrac(ithprime(i)^(1/2), 'periodic','quotients'); printf(`%d,`, nops(cfr[2])) od:

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 *)

More terms from James Sellers, May 05 2000

A036690 Product of a prime and the following number.

Original entry on oeis.org

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

Offset: 1

Felice Russo

The infinite sum over the reciprocals is given in A179119. - Wolfdieter Lang, Jul 10 2019

1/a(n) is the asymptotic density of numbers whose prime(n)-adic valuation is positive and even. - Amiram Eldar, Jan 23 2021

			a(3)=30 because prime(3)=5 and prime(3)+1=6, hence 5*6 = 30.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A036689, A034953, A065463, A179119, A306633.

[p^2+p: p in PrimesUpTo(250)]; // Vincenzo Librandi, Dec 19 2010

Table[(Prime[n] + 1) Prime[n], {n, 1, 100}] (* Artur Jasinski, Feb 06 2007 *)

a(n)=my(p=prime(n)); p*(p+1) \\ Charles R Greathouse IV, Mar 27 2020

a(n) = prime(n)*(prime(n)+1).
a(n) = A060800(n) - 1.
a(n) = 2*A034953(n). - Artur Jasinski, Feb 06 2007
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(2)/zeta(3) (A306633).
Product_{n>=1} (1 - 1/a(n)) = A065463. (End)
Sum_{n>=1} 1/a(n) = A179119. - R. J. Mathar, Mar 31 2025

A127917 Product of three numbers: n-th prime, previous number, and following number.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Feb 06 2007

a(n) is the order of the matrix group SL(2,prime(n)). - Tom Edgar, Sep 28 2015

G. C. Greubel, Table of n, a(n) for n = 1..10000
J. B. Marshall, On the extension of Fermat's theorem to matrices of order n, Proceedings of the Edinburgh Mathematical Society 6 (1939) 85-91. See (11) page 90-91 when p=2.
Index to sequences related to prime powers.

Cf. A036689, A034953, A084920, A117762.

Cf. A065470, A065487.

[6] cat [NthPrime(n)*(NthPrime(n)^2-1): n in [2..40]]; // Vincenzo Librandi, Sep 29 2015

Table[(Prime[n] + 1) Prime[n](Prime[n] - 1), {n, 1, 100}]
Table[p(p^2-1),{p,Prime[Range[40]]}] (* Harvey P. Dale, Apr 26 2025 *)

forprime(p=2,1e3,print1(6*binomial(p+1,3)", ")) \\ Charles R Greathouse IV, Jun 16 2011

a(n) = prime(n)*(prime(n)^2-1);
vector(40, n, a(n)) \\ Altug Alkan, Sep 28 2015

a(n) = prime(n)*(prime(n)^2-1). - Tom Edgar, Sep 28 2015
a(n) = 2 * A117762(n), for n > 1. - Altug Alkan, Sep 28 2015
From Amiram Eldar, Nov 22 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A065487.
Product_{n>=1} (1 - 1/a(n)) = A065470. (End)

A011756 a(n) = prime(n*(n+1)/2).

Original entry on oeis.org

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

Offset: 1

Jeff Burch

nonn

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Triangular Number.

Primes in leading diagonal of triangle in A078721.
Cf. A000040, A000217, A034953.
Cf. A195678.
Cf. A000720.

a011756 n = a011756_list !! (n-1)
a011756_list = map a000040 $ tail a000217_list
-- Reinhard Zumkeller, Sep 23 2011

[NthPrime(n*(n+1) div 2): n in [1..100] ]; // Vincenzo Librandi, Apr 11 2011

seq(ithprime(n*(n+1)/2),n=1..50); # Muniru A Asiru, Jul 22 2018

Prime[#]&/@Accumulate[Range[50]] (* Harvey P. Dale, Mar 23 2015 *)

a(n) = prime(n*(n+1)/2); \\ Michel Marcus, Jul 22 2018

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

a(n) = A000040(A000217(n)). - David James Sycamore, Sep 03 2024

A127918 Half of product of three numbers: n-th prime, previous and following number.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Feb 06 2007

Apart from the first term, the same as A117762. - R. J. Mathar, Jun 14 2008
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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A036689, A034953, A127917, A127920.

[(NthPrime(n)+1)*NthPrime(n)*(NthPrime(n)-1)/2: n in [1..40]]; // Vincenzo Librandi, Apr 09 2017

Table[(Prime[n] + 1) Prime[n](Prime[n] - 1)/2, {n, 1, 100}]

forprime(p=2,1e3,print1(3*binomial(p+1,3)", ")) \\ Charles R Greathouse IV, Jun 16 2011

A127919 1/3 of product of three numbers: the n-th prime, the previous number and the following number.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Feb 06 2007

Number of irreducible monic cubic polynomials over GF(prime(n)). - Robert Israel, Jan 06 2015

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A036689, A034953, A127917, A127918, A127920.

[(p^3 - p) div 3: p in PrimesUpTo(150)]; // Vincenzo Librandi, Jan 08 2015

seq((ithprime(n)^3 - ithprime(n))/3, n=1..100); # Robert Israel, Jan 06 2015

Table[(Prime[n] + 1) Prime[n] (Prime[n] - 1)/3, {n, 100}]

forprime(p=2,1e3,print1(2*binomial(p+1,3)", ")) \\ Charles R Greathouse IV, Jun 16 2011

a(n) = (prime(n)^3 - prime(n))/3. - Wesley Ivan Hurt, Oct 15 2023

cp's OEIS Frontend

A085739 Partial sums of A034953(n).

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A006254 Numbers k such that 2k-1 is prime.

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A067076 Numbers k such that 2*k + 3 is a prime.

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A008837 a(n) = p*(p-1)/2 for p = prime(n).

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A054269 Length of period of continued fraction for sqrt(prime(n)).

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A036690 Product of a prime and the following number.

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