A111333 -id:A111333 - cp's OEIS frontend

A081385 Incorrect version of A006254 (or A111333).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 17, 18, 19, 20, 26, 27, 28, 29, 30, 36, 37, 38, 39, 40, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65

Offset: 1

Michael Joseph Halm, Apr 20 2003

dead

Name was: Numbers just more than half-prime.

			a(11) = 16 because [(31)/2 + 1/2]=16.

M. J. Halm, More Sequences, Mpossibilities 83, 2003.

M. J. Halm, Sequences

Cf. A000040.

a(n) = ceil(prime(n)/2); \\ Michel Marcus, Aug 13 2013

a(n) = [p(n)/2 + 1/2].

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

A130290 Number of nonzero quadratic residues modulo the n-th prime.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156, 158

Offset: 1

M. F. Hasler, May 21 2007

Row lengths for formatting A063987 as a table: The number of nonzero quadratic residues modulo a prime p equals floor(p/2), or (p-1)/2 if p is odd. The number of squares including 0 is (p+1)/2, if p is odd (rows prime(i) of A096008 formatted as a table). In fields of characteristic 2, all elements are squares. For any m > 0, floor(m/2) is the number of even positive integers less than or equal to m, so a(n) also equals the number of even positive integers less than or equal to the n-th prime. For all n > 0, A130290(n+1) = A005097(n) = A102781(n+1) = A102781(n+1) = A130291(n+1)-1 = A111333(n+1)-1 = A006254(n)-1.
From Vladimir Shevelev, Jun 18 2016: (Start)
a(1)+2 and, for n >= 2, a(n)+1 is the smallest k such that there exists 0 < k_1 < k with the condition k_1^2 == k^2 (mod prime(n)).
Indeed, for n >= 2, if prime(n) = 4*t+1 then k = 2*t+1 = a(n)+1, since (2*t+1)^2 == (2*t)^2 (mod prime(n)) and there cannot be a smaller value of k; if prime(n) = 4*t-1, then k = 2*t = a(n)+1, since (2*t)^2 == (2*t-1)^2 (mod prime(n)). (End)
a(n) is the number of pairs (a,b) such that a + b = prime(n) with 1 <= a <= b. - Nicholas Leonard, Oct 02 2022

			a(1)=1 since the only nonzero element of Z/2Z equals its square.
a(3)=2 since 1=1^2=(-1)^2 and 4=2^2=(-2)^2 are the only nonzero squares in Z/5Z.
a(1000000) = 7742931 = (prime(1000000)-1)/2.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Quadratic Residue
Wikipedia, Quadratic Residue

Essentially the same as A005097.
Cf. A102781 (Number of even numbers less than the n-th prime), A063987 (quadratic residues modulo the n-th prime), A006254 (Numbers n such that 2n-1 is prime), A111333 (Number of odd numbers <= n-th prime), A000040 (prime numbers), A130291.
Appears in A217983. - Johannes W. Meijer, Oct 25 2012

[Floor((NthPrime(n))/2): n in [1..60]]; // Vincenzo Librandi, Jan 16 2013

A130290 := proc(n): if n =1 then 1 else (A000040(n)-1)/2 fi: end: A000040 := proc(n): ithprime(n) end: seq(A130290(n), n=1..55); # Johannes W. Meijer, Oct 25 2012

Quotient[Prime[Range[66]], 2] (* Vladimir Joseph Stephan Orlovsky, Sep 20 2008 *)

PARI
```
A130290(n) = prime(n)>>1
```

from sympy import prime
def A130290(n): return prime(n)//2  # Chai Wah Wu, Jun 04 2022

a(n) = floor( A000040(n)/2 ) = #{ even positive integers <= A000040(n) }
a(n) = A055034(A000040(n)), n>=1. - Wolfdieter Lang, Sep 20 2012
a(n) = A005097(n-[n>1]) = A005097(max(n-1,1)). - M. F. Hasler, Dec 13 2019

A049990 a(n) is the number of arithmetic progressions of 2 or more positive integers, nondecreasing with sum n.

Original entry on oeis.org

0, 1, 2, 3, 3, 6, 4, 6, 8, 8, 6, 13, 7, 10, 15, 12, 9, 19, 10, 16, 20, 15, 12, 26, 16, 17, 25, 21, 15, 34, 16, 22, 30, 22, 24, 40, 19, 24, 35, 32, 21, 45, 22, 30, 47, 29, 24, 51, 28, 37, 46, 35, 27, 56, 36, 40, 51, 36, 30, 70, 31, 38, 61, 43

Offset: 1

Clark Kimberling

nonn

			a(6) counts these 6 partitions of 6: [5,1], [4,2], [3,3], [3,2,1], [2,2,2], [1,1,1,1,1,1].

Clark Kimberling, Table of n, a(n) for n = 1..1000
Sadek Bourbaki and Nevrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
Wikipedia, Arithmetic progression.

Cf. A014405, A014406, A049980, A049981, A049982, A049983, A049987, A049988, A049989, A049990, A049991, A111333, A127938.

(* Program 1 *)
Map[Length[Map[#[[2]] &, Select[Map[{Apply[SameQ, Differences[#]], #} &,
IntegerPartitions[#]], #[[1]] &]]] &, Range[40]] - 1
(* Peter J. C. Moses, Dec 24 2016 *)
(* Program 2 *)
enumerateArithmeticPartitions[n_] := Module[{allDivs, oddDivs},
{allDivs, oddDivs} = {#, Select[#, OddQ]} &[Divisors[n]]; Map[Reverse, Union[Flatten[Table[If[OddQ[cDiff], (Flatten[
Map[{If[(2 n - #) cDiff <= # (# - 2), {Table[(cDiff + # - 2 cDiff n/#)/2 +
cDiff term, {term, 0, 2 n/# - 1}]}, {}], If[# (# - 1) cDiff <= 2 (n - #),
{Table[(cDiff + 2 n/# - # cDiff)/2 + cDiff term, {term, 0, # - 1}]},
{}]} &, oddDivs], 2]), (Flatten[Map[If[(n - #) cDiff <= 2 # (# - 1),
{Table[(cDiff + 2 # - n cDiff/#)/2 + cDiff term, {term, 0, n/# - 1}]}, {}] &,
allDivs], 1])], {cDiff, 0, n - 2}], 1]]]];
Join[{0}, Map[Length[enumerateArithmeticPartitions[#]] - 1 &, Range[2, 300]]]
n = 12; enumerateArithmeticPartitions[12] (* shows the desired partition of n *)
(* Peter J. C. Moses, Dec 24 2016 *)

a(A000040(n)) = A111333(n). - Clark Kimberling, Dec 26 2016
From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = A049988(n) - 1. [Note that A049988 has offset 0.]
G.f.: Sum_{k>=2} x^k/(1-x^(k*(k-1)/2))/(1-x^k). [Leroy Quet from A049988]
(End)

A246372 Numbers n such that 2n-1 = product_{k >= 1} (p_k)^(c_k), then n <= product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 16, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 40, 42, 44, 45, 46, 47, 48, 49, 51, 52, 54, 55, 56, 57, 60, 62, 64, 65, 66, 67, 69, 70, 71, 72, 75, 76, 78, 79, 80, 81, 82, 84, 85, 87, 89, 90, 91, 92, 93, 96, 97, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 110

Offset: 1

Antti Karttunen, Aug 24 2014

nonn

Numbers n such that A064216(n) >= n.

Numbers n such that A064989(2n-1) >= n.

			1 is present, as 2*1 - 1 = empty product = 1.
2 is present, as 2*2 - 1 = 3 = p_2, and p_{2-1} = p_1 = 2 >= 2.
3 is present, as 2*3 - 1 = 5 = p_3, and p_{3-1} = p_2 = 3 >= 3.
5 is not present, as 2*5 - 1 = 9 = p_2 * p_2, and p_1 * p_1 = 4, with 4 < 5.
6 is present, as 2*6 - 1 = 11 = p_5, and p_{5-1} = p_4 = 7 >= 6.
25 is present, as 2*25 - 1 = 49 = p_4^2, and p_3^2 = 5*5 = 25 >= 25.
35 is present, as 2*35 - 1 = 69 = 3*23 = p_2 * p_9, and p_1 * p_8 = 2*19 = 38 >= 35.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement: A246371
Union of A246362 and A048674.
Subsequences: A006254 (A111333), A246373 (the primes present in this sequence).
Cf. A000040, A064216, A064989, A246352.

default(primelimit, 2^30);
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A064216(n) = A064989((2*n)-1);
isA246372(n) = (A064216(n) >= n);
n = 0; i = 0; while(i < 10000, n++; if(isA246372(n), i++; write("b246372.txt", i, " ", n)));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A246372 (MATCHING-POS 1 1 (lambda (n) (>= (A064216 n) n))))

A049991 a(n) is the number of arithmetic progressions of 2 or more positive integers, nondecreasing with sum <= n.

Original entry on oeis.org

0, 1, 3, 6, 9, 15, 19, 25, 33, 41, 47, 60, 67, 77, 92, 104, 113, 132, 142, 158, 178, 193, 205, 231, 247, 264, 289, 310, 325, 359, 375, 397, 427, 449, 473, 513, 532, 556, 591, 623, 644, 689, 711, 741, 788, 817, 841, 892, 920, 957, 1003, 1038, 1065, 1121, 1157, 1197, 1248, 1284, 1314, 1384, 1415

Offset: 1

Clark Kimberling

nonn

Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.

Cf. A014405, A014406, A049980, A049981, A049982, A049983, A049987, A049988, A049989, A049990, A049991, A111333, A127938.

From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = Sum_{k = 1..n} A049990(k).
G.f.: (g.f. of A049990)/(1-x). (End)

More terms from Petros Hadjicostas, Sep 29 2019

A177226 Triangle read by rows: T(n, k) = 2^(prime(n) - prime(k)) mod prime(n), 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 2, 4, 1, 6, 3, 9, 5, 1, 7, 10, 9, 12, 4, 1, 9, 13, 16, 4, 13, 16, 1, 10, 5, 6, 11, 9, 7, 4, 1, 12, 6, 13, 9, 2, 12, 18, 16, 1, 15, 22, 20, 5, 13, 25, 7, 9, 6, 1, 16, 8, 2, 16, 1, 8, 16, 4, 8, 4, 1, 19, 28, 7, 11, 3, 10, 33, 36, 30, 34, 27, 1, 21, 31, 18, 25, 40, 10, 16, 4, 31, 37, 40, 16, 1

Offset: 1

Juri-Stepan Gerasimov, Dec 10 2010

			Triangle begins:
   1;
   2,  1;
   3,  4,  1;
   4,  2,  4,  1;
   6,  3,  9,  5,  1;
   7, 10,  9, 12,  4,  1;
   9, 13, 16,  4, 13, 16,  1;
  10,  5,  6, 11,  9,  7,  4,  1;
  12,  6, 13,  9,  2, 12, 18, 16,  1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000079, A111333, A173655, A174497, A174947, A174996, A292411.

A177226:= func< n,k | Modexp(2, NthPrime(n) - NthPrime(k), NthPrime(n)) >;
[A177226(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 09 2024

Flatten[Table[PowerMod[2,Prime[n]-Prime[k],Prime[n]],{n,20},{k,n}]] (* Harvey P. Dale, May 10 2014 *)

def A177226(n,k): return pow(2, nth_prime(n) - nth_prime(k), nth_prime(n))
flatten([[A177226(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Apr 09 2024

From G. C. Greubel, Apr 09 2024: (Start)
T(n, 1) = A111333(n).
T(n, 2) = A292411(n). (End)

Corrected by D. S. McNeil, Dec 10 2010

A245685 Sigma(2p)/2, for odd primes p.

Original entry on oeis.org

6, 9, 12, 18, 21, 27, 30, 36, 45, 48, 57, 63, 66, 72, 81, 90, 93, 102, 108, 111, 120, 126, 135, 147, 153, 156, 162, 165, 171, 192, 198, 207, 210, 225, 228, 237, 246, 252, 261, 270, 273, 288, 291, 297, 300, 318, 336, 342, 345, 351, 360, 363, 378, 387, 396, 405

Offset: 1

Hartmut F. W. Hoft, Jul 29 2014

The symmetric representation of sigma(2*p), p > 3 prime, consists of two sections each with three contiguous legs of width one (for a proof see the link).
The two ratios of successive legs in the symmetric representation of sigma(2*p) are integers 3 and 2, respectively, for all primes p > 3 satisfying p = -1(mod 6); see also A003627.  If one ratio is an integer then so is the other.
The sequence 2*p for primes p > 3 is a subsequence of A239929, numbers n whose symmetric representation of sigma(n) has two parts.
Since sigma(2*p) = 3*(p+1), each element of the sequence is a multiple of 3; furthermore, a(n)/3 = A006254(n) = A111333(n+1).

			a(4) = T(22, 1) - T(22, 4) = 22 - 4 = 18 = sigma(22)/2
The last image in the Example section of A237593 includes the first four symmetric representations for this sequence, i.e., when 2*p = 10, 14, 22 & 26; see also the link for an image of the first 10 symmetric representations.

Hartmut F. W. Hoft, Proofs of properties of sigma(2p)
Hartmut F. W. Hoft, Visualization of sigma(2p)

Cf. A000203, A006254, A111333, A237048, A237270, A237271, A237591, A237593, A239929.

[3*(NthPrime(n+1)+1)/2: n in [1..60]]; // Vincenzo Librandi, Sep 19 2014

a[n_]:=3(Prime[n+1]+1)/2
Map[a,Range[55]] (* data *)
DivisorSigma[1,2#]/2&/@Prime[Range[2,60]] (* Harvey P. Dale, Jan 07 2023 *)

vector(100,n,3*(prime(n+1)+1)/2) \\ Derek Orr, Sep 19 2014

vector(60, n, sigma(2*prime(n+1))/2) \\ Michel Marcus, Nov 25 2014

a(n) = T(2*prime(n+1), 1) - T(2*prime(n+1), 4) = 3*(prime(n+1)+1)/2 = sigma(2*prime(n+1))/2 where T(n,k) is defined in A235791.

a(n)=A247159(n+1)/2. - Omar E. Pol, Nov 22 2014

A130291 Number of quadratic residues (including 0) modulo the n-th prime.

Original entry on oeis.org

2, 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

M. F. Hasler, May 21 2007

The number of squares (quadratic residues including 0) modulo a prime p (sequence A096008 with every "1" prefixed by a "0") equals 1+floor(p/2), or ceiling(p/2) = (p+1)/2 if p is odd. (In fields of characteristic 2, all elements are squares.) See A130290(n)=A130291(n)-1 for number of nonzero residues. For all n>0, A130291(n+1) = A111333(n+1) = A006254(n) = A005097(n)-1 = A102781(n+1)-1 = A102781(n+1)-1 = A130290(n+1)-1.

			a(1)=2 since both elements of Z/2Z are squares.
a(3)=0 since 0=0^2, 1=1^2=(-1)^2 and 4=2^2=(-2)^2 are squares in Z/5Z.
a(1000000) = 7742932 = (p[1000000]+1)/2.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Quadratic Residue
Wikipedia, Quadratic Residue

Essentially the same as A006254.

Cf. A005097 (Odd primes - 1)/2, A102781 (Integer part of n#/(n-2)#/2#), A102781 (Number of even numbers less than the n-th prime), A063987 (quadratic residues modulo the n-th prime), A006254 (Numbers n such that 2n-1 is prime), A111333 (Number of odd numbers <= n-th prime), A000040 (prime numbers), A130290 (number of nonzero residues modulo primes).

[Floor((NthPrime(n))/2)+1: n in [1..60]]; // Vincenzo Librandi, Jan 16 2013

Quotient[Prime[Range[200]], 2] + 1 (* Vincenzo Librandi, Jan 16 2013 *)

PARI
```
A130291(n) = 1+prime(n)>>1
```

a(n) = floor( A000040(n)/2 )+1

A008508 Number of odd primes less than n-th odd composite number.

Original entry on oeis.org

3, 5, 7, 8, 8, 10, 10, 11, 13, 14, 14, 15, 15, 17, 17, 18, 20, 20, 21, 22, 22, 23, 23, 23, 24, 26, 28, 29, 29, 29, 29, 29, 29, 30, 31, 31, 33, 33, 33, 33, 35, 35, 36, 36, 37, 38, 38, 39, 39, 41, 41, 41, 41, 43, 45, 45, 45, 45, 45, 46

Offset: 1

Gary Findley (chfindley(AT)alpha.nlu.edu), Mar 15 1996

nonn

			The first odd composite is 9, and there are 4 primes below: 2, 3, 5, and 7; so there are 3 odd primes, hence a(1)=3.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000720, A053726, A071904, A111333.

PrimePi[#] - 1 & /@ Select[Range@ 213, CompositeQ@ # && OddQ@ # &] (* Michael De Vlieger, Apr 17 2015 *)

lista(nn) = {forcomposite (n=1, nn, if (n % 2, print1(primepi(n)-1, ", ")););} \\ Michel Marcus, Apr 18 2015

from sympy import primepi
def A008508(n):
    if n == 1: return 3
    m, k = n, (r:=primepi(n)) + n + (n>>1)
    while m != k:
        m, k = k, (r:=primepi(k)) + n + (k>>1)
    return r-1 # Chai Wah Wu, Aug 01 2024

From Antti Karttunen, Apr 17 2015: (Start)
a(n) = A000720(A071904(n)) - 1 (by the definition).
a(n) = A053726(n) - n - 1.
(End)

cp's OEIS Frontend

A081385 Incorrect version of A006254 (or A111333).

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

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Magma

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Extensions

A130290 Number of nonzero quadratic residues modulo the n-th prime.

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Magma

Maple

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A049990 a(n) is the number of arithmetic progressions of 2 or more positive integers, nondecreasing with sum n.

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Mathematica

Formula

A246372 Numbers n such that 2n-1 = product_{k >= 1} (p_k)^(c_k), then n <= product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

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A049991 a(n) is the number of arithmetic progressions of 2 or more positive integers, nondecreasing with sum <= n.

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A177226 Triangle read by rows: T(n, k) = 2^(prime(n) - prime(k)) mod prime(n), 1 <= k <= n.

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Magma

Mathematica