A006254 -id:A006254 - cp's OEIS frontend

A124494 Numbers k for which 2k-1, 4k-1, 8k-1 and 16k-1 are primes.

3, 45, 90, 180, 255, 615, 705, 1350, 1770, 3225, 5295, 5775, 5955, 6060, 8580, 9855, 9945, 11175, 13095, 13740, 15195, 21825, 26820, 26925, 27615, 28920, 30075, 30705, 31710, 33375, 35700, 37350, 37665, 41250, 43770, 49185, 50700, 52185, 53640

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006254, A124485, A125113, A124514, A124515, A124516.

Subsequence of A124493. Supersequence of A124017.

Select[3*Range[20000], And @@ PrimeQ /@ ({2, 4, 8, 16}*# - 1) &] (* Ray Chandler, Nov 22 2006 *)
Select[Range[3,55000,3],AllTrue[2^Range[4] #-1,PrimeQ]&] (* or, faster  *) Join[{3},Select[Range[ 15,55000,15],AllTrue[ 2^Range[4] #-1,PrimeQ]&]] (* Harvey P. Dale, Feb 02 2025 *)

Extended by Ray Chandler, Nov 22 2006

A124515 Numbers k for which 2k-1, 4k-1, 8k-1, 16k-1, 32k-1, 64k-1, 128k-1 and 256k-1 are primes.

Original entry on oeis.org

9549960, 26277285, 42932385, 85864770, 99239790, 113183070, 152596290, 172159515, 198479580, 237059175, 287482065, 305192580, 342533490, 382203030, 542591115, 563002110, 597825570, 686106720, 742227135, 786875025, 1135145760

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A006254, A124485, A124493, A124494, A124017, A125113, A124514, A124516.

Select[15*Range[76000000], And @@ PrimeQ /@ ({2, 4, 8, 16, 32, 64, 128, 256}*# - 1) &] (* Ray Chandler, Nov 22 2006 *)

Extended by Ray Chandler, Nov 22 2006

A244983 Permutation of natural numbers: a(1) = 1, a(n) = (1 + A122111(A070003(n-1))) / 2.

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 14, 13, 6, 11, 41, 23, 18, 7, 17, 38, 25, 68, 32, 28, 122, 63, 9, 20, 113, 53, 39, 365, 95, 50, 33, 74, 203, 61, 188, 88, 10, 26, 1094, 158, 83, 46, 608, 313, 3281, 338, 123, 149, 59, 43, 221, 116, 284, 72, 263, 138, 1013, 12, 9842, 29, 1823, 248, 98, 563, 172, 60

Offset: 1

Antti Karttunen, Jul 21 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A244984.
Cf. A070003, A006254, A244986.
Related or similar permutations: A122111, A244981-A244982, A243505-A243506, A243065-A243066.

(define (A244983 n) (if (= 1 n) 1 (/ (+ 1 (A122111 (A070003 (- n 1)))) 2)))

a(1) = 1, a(n) = (1 + A122111(A070003(n-1))) / 2.

For all n >= 1, a(A244986(n+1)) = A006254(n).

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

A285716 a(n) = A080791(A245611(n)).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 0, 2, 1, 0, 3, 0, 1, 1, 0, 1, 1, 1, 0, 2, 0, 0, 2, 0, 0, 1, 0, 1, 2, 1, 1, 1, 2, 0, 1, 0, 1, 3, 0, 0, 1, 1, 1, 2, 0, 0, 2, 1, 0, 1, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 0, 1, 1, 1, 3, 0, 0, 2, 0, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 0, 3, 0, 0, 2, 0, 1, 1, 0

Offset: 1

Antti Karttunen, Apr 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

One less than A091304 after the initial term.
Cf. A000120, A080791, A244153, A245611, A278223, A285712, A285714, A285715.
Cf. A006254 (gives the positions of zeros after initial a(1)=0.)

a[n_] := PrimeOmega[2*n - 1] - 1; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jul 23 2023 *)

;; First implementation uses memoization-macro definec:
(definec (A285716 n) (if (<= n 2) 0 (+ (if (= 2 (modulo n 3)) 1 0) (A285716 (A285712 n)))))
(define (A285716 n) (A080791 (A245611 n)))

a(1) = 0, a(2) = 1, for n > 2, a(n) = a(A285712(n)) + [n == 2 mod 3]. (Where [] is Iverson bracket, giving here 1 only if n is of the form 3k+2, and 0 otherwise.)
a(n) = A080791(A245611(n)).
For all n >= 2,  a(n) = A091304(n)-1 = A000120(A244153(n))-1. - Antti Karttunen, May 31 2017

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

A139757 a(n) = (n+1)*(2n+1)^2.

Original entry on oeis.org

1, 18, 75, 196, 405, 726, 1183, 1800, 2601, 3610, 4851, 6348, 8125, 10206, 12615, 15376, 18513, 22050, 26011, 30420, 35301, 40678, 46575, 53016, 60025, 67626, 75843, 84700, 94221, 104430, 115351, 127008, 139425, 152626, 166635, 181476

Offset: 0

Odimar Fabeny, May 19 2008

Also the detour index of the (n+1)-antiprism graph and (n+1)-cocktail party graphs for n>=2. - Eric W. Weisstein, Jul 15 2011 and Dec 20 2017

Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Cocktail Party Graph
Eric Weisstein's World of Mathematics, Detour Index
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000217, A006254.

[(n+1)*(2*n+1)^2 : n in [0..30]]; // Wesley Ivan Hurt, Sep 29 2014

A139757:=n->(n+1)*(2*n+1)^2: seq(A139757(n), n=0..30); # Wesley Ivan Hurt, Sep 29 2014

Table[(n + 1) (2 n + 1)^2, {n, 0, 30}] (* Wesley Ivan Hurt, Sep 29 2014 *)
LinearRecurrence[{4, -6, 4, -1}, {18, 75, 196, 405}, {0, 20}] (* Eric W. Weisstein, Dec 20 2017 *)
CoefficientList[Series[(1 + 14 x + 9 x^2)/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 20 2017 *)

a(n) = (n+1)*(2*n+1)^2; \\ Altug Alkan, Dec 20 2017

a(n) = (2n+1) * A000217(2n+1).
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4); G.f.: (1+14*x+9*x^2)/(x-1)^4. - R. J. Mathar, Sep 19 2010
a(n) = Sum_{i=1..2n-1} (n^2 + n*i - i). - Wesley Ivan Hurt, Sep 29 2014
From Amiram Eldar, Jun 28 2020: (Start)
Sum_{n>=0} 1/a(n) = Pi^2/4 - log(4).
Sum_{n>=0} (-1)^n/a(n) = 2*G + log(2) - Pi/2, where G is the Catalan constant (A006752). (End)

Missing a(0) inserted by R. J. Mathar, Sep 19 2010

A066886 Sum of the elements in any transversal of a prime(n) X prime(n) array containing the numbers from 1 to prime(n)^2 in standard order.

Original entry on oeis.org

5, 15, 65, 175, 671, 1105, 2465, 3439, 6095, 12209, 14911, 25345, 34481, 39775, 51935, 74465, 102719, 113521, 150415, 178991, 194545, 246559, 285935, 352529, 456385, 515201, 546415, 612575, 647569, 721505, 1024255, 1124111, 1285745

Offset: 1

Enoch Haga, Jan 22 2002

a(n) is the sum of the primes in a prime(n) X prime(n) example of Haga's conjecture (see link below).

Robert Israel, Table of n, a(n) for n = 1..10000
Carlos Rivera, The prime puzzles & problems connection, conjecture 26, The Prime Puzzles and Problems Connection.

Cf. A005097, A006003, A006254, A066883, A066885.

map(t -> t*(t^2+1)/2, [seq(ithprime(i),i=1..100)]); # Robert Israel, Apr 04 2018

a[n_] := Prime[n] (Prime[n]^2 + 1)/2; Table[a[n], {n, 50}]

apply(x->(x*(x^2+1)/2), primes(100)) \\ Michel Marcus, Apr 04 2018

a(n) = prime(n)*(prime(n)^2+1)/2, where prime(n) is the n-th prime.
a(n) = A006003(prime(n)). - Michel Marcus, Apr 04 2018
a(n) = A006254(n-1)^4 - A005097(n-1)^4, for n>1. - Dimitris Valianatos, Apr 10 2018

Edited by Dean Hickerson, Jun 08 2002

A163427 Primes p such that (p+1)^3/8+(p-1)/2 is also prime.

Original entry on oeis.org

5, 7, 13, 19, 29, 31, 41, 53, 71, 101, 103, 109, 173, 191, 199, 223, 229, 233, 239, 257, 269, 277, 331, 383, 397, 431, 491, 569, 571, 599, 619, 631, 719, 733, 751, 757, 761, 823, 857, 859, 863, 887, 907, 937, 967, 971, 977, 1009, 1019, 1063, 1069, 1123, 1163

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

Primes A000040(k) such that (A006254(k-1))^3+ A005097(k-1) is also prime.

			For p=5, (5+1)^3/8+(5-1)/2=27+2=29, prime, which adds p=5 to the sequence.
For p=7, (7+1)^3/8+(7-1)/2=67, prime, which adds p=7 to the sequence.

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

Cf. A162652, A163418, A163419, A163420, A163421, A163422, A163424, A163425, A163426.

[p: p in PrimesInInterval(3, 1200) | IsPrime((p+1)^3 div 8+(p-1) div 2)]; // Vincenzo Librandi, Apr 09 2013

f[n_]:=((p+1)/2)^3+((p-1)/2); lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst, p]],{n,6!}];lst
Select[Prime[Range[100]], PrimeQ[(# + 1)^3 / 8 + (# - 1) / 2]&] (* Vincenzo Librandi, Apr 09 2013 *)

(a(n)+1)^3/8+(a(n)-1)/2 = A163426(n).

Edited by R. J. Mathar, Aug 24 2009

A201912 Irregular triangle of 2^k mod prime(n).

Original entry on oeis.org

0, 1, 2, 1, 2, 4, 3, 1, 2, 4, 1, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1, 2, 4, 8, 16, 9, 18, 13, 3, 6, 12, 1, 2, 4, 8, 16, 3, 6, 12, 24

Offset: 1

T. D. Noe, Dec 17 2011

The row lengths are in A014664. For n > 1, the first term of each row is 1 and the last term is 2*prime(n)-1, which is A006254. Many sequences are in this one.

			The first 11 rows are:
2:  0;
3:  1, 2;
5:  1, 2, 4, 3;
7:  1, 2, 4;
11: 1, 2, 4, 8,  5, 10,  9,  7,  3,  6;
13: 1, 2, 4, 8,  3,  6, 12, 11,  9,  5, 10,  7;
17: 1, 2, 4, 8, 16, 15, 13,  9;
19: 1, 2, 4, 8, 16, 13,  7, 14,  9, 18, 17, 15, 11,  3,  6, 12,  5, 10;
23: 1, 2, 4, 8, 16,  9, 18, 13,  3,  6, 12;
29: 1, 2, 4, 8, 16,  3,  6, 12, 24, 19,  9, 18,  7, 14, 28, 27, 25, 21, 13, 26, 23, 17, 5, 10, 20, 11, 22, 15;
31: 1, 2, 4, 8, 16;

T. D. Noe, Rows n = 1..60, flattened

Cf. A062117, A201908, A201909, A201910, A201911.

Cf. similar sequences of the type 2^n mod p, where p is a prime: A000034 (p=3), A070402 (p=5), A069705 (p=7), A036117 (p=11), A036118 (p=13), A062116 (p=17), A036120 (p=19), A070335 (p=23), A036122 (p=29), A269266 (p=31), A036124 (p=37), A070348 (p=41), A070349 (p=43), A070351 (p=47), A036128 (p=53), A036129 (p=59), A036130 (p=61), A036131 (p=67), A036135 (p=83), A036138 (p=101), A036140 (p=107), A036144 (p=131), A036146 (p=139), A036147 (p=149), A036150 (p=163), A036152 (p=173), A036153 (p=179), A036154 (p=181), A036157 (p=197), A036159 (p=211), A036161 (p=227).

P:=Filtered([1..350],IsPrime);;
R:=List([1..Length(P)],n->OrderMod(2,P[n]));;
Flat(Concatenation([0],List([2..10],n->List([0..R[n]-1],k->PowerMod(2,k,P[n]))))); # Muniru A Asiru, Feb 01 2019

nn = 10; p = 2; t = p^Range[0,Prime[nn]]; Flatten[Table[If[Mod[n, p] == 0, {0}, tm = Mod[t, n]; len = Position[tm, 1, 1, 2][[-1,1]]; Take[tm, len-1]], {n, Prime[Range[nn]]}]]

cp's OEIS Frontend

A124494 Numbers k for which 2*k-1, 4*k-1, 8*k-1 and 16*k-1 are primes.

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A124515 Numbers k for which 2*k-1, 4*k-1, 8*k-1, 16*k-1, 32*k-1, 64*k-1, 128*k-1 and 256*k-1 are primes.

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A244983 Permutation of natural numbers: a(1) = 1, a(n) = (1 + A122111(A070003(n-1))) / 2.

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

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A285716 a(n) = A080791(A245611(n)).

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

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A139757 a(n) = (n+1)*(2n+1)^2.

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A066886 Sum of the elements in any transversal of a prime(n) X prime(n) array containing the numbers from 1 to prime(n)^2 in standard order.

A124494 Numbers k for which 2k-1, 4k-1, 8k-1 and 16k-1 are primes.

A124515 Numbers k for which 2k-1, 4k-1, 8k-1, 16k-1, 32k-1, 64k-1, 128k-1 and 256k-1 are primes.