A005097 -id:A005097 - cp's OEIS frontend

A130800 Numbers k such that both 2k+1 and 3k+1 are primes.

2, 6, 14, 20, 26, 36, 50, 54, 74, 90, 116, 140, 146, 174, 200, 204, 210, 224, 230, 270, 284, 306, 330, 336, 350, 354, 384, 404, 410, 426, 440, 476, 510, 516, 554, 564, 596, 600, 624, 644, 650, 704, 714, 726, 740, 746, 834, 846, 894, 930, 944, 950, 1026, 1040

Offset: 1

Max Alekseyev, Jul 18 2007

nonn

Also: k such that A033570(k) is semiprime. All terms are congruent to 0 or 2 modulo 6. - M. F. Hasler, Dec 13 2019

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Intersection of A005097 and A024892. - M. F. Hasler, Dec 13 2019

Cf. A033570; A255584: semiprimes of the form (4*n+1)*(6*n+1).

[n: n in [0..500] | IsPrime(2*n+1) and IsPrime(3*n+1)]; // Vincenzo Librandi, Nov 23 2010

Select[Range[1100],AllTrue[{2,3}#+1,PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 17 2016 *)

select( is_A130800(n)=isprime(2*n+1)&&isprime(3*n+1), [1..1111]) \\ M. F. Hasler, Dec 13 2019

a(n) = 2*A255607(n). - M. F. Hasler, Dec 13 2019

More terms from Vincenzo Librandi, Mar 26 2010

A266409 a(n) = (A003309(n+2)-1) / 2; numbers n such that 2n+1 is a Ludic number (in A003309).

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 11, 12, 14, 18, 20, 21, 23, 26, 30, 33, 35, 38, 41, 44, 45, 48, 53, 57, 59, 60, 63, 65, 71, 74, 78, 80, 86, 87, 89, 90, 96, 104, 105, 110, 111, 113, 116, 117, 119, 123, 128, 132, 138, 141, 143, 150, 153, 156, 164, 165, 168, 170, 176, 179, 180, 188, 191, 194, 198, 203, 207, 209, 210, 215

Offset: 1

Antti Karttunen, Jan 28 2016

nonn

Ludic numbers from A003309(2) = 3 onward, decremented by one, then halved.

Antti Karttunen, Table of n, a(n) for n = 1..10226

Complement: A266410.
Cf. A003309, A192490.
Cf. A266350 (least monotonic left inverse).
Cf. permutations A266418, A266638.
Cf. also A005097.

a(n) = (A003309(n+2)-1) / 2.
Other identities. For all n >= 1:
A266350(a(n)) = n.

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

A269252 Define a sequence by s(k) = n*s(k-1) - s(k-2), with s(0) = 1, s(1) = n - 1. a(n) is the smallest index k such that s(k) is prime, or -1 if no such k exists.

Original entry on oeis.org

-1, -1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 14, 1, 2, 2, 3, 1, 2, 5, 2, 36, 2, 1, 2, 1, 15, -1, 6, 2, 3, 1, 2, 2, 6, 1, 3, 1, 2, 2, 2, 1, 2, 3, 2, -1, 3, 1, 2, 2, 2, 6, 3, 1, 2, 1, 30, 3, 2, 2, 2, 1, 2, 5, 2, 1, 5, 1, 6, 3, 2, 6, 3, 1, 8, 6, 14, 1, 3

Offset: 1

Arkadiusz Wesolowski, Jul 09 2016

sign

For n >= 2, positive integer k yielding the smallest prime of the form (x^y + 1/x^y)/(x + 1/x), where x = (sqrt(n+2) +/- sqrt(n-2))/2 and y = 2*k + 1, or -1 if no such k exists.
Every positive term belongs to A005097.
For detailed theory, see [Hone]. - L. Edson Jeffery, Feb 09 2018

			Let b(k) be the recursive sequence defined by the initial conditions b(0) = 1, b(1) = 10, and the recursive equation b(k) = 11*b(k-1) - b(k-2). a(11) = 2 because b(2) = 109 is the smallest prime in b(k).
Let c(k) be the recursive sequence defined by the initial conditions c(0) = 1, c(1) = 12, and the recursive equation c(k) = 13*c(k-1) - c(k-2). a(13) = 3 because c(3) = 2003 is the smallest prime in c(k).

C. K. Caldwell, Top Twenty page, Lehmer number
Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.
Wikipedia, Lehmer number

Cf. A005097, A269251, A269253, A269254, A299045 (array used to compute this sequence).

Cf. A294099, A298675, A298677, A298878, A299071, A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501.

lst:=[]; for n in [1..85] do if n in [1, 2, 34, 52] then Append(~lst, -1); else a:=1; c:=1; t:=0; repeat b:=n*a-c; c:=a; a:=b; t+:=1; until IsPrime(a); Append(~lst, t); end if; end for; lst;

s[k_, m_] := s[k, m] = Which[k == 0, 1, k == 1, 1 + m, True, m s[k - 1, m] - s[k - 2, m]]; Table[SelectFirst[Range[120], PrimeQ@ Abs@ s[#, -n] &] /. k_ /; MissingQ@ k -> -1, {n, 85}] (* Michael De Vlieger, Feb 03 2018 *)

If n is prime then a(n+1) = 1.

A289585 Quotients as they appear as k increases when tau(k) divides phi(k).

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 1, 5, 6, 2, 8, 1, 9, 3, 11, 1, 3, 2, 14, 1, 15, 5, 4, 6, 18, 6, 2, 20, 21, 4, 23, 14, 8, 4, 26, 10, 3, 9, 7, 29, 30, 6, 12, 33, 11, 3, 35, 2, 36, 9, 6, 15, 3, 39, 10, 41, 2, 16, 14, 5, 44, 2, 18, 15, 18, 48, 7, 10, 50, 4, 51, 6, 6, 13, 53, 3, 54, 5, 18, 56, 22, 12, 24, 2

Offset: 1

Bernard Schott, Jul 08 2017

nonn

Numbers k such that tau(k) divides phi(k) are in A020491.
Only for seven integers which are in A020488, we have a(n) = 1.
The integers such that a(n) = 2, 3, 4 are respectively in A062516, A063469, A063470.
When p is an odd prime then phi(p) = p-1, tau(p) = 2, so phi(p)/tau(p) = (p-1)/2 and A005097 is an infinite subsequence.
For k = A058891(m+1), that is 2^A000225(m), with m>=2, the corresponding quotient phi(k)/tau(k) is the integer A076688(m). - Bernard Schott, Aug 15 2020

			a(10) = 2 because A020491(10) = 15 and phi(15)/tau(15) = 8/4 = 2.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A000010, A000005, A020491, A015733, A020488, A058891, A062516, A063469, A063470, A175667, A112955, A112954.
Cf. A000225, A058891, A076688.
Cf. A005097 (a subsequence).
Cf. A290634 (complement).

for n from 1 to 50 do q:=phi(n)/tau(n);
if q=floor(q) then print(n,q,phi(n),tau(n)) else fi; od:

f[n_] := Block[{d = EulerPhi[n]/DivisorSigma[0, n]}, If[ IntegerQ@d, d, Nothing]]; Array[f, 120] (* Robert G. Wilson v, Jul 09 2017 *)

lista(nn) = {for (n=1, nn, q = eulerphi(n)/numdiv(n); if (denominator(q)==1, print1(q, ", ")););} \\ Michel Marcus, Jul 10 2017

a(n) = A000010(A020491(n)) / A000005(A020491(n)). - David A. Corneth, Jul 09 2017

A364678 Maximum number of primes between consecutive multiples of n, as permitted by divisibility considerations.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 5, 7, 7, 6, 7, 7, 7, 7, 8, 7, 8, 9, 8, 10, 8, 10, 10, 10, 11, 11, 11, 10, 11, 11, 11, 12, 12, 12, 12, 13, 12, 13, 14, 13, 13, 14, 14, 15, 15, 14, 15, 15, 15, 16, 15, 15, 16, 16, 17, 16, 17, 18, 18, 18, 18, 18, 17, 19, 19, 19, 19, 20, 20, 19, 19, 20, 21, 21

Offset: 1

Brian Kehrig, Aug 24 2023

nonn

Alternatively: a(n) = the maximum number of elements of an admissible k-tuple strictly contained in (0,n) such that all elements are relatively prime to n. Recall that an admissible tuple is defined as a tuple of integers with the property that all primes p have at least one residue class that has no intersection with the tuple.
For n > 1, we have a(n) <= A023193(n-1), with equality if (but not only if) n is prime or a power of 2. The smallest n for which it is not an equality is n=14.
Conjecture 1: Every nonnegative integer appears in this sequence.
Conjecture 2: For all n, there is an infinitude of k's such that there are a(n) primes between n*k and n*(k+1).
Conjecture 2 resembles the k-tuples conjecture a.k.a. the first Hardy-Littlewood conjecture, although it is not the same.
A notable value is a(35) = 8. Compare with A000010(210) = 48. This says that between any two consecutive multiples of 210 the 48 values that are not divisible by 2, 3, 5 or 7 are equally distributed between 6 equal divisions of 210; that is, 8 are in the interval [0, 34], 8 in the interval [35, 69], etc. - Peter Munn, Feb 16 2024

			Between two multiples of 15 (n and n+15), only n+1, n+2, n+4, n+7, n+8, n+11, n+13, and n+14 could possibly be prime based on divisibility by 3 and 5. However, 4 of these are even and 4 are odd, so at most 4 of them can be prime. Thus, a(15)=4.

Brian Kehrig, Table of n, a(n) for n = 1..1000
Brian Kehrig, Python code for sequence

Cf. A000010, A023193.

Multiples of n following which the maximum number of primes occur for particular n: A005097 (2), A144769 (3), A123986 (4), A056956 (6), A007811 (10), A123985 (12), A309871 (18).

Python
```
# see Links section
```

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

A086668 Number of divisors d of n such that 2d+1 is a prime.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 1, 3, 3, 3, 2, 4, 1, 3, 4, 3, 1, 6, 1, 4, 3, 3, 2, 5, 2, 3, 3, 3, 2, 7, 1, 3, 4, 2, 3, 7, 1, 2, 3, 5, 2, 6, 1, 4, 5, 3, 1, 6, 1, 4, 3, 3, 2, 7, 3, 5, 2, 3, 1, 8, 1, 2, 5, 3, 3, 6, 1, 3, 4, 5, 1, 8, 1, 3, 5, 2, 2, 7, 1, 5, 4, 3, 2, 6, 2, 3, 3, 5, 2, 10, 1, 3, 2, 2, 3, 7, 1, 4, 6, 5

Offset: 1

Jon Perry, Jul 27 2003

nonn

From Antti Karttunen, Jun 15 2018: (Start)
Number of terms of A005097 that divide n.
For all n >= 1, a(n) > A156660(n). Specifically, a(p) = 2 for all p in A005384 (Sophie Germain primes), although 2's occur in other positions as well.
(End)

			10 has divisors 1,2,5 and 10 of which 2.1+1, 2.2+1 and 2.5+1 are prime, so a(10)=3

Antti Karttunen, Table of n, a(n) for n = 1..65537

One less than A046886.

Cf. A005097, A005384, A101264, A305818.

Table[Count[Divisors[n],?(PrimeQ[2#+1]&)],{n,100}] (* _Harvey P. Dale, Apr 29 2015 *)

for (n=2,100,s=0; fordiv(i=n,i,s+=isprime(2*i+1)); print1(","s))

A086668(n) = sumdiv(n,d,isprime(d+d+1)); \\ Antti Karttunen, Jun 15 2018

From Antti Karttunen, Jun 15 2018: (Start)
a(n) = Sum_{d|n} A101264(d).
a(n) = A305818(n) + A101264(n).
(End)

Definition modified by Harvey P. Dale, Apr 29 2015

A138239 Triangle read by rows: T(n,k) = A000040(k) if A002445(n) mod A000040(k) = 0, otherwise 1.

Original entry on oeis.org

1, 2, 3, 2, 3, 5, 2, 3, 1, 7, 2, 3, 5, 1, 1, 2, 3, 1, 1, 11, 1, 2, 3, 5, 7, 1, 13, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 3, 5, 1, 1, 1, 17, 1, 1, 2, 3, 1, 7, 1, 1, 1, 19, 1, 1, 2, 3, 5, 1, 11, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 23, 1, 1, 1, 2, 3, 5, 7, 1, 13, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Mats Granvik, Mar 07 2008

Row products give A002445.

A prime number appears in a column at every A130290-th row from the (A130290+1)-th row onwards. The prime numbers are, so to speak, equidistantly distributed in the columns. A130290 is essentially A005097. Counting terms > 1 in the rows gives A046886.

			First few rows of the triangle and row products are:
1 = 1
2*3 = 6
2*3*5 = 30
2*3*1*7 = 42
2*3*5*1*1 = 30
2*3*1*1*11*1 = 66
2*3*5*7*1*13*1 = 2730

Michel Marcus, Rows n=0..100 of triangle, flattened

Cf. A002445, A005097, A046886, A130290.

T:= (n, k)-> (p-> `if`(irem(denom(bernoulli(2*n)), p)=0, p, 1))(ithprime(k)):
seq(seq(T(n, k), k=1..n+1), n=0..20);  # Alois P. Heinz, Aug 27 2017

t[n_, k_] := If[Mod[Denominator[BernoulliB[2n]], (p = Prime[k])] == 0, p, 1];
Flatten[Table[t[n, k], {n, 0, 13}, {k, 1, n+1}]][[1 ;; 102]] (* Jean-François Alcover, Jun 16 2011 *)

tabl(nn) = {for (n=0, nn, dbn = denominator(bernfrac(2*n)); for (k=1, n+1, if (! (dbn % prime(k)), w = prime(k), w = 1); print1(w, ", "); ); print; ); } \\ Michel Marcus, Aug 27 2017

Definition edited by N. J. A. Sloane, Mar 18 2010

Offset corrected by Alois P. Heinz, Aug 27 2017

A154111 Numbers n such that (n+1)^2 - n^3 is a (positive or negative) prime.

Original entry on oeis.org

1, 3, 5, 6, 8, 11, 12, 15, 18, 20, 27, 33, 35, 39, 41, 45, 48, 50, 54, 65, 66, 68, 86, 87, 92, 96, 99, 107, 116, 122, 123, 126, 138, 140, 149, 150, 156, 159, 161, 164, 165, 167, 170, 177, 182, 185, 188, 191, 192, 198, 200, 207, 209, 219, 228, 237, 239, 240, 242, 252

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 04 2009

G. C. Greubel, Table of n, a(n) for n = 1..2000

Cf. A140719, A005097.

[n: n in [0..500] | IsPrime((n+1)^2-n^3)]; // Vincenzo Librandi, Nov 26 2010

Select[Range[300],PrimeQ[(#+1)^2-#^3]&] (* Harvey P. Dale, Dec 06 2015 *)

is(n)=isprime(abs((n+1)^2 - n^3)) \\ Charles R Greathouse IV, Sep 02 2016

cp's OEIS Frontend

A130800 Numbers k such that both 2k+1 and 3k+1 are primes.

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A266409 a(n) = (A003309(n+2)-1) / 2; numbers n such that 2n+1 is a Ludic number (in A003309).

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

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A269252 Define a sequence by s(k) = n*s(k-1) - s(k-2), with s(0) = 1, s(1) = n - 1. a(n) is the smallest index k such that s(k) is prime, or -1 if no such k exists.

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A289585 Quotients as they appear as k increases when tau(k) divides phi(k).

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Maple

Mathematica

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A364678 Maximum number of primes between consecutive multiples of n, as permitted by divisibility considerations.

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Python

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.

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Maple

Mathematica