A090698 -id:A090698 - cp's OEIS frontend

A089001 Numbers n such that 2*n^2 + 1 is prime.

1, 3, 6, 9, 21, 24, 27, 30, 33, 36, 42, 45, 66, 72, 75, 87, 93, 96, 99, 102, 105, 123, 132, 135, 153, 156, 162, 177, 186, 189, 201, 204, 219, 228, 237, 240, 255, 264, 273, 285, 297, 300, 306, 321, 324, 327, 351, 357, 360, 366, 375, 387, 393, 399, 405, 417, 423

Offset: 1

N. J. A. Sloane, Dec 20 2003

All terms except the first one are multiples of 3. - Zak Seidov, Feb 24 2006
And because of this, all the primes except for the first one are congruent to 1 (mod 6). - Robert G. Wilson v, Aug 05 2014
For any n in this sequence, 3*(2*n^2 + 1) has the same nonzero digits as its prime factors in base 2n. - Ely Golden, Dec 12 2016

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

Cf. A089008, A090612, A090698.

[n: n in [1..500] | IsPrime(2*n^2+1)]; // Vincenzo Librandi, Jan 07 2013

Select[Range[500], PrimeQ[2#^2 + 1]&] (* Vincenzo Librandi, Jan 07 2013 *)

is(n)=isprime(2*n^2+1) \\ Charles R Greathouse IV, Feb 17 2017

a(n)=((A090698(n)-1)/2)^(1/2).

Starting with n=2, a(n)=3*A089008(n-1). - Zak Seidov, Feb 24 2006

A090612 Numbers k such that the k-th prime is of the form 2*j^2 + 1.

Original entry on oeis.org

2, 8, 21, 38, 153, 191, 232, 279, 327, 378, 493, 559, 1086, 1272, 1360, 1769, 1989, 2111, 2224, 2344, 2471, 3272, 3721, 3863, 4838, 5006, 5359, 6291, 6871, 7077, 7909, 8127, 9245, 9928, 10654, 10889, 12164, 12957, 13764, 14881, 16034, 16343, 16944

Offset: 1

Ray Chandler, Dec 21 2003

nonn

A090698 indexed by A000040.

			From _Jon E. Schoenfield_, Jan 24 2018: (Start)
prime(8) = 19 = 2*3^2 + 1, so 8 is in the sequence.
prime(21) = 73 = 2*6^2 + 1, so 21 is in the sequence.
prime(33) = 137 = 2*68 + 1, and 68 is not a square, so 33 is not in the sequence. (End)

Robert Israel, Table of n, a(n) for n = 1..1525

Cf. A089001, A089008, A090698.

N:= 1000; # to get all entries corresponding to primes <= 2*N^2+1.
R:= select(isprime,[seq(2*k^2+1,k=1..N)]):
A090612:= map(numtheory[pi],R); # Robert Israel, May 09 2014

Select[Range[18000],IntegerQ[Sqrt[(Prime[#]-1)/2]]&] (* Harvey P. Dale, Apr 25 2016 *)

a(n)=k such that A000040(k) = A090698(n) = 2*A089001(n)^2 + 1.

A089008 Numbers k such that 18*k^2 + 1 is prime.

Original entry on oeis.org

1, 2, 3, 7, 8, 9, 10, 11, 12, 14, 15, 22, 24, 25, 29, 31, 32, 33, 34, 35, 41, 44, 45, 51, 52, 54, 59, 62, 63, 67, 68, 73, 76, 79, 80, 85, 88, 91, 95, 99, 100, 102, 107, 108, 109, 117, 119, 120, 122, 125, 129, 131, 133, 135, 139, 141, 142, 143, 147, 150, 152, 154, 156

Offset: 1

N. J. A. Sloane, Dec 20 2003

nonn

There are 8 consecutive terms at n=13537 and n=105819293 for n < 10^9. - Jean C. Lambry, Oct 19 2015

Since 18*k^2 + 1 is divisible by 17 for k == 4, 13 (mod 17), the maximum possible number of consecutive terms is 8, in which case the first term must be congruent to 5 modulo 17 and 7 or 8 modulo 11. - Jianing Song, Nov 14 2021

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A089001, A090612, A090698.

Select[Range[200],PrimeQ[18#^2+1]&]  (* Harvey P. Dale, Apr 25 2011 *)

for(n=0, 1e3, if(isprime(k=(18*n^2 + 1)), print1(n", "))) \\ Altug Alkan, Oct 19 2015

a(n) = A089001(n+1)/3.

A249410 Primes p such that sigma(p-1) is odd.

Original entry on oeis.org

2, 3, 5, 17, 19, 37, 73, 101, 163, 197, 257, 401, 577, 677, 883, 1153, 1297, 1459, 1601, 1801, 2179, 2593, 2917, 3137, 3529, 4051, 4357, 5477, 7057, 8101, 8713, 8837, 10369, 11251, 12101, 13457, 14401, 15139, 15377, 15877, 16901, 17299, 17957, 18433, 19603

Offset: 1

Juri-Stepan Gerasimov, Oct 27 2014

nonn

Subsequence of A058501.
Union of A002496 and A090698. - Ivan Neretin, Dec 04 2018
Except for the terms 2 and 3, union of the primes of the form 4*k^2 + 1 and the primes of the form 18*k^2 + 1. - Jianing Song, Nov 14 2021

			2 is in this sequence because 2 is prime and sigma(2-1) = 1 is odd.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000203, A058501, A090698.

Filtered(Filtered([1..25000],i->IsPrime(i)),p->IsOddInt(Sigma(p-1))); # Muniru A Asiru, Dec 05 2018

Select[Range[20000], PrimeQ[#] && OddQ[DivisorSigma[1, #-1]] &] (* Amiram Eldar, Dec 04 2018 *)

lista(nn) = {forprime(p=2, nn, if (sigma(p-1) % 2, print1(p, ", ")););} \\ Michel Marcus, Oct 30 2014

list(lim)=my(v=List([2]),t); forstep(n=2,sqrt(lim),2, if(isprime(t=n^2+1), listput(v,t))); for(n=1,sqrtint(lim\2), if(isprime(t=2*n^2+1), listput(v,t))); Set(v) \\ Charles R Greathouse IV, Nov 04 2014

More terms from Michel Marcus, Oct 30 2014

A341938 Numbers m such that the geometric mean of tau(m) and phi(m) is an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

Original entry on oeis.org

1, 3, 8, 10, 18, 19, 24, 30, 34, 45, 52, 54, 57, 73, 74, 85, 102, 125, 135, 140, 152, 153, 156, 163, 182, 185, 190, 202, 219, 222, 252, 255, 333, 342, 360, 375, 394, 416, 420, 436, 451, 455, 456, 459, 476, 489, 505, 514, 546, 555, 570, 584, 606, 625, 629, 640, 646, 679, 680, 730

Offset: 1

Bernard Schott, Feb 24 2021

nonn

The first 11 terms of this sequence are also the first 11 terms of A341939: m such that phi(m)/tau(m) is the square of an integer. Indeed, if phi(m)/tau(m) is a perfect square then phi(m)*tau(m) is also a square, but the converse is false. These counterexamples are in A341940, the first one is a(12) = 54.
If k and q are terms and coprimes, then k*q is another term.
Some subsequences (see examples):
-> The seven terms that satisfy tau(m) = phi(m) form the subsequence A020488.
-> Primes p of the form 2*k^2 + 1 (A090698) form another subsequence because tau(p) = 2 and phi(p) = p-1 = 2*k^2, so tau(p)*phi(p) = (2*k)^2.
-> Cubes p^3 where p is a prime of the form k^2+1 (A002496) form another subset with tau(p^3)*phi(p^3) = (2*k*p)^2.

			phi(18) = tau(18) = 6, so phi(18)*tau(18) = 6^2.
phi(19) = 18, tau(19) = 2, so phi(19)*tau(19) = 36 = 6^2.
phi(34) = 16, tau(34) = 4, so phi(34)*tau(34) = 16*4 = 64 = 8^2.
phi(125) = 100, tau(125) = 4, so phi(125)*tau(125) = 400 = 20^2.

Similar for: A011257 (phi*sigma square), A327830 (sigma*tau square).
Subsequences: A020488, A090698.
Cf. A341939, A341940.
Cf. A000005 (tau), A000010 (phi).

with(numtheory): filter:= n -> issqr(phi(n)*tau(n)) : select(filter, [$1..750]);

Select[Range[1000], IntegerQ @ GeometricMean[{DivisorSigma[0, #], EulerPhi[#]}] &] (* Amiram Eldar, Feb 24 2021 *)

isok(m) = issquare(numdiv(m)*eulerphi(m)); \\ Michel Marcus, Feb 24 2021

A341939 Numbers m such that phi(m)/tau(m) is a square of an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

Original entry on oeis.org

1, 3, 8, 10, 18, 19, 24, 30, 34, 45, 52, 57, 73, 74, 85, 102, 125, 135, 140, 152, 153, 156, 163, 182, 185, 190, 202, 219, 222, 252, 255, 333, 342, 360, 375, 394, 416, 420, 436, 451, 455, 456, 459, 476, 489, 505, 514, 546, 555, 570, 584, 606, 625, 629, 640, 646, 679, 680, 730

Offset: 1

Bernard Schott, Feb 24 2021

nonn

The first 11 terms of this sequence are also the first 11 terms of A341938: m such that phi(m)*tau(m) is a square, then, a(12) = 57 while A341938(12) = 54. Indeed, if phi(m)/tau(m) is a perfect square then phi(m)*tau(m) is also a square, but the converse is false. These counterexamples are in A341940, the first one is 54 (last example).
Some subsequences (see examples):
-> The seven terms that satisfy also tau(m) = phi(m) form the subsequence A020488 with phi(m)/tau(m) = 1^2.
-> Primes p of the form 2*k^2 + 1 (A090698) form another subsequence because tau(p) = 2 and phi(p) = p-1 = 2*k^2, so phi(p)/tau(p) = k^2.
-> Cubes p^3 where p is a prime of the form k^2+1 (A002496) form another subset because if p = 2, phi(8)/tau(8)=1, and if p odd, phi(p^3)/tau(p^3) = (k*p/2)^2 with k even.

			phi(30) = 8, tau(30) = 8 so phi(30)/tau(30) = 1^2, and 30 is a term.
phi(45) = 24, tau(45) = 6, so phi(45)/tau(45) = 4 = 2^2, and 85 is a term.
phi(125) = 100, tau(125) = 4, so phi(125)/tau(125) = 25 = 5^2, and 125 is a term.
phi(54) = 18, tau(54) = 8, and phi(54)/tau(54) = 18/8 = 9/4 = (3/2)^2 and 54 is not a term while phi(54)*tau(54) = 12^2.

Intersection of A020491 and A341938.
Similar for: A144695 (sigma(n)/tau(n) perfect square), A293391 (sigma(n)/phi(n) perfect square).
Subsequences: A090698, A020488.
Cf. A069237, A289585, A341940.
Cf. A000005 (phi), A000010(tau).

with(numtheory): filter:= q -> phi(q)/tau(q) = floor(phi(q)/tau(q)) and issqr(phi(q)/tau(q)) : select(filter, [$1..750]);

Select[Range[1000], IntegerQ @ Sqrt[EulerPhi[#]/DivisorSigma[0, #]] &] (* Amiram Eldar, Feb 24 2021 *)

isok(m) = my(x=eulerphi(m)/numdiv(m)); (denominator(x)==1) && issquare(x); \\ Michel Marcus, Feb 24 2021

A129827 Numbers k such that Euler's totient phi(k) divided by 2 is a perfect square.

Original entry on oeis.org

3, 4, 6, 15, 16, 19, 20, 24, 27, 30, 38, 51, 54, 64, 68, 73, 80, 91, 95, 96, 102, 111, 117, 120, 135, 146, 148, 152, 163, 182, 190, 216, 222, 228, 234, 243, 252, 255, 256, 270, 272, 275, 303, 320, 323, 326, 340, 365, 375, 384, 404, 408, 455, 459, 480, 486, 500

Offset: 1

Walter Nissen, May 20 2007

nonn

Primes in this sequence are of the form 2*m^2+1 (see A090698). - Bernard Schott, Mar 07 2020

If k is an odd term, so is 2*k. If k is an even term, so is 4*k. - Waldemar Puszkarz, Oct 15 2024

			a(4) is 15 because phi(15) = 8, which is twice the square of 2.

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

Cf. A000010, A000290, A090698 (subsequence).

Select[Range[500], IntegerQ @ Sqrt[EulerPhi[#]/2] &] (* Amiram Eldar, Mar 07 2020 *)

isok(n) = issquare(eulerphi(n)/2) \\ Michel Marcus, Jul 23 2013

from sympy import totient
from sympy.ntheory.primetest import is_square
for i in range(3, 501):
    if is_square(int(totient(i)/2)):
        print(i, end=", ") # Waldemar Puszkarz, Oct 15 2024

A247271 Numbers n such that n^2+1 and 2*n^2+1 are both prime numbers.

Original entry on oeis.org

1, 6, 24, 36, 66, 156, 204, 240, 264, 300, 306, 474, 570, 636, 750, 864, 936, 960, 1146, 1176, 1290, 1494, 1524, 1716, 1974, 2034, 2136, 2310, 2406, 2706, 2736, 2964, 3156, 3240, 3624, 3756, 3774, 3900, 3984, 4026, 4080, 4524, 4530, 4554, 4590, 4644, 4650, 4716

Offset: 1

Michel Lagneau, Sep 11 2014

Numbers n such that A002522(n) and A058331(n) are prime numbers.
a(n)==0 mod 6 because the primes n^2+1 and 2*n^2+1 are congruent to 1 (mod 6).
The corresponding pairs of primes (n^2+1,2*n^2+1) are (2,3), (37,73), (577, 1153), (1297,2593), (4357,8713), (24337,48673), ...

			a(2)=6 because A002522(6)=37 and A058331(6)=73 are both prime numbers.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A002496, A090698, A002522, A058331.

[n: n in [0..5000] | IsPrime(n^2+1) and IsPrime(2*n^2+1)]; // Vincenzo Librandi, Sep 14 2014

A247271:=n->`if`(isprime(n^2+1) and isprime(2*n^2+1), n, NULL): seq(A247271(n), n=1..10^4); # Wesley Ivan Hurt, Sep 12 2014

lst={}; Do[p=n^2+1; q=2n^2+1; If[PrimeQ[p] && PrimeQ[q], AppendTo[lst, n]], {n, 5000}]; lst

for(n=1,10^4,if(isprime(n^2+1)&&isprime(2*n^2+1),print1(n,", "))) \\ Derek Orr, Sep 11 2014

A256917 Primes which are not the sums of two consecutive nonsquares.

Original entry on oeis.org

2, 3, 7, 17, 19, 31, 71, 73, 97, 127, 163, 199, 241, 337, 449, 577, 647, 881, 883, 967, 1151, 1153, 1249, 1459, 1567, 1801, 2179, 2311, 2591, 2593, 2887, 3041, 3361, 3527, 3529, 3697, 4049, 4051, 4231, 4801, 4999, 5407, 6271, 6961, 7687, 7937, 8191, 8713, 9521, 10369, 10657

Offset: 1

Juri-Stepan Gerasimov, Apr 23 2015

The union of 2 and A066436 and A090698.

The sums of two consecutive nonsquares are 5, 8, 11, 13, 15, 18, 21, 23, 25, 27, 29, 32, 35, 37, ...

			2, 3, 7 are in this sequence because first three sums of two consecutive nonsquares are 5, 8, 11 and 2, 3, 7 are primes.

Colin Barker and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 800 terms from Barker)

Cf. A000037, A066436, A090698, A154670.

Union[{2},Select[Table[2n^2-1,{n,0,1000}],PrimeQ],Select[Table[2n^2+1,{n,0,1000}],PrimeQ]] (* Ivan N. Ianakiev, Apr 24 2015 *)
Module[{nn=11000,ns},ns=Total/@Partition[Select[Range[nn],!IntegerQ[Sqrt[#]]&],2,1]; Complement[ Prime[Range[PrimePi[Last[ns]]]],ns]] (* Harvey P. Dale, Mar 06 2024 *)

a256917(maxp) = {
  ps=[2];
  k=1; while((t=2*k^2-1)<=maxp, k++; if(isprime(t), ps=setunion(ps, [t])));
  k=1; while((t=2*k^2+1)<=maxp, k++; if(isprime(t), ps=setunion(ps, [t])));
  ps
}
a256917(11000) \\ Colin Barker, Apr 23 2015

list(lim)=my(v=List([2]),t); for(k=2,sqrtint((lim+1)\2), if(isprime(t=2*k^2-1), listput(v,t))); for(k=1,sqrtint((lim-1)\2), if(isprime(t=2*k^2+1), listput(v,t))); Set(v) \\ Charles R Greathouse IV, Apr 23 2015

A257163 Primes of the form 3n^2 + 2.

Original entry on oeis.org

2, 5, 29, 149, 509, 677, 1877, 3677, 8429, 9749, 11909, 13469, 17789, 22709, 27077, 28229, 45389, 46877, 53069, 70229, 72077, 81677, 100469, 102677, 114077, 128549, 141269, 154589, 180077, 192029, 195077, 207509, 223589, 230189, 261077, 312989, 340709, 352949, 395309, 399677, 426389

Offset: 1

Juri-Stepan Gerasimov, Apr 16 2015

Two together with A027864(n).

Generated by n = 0, 1, 3, 7, 13, 15, 25, 35, 53, 57, ...

			2 is in this sequence because 3*0^2 + 2 = 2 and 2 is prime.

Cf. A103564, A027864. Primes of the form k*n^2 + k - 1: A090698, this sequence, A121825, A201483, A201600, A201607, A201704.

[a: n in [0..400] | IsPrime(a) where a is (3*n^2+2)];

Select[Table[3 n^2 + 2, {n, 0, 400}], PrimeQ] (* Vincenzo Librandi, Apr 17 2015 *)

select(isprime, vector(100, n, 3*n^2-6*n+5)) \\ Charles R Greathouse IV, Apr 17 2015

cp's OEIS Frontend

A089001 Numbers n such that 2*n^2 + 1 is prime.

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Magma

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A090612 Numbers k such that the k-th prime is of the form 2*j^2 + 1.

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Maple

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A089008 Numbers k such that 18*k^2 + 1 is prime.

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A249410 Primes p such that sigma(p-1) is odd.

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A341938 Numbers m such that the geometric mean of tau(m) and phi(m) is an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

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A341939 Numbers m such that phi(m)/tau(m) is a square of an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

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A129827 Numbers k such that Euler's totient phi(k) divided by 2 is a perfect square.

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