A052291 -id:A052291 - cp's OEIS frontend

A378146 Primes p such that 16*p^4 + 1 is prime.

2, 3, 17, 23, 37, 41, 53, 59, 71, 97, 127, 139, 167, 233, 263, 277, 283, 379, 389, 457, 521, 563, 571, 601, 619, 661, 691, 743, 797, 809, 811, 823, 853, 859, 877, 967, 971, 997, 1051, 1063, 1103, 1187, 1277, 1289, 1321, 1367, 1399, 1433, 1451, 1499

Offset: 1

Juri-Stepan Gerasimov, Nov 17 2024

nonn

Primes p such that (2*p)^(2^k) + 1 is prime: A005384 (k = 0), A052291 (k = 1), this sequence (k = 2).

Cf. A378134, A378143.

[p: p in PrimesUpTo(1500) | IsPrime(16*p^4+1)];

Select[Prime[Range[250]], PrimeQ[16*#^4 + 1] &] (* Amiram Eldar, Nov 17 2024 *)

list(lim)=my(v=List()); forprime(p=2,lim, if(isprime(16*p^4+1), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Nov 17 2024

a(n) >> n log^2 n. - Charles R Greathouse IV, Nov 17 2024

A060429 a(n) = 4*prime(n)^2+1.

Original entry on oeis.org

17, 37, 101, 197, 485, 677, 1157, 1445, 2117, 3365, 3845, 5477, 6725, 7397, 8837, 11237, 13925, 14885, 17957, 20165, 21317, 24965, 27557, 31685, 37637, 40805, 42437, 45797, 47525, 51077, 64517, 68645, 75077, 77285, 88805

Offset: 1

Jason Earls, Apr 06 2001

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A001248, A052291, A052292.

A060429:=n->4*ithprime(n)^2+1; seq(A060429(n), n=1..100); # Wesley Ivan Hurt, Dec 17 2013

Table[4*Prime[n]^2+1, {n, 100}] (* Wesley Ivan Hurt, Dec 17 2013 *)

{ n=0; forprime (p=1, 542, write("b060429.txt", n++, " ", 4*p^2 + 1); ) } \\ Harry J. Smith, Jul 05 2009

a(n) = 4*A001248(n) + 1. - Vincenzo Librandi, Dec 17 2013

A121834 Primes p of the form 4n^2 + 1 such that 4p^2+1 is also prime.

Original entry on oeis.org

5, 37, 677, 1297, 2917, 8837, 13457, 50177, 147457, 156817, 246017, 341057, 414737, 746497, 1136357, 1726597, 1833317, 2119937, 2802277, 2808977, 3013697, 3549457, 3865157, 3896677, 4104677, 4384837, 5354597, 5410277, 5779217, 6031937, 6635777, 7001317

Offset: 1

Zak Seidov, Aug 28 2006

nonn

Intersection of A001912 and A121326. Except for the first term all other terms are == 7 (mod 10). Also all the primes 4*p^2+1 are == 7 (mod 10). - Zak Seidov, Mar 05 2015

Cf. A002496, A052291, A121326.

fpQ[n_]:=PrimeQ[n]&&PrimeQ[4n^2+1]; Select[4Range[2000]^2+1,fpQ] (* Harvey P. Dale, Nov 07 2016 *)

isA121834(n)={ if( (n-1) %4, return(0) ; ) ; if( issquare((n-1)/4), if( isprime(4*n^2+1), return(1), return(0) ), return(0) ; ) ; } { for(i=1,1000000, p=prime(i) ; if( isA121834(p), print1(p,",") ; ) ; ) ; } /* R. J. Mathar, Sep 01 2006*/

More terms from R. J. Mathar, Sep 01 2006

A122429 Primes p such that q = 4p^2 + 1, r = 4q^2 + 1 and s = 4r^2 + 1 are all primes.

Original entry on oeis.org

13, 9833, 41647, 151607, 264757, 356123, 361223, 446863, 449093, 457813, 531383, 641057, 655927, 841697, 855947, 899263, 913687, 1052813, 1081757, 1379383, 1506493, 1575757, 1685087, 1821013, 1821377, 1981517, 2054233, 2142037

Offset: 1

Zak Seidov, Oct 20 2006

nonn

Next terms up to 400000th prime are 2286877, 2524157, 2595247, 2621737, 2931583, 3023437, 3425843, 3428567, 3538517, 3705187, 3777883, 3799717, 3875143, 3913727, 3973553, 4019833, 4167073, 4249523, 4488167, 4651873, 4822193, 4914937, 5054167, 5108293, 5140147, 5465303, 5520007, 5542003. - Zak Seidov, Jan 16 2009

Subsequence of A122424. - Pierre CAMI, Jul 21 2014

			13 is there because 13, 677, 1833317 and 13444204889957 are prime.

Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Inc., 2005, p.74.

Pierre CAMI, Table of n, a(n) for n = 1..3636

Cf. A052291, A005574, A001912, A122424.

Reap[Do[p=Prime[n];q=4p^2+1;r=4q^2+1;s=4r^2+1;If[PrimeQ[{q,r,s}]=={True, True,True},Sow[p]],{n,15000}]][[2,1]]
Select[Prime[Range[200000]],AllTrue[NestList[4#^2+1&,#,3],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 22 2015 *)

f(x)=4*x^2+1;
forprime(p=1, 10^8, if(isprime(f(p))&&isprime(f(f(p)))&&isprime(f(f(f(p)))), print1(p, ", "))) \\ Derek Orr, Jul 31 2014

More terms from Don Reble, Oct 24 2006

Edited by R. J. Mathar, Nov 02 2009

A188583 Primes p such that 6*p^3+1 is also prime.

Original entry on oeis.org

3, 5, 13, 41, 97, 131, 223, 283, 353, 397, 461, 467, 523, 577, 661, 677, 691, 773, 811, 887, 937, 997, 1091, 1223, 1277, 1321, 1447, 1487, 1567, 1571, 1637, 1721, 1741, 1777, 1823, 1861, 2161, 2243, 2273, 2341, 2351, 2357, 2371, 2383, 2467, 2551

Offset: 1

Bruno Berselli, Apr 22 2011

nonn

			For prime  p = 283,  6*p^3+1 = 135991123  is prime.

Bruno Berselli, Table of n, a(n) for n = 1..900

Cf. A052291, A005384; A007693, A153812.

[ p: p in PrimesUpTo(2600) | IsPrime(6*p^3+1) ];

Select[Prime[Range[500]],PrimeQ[6#^3+1]&] (* Harvey P. Dale, May 13 2011 *)

A306882 Even numbers k such that phi(m) = k^2 has no solution.

Original entry on oeis.org

22, 34, 38, 46, 58, 62, 76, 78, 82, 86, 92, 98, 102, 106, 118, 122, 138, 142, 152, 154, 158, 164, 166, 172, 178, 182, 190, 194, 202, 212, 214, 218, 226, 238, 244, 254, 258, 262, 266, 274, 278, 282, 298, 302, 304, 310, 316, 318, 322, 328, 332, 334, 338, 344, 346, 356, 358, 362

Offset: 1

Bernard Schott, Mar 15 2019

nonn

In the link, P. Pollack and C. Pomerance "show that almost all squares are missing from the range of Euler's phi-function".
Except for m=1 and m=2, phi(m) is always even, so, the odd numbers >= 3 are not included in the data for clarity.
Includes 2*p if p is a prime not in A052291. - Robert Israel, Apr 10 2019

			phi(489) = 18^2, phi(401) = 20^2, phi(577) = 24^2, phi(677) = 26^2, but there is no integer m such that phi(m) = 22^2 = 484.

Robert Israel, Table of n, a(n) for n = 1..10000
P. Pollack and C. Pomerance, Square values of Euler's function, preprint (2013); Bulletin of the London Mathematical Society, Volume 46, Issue 2, 1 April 2014, Pages 403-414.

Cf. A000010, A002202, A052291, A058277, A062732, A221284, A221285.

select(t -> numtheory:-invphi(t^2)=[], [seq(i,i=2..400,2)]);  # Robert Israel, Apr 10 2019

isok(n) = !(n%2) && !istotient(n^2); \\ Michel Marcus, Mar 15 2019

A309498 Least number k > 0 such that 4p^2k^2 + 1 is prime, where p = prime(n) is the n-th prime.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 15, 4, 2, 2, 1, 13, 5, 1, 6, 2, 13, 1, 2, 1, 3, 9, 5, 10, 5, 1, 5, 2, 9, 6, 8, 4, 2, 7, 3, 1, 1, 10, 11, 2, 7, 2, 1, 4, 5, 13, 4, 4, 3, 1, 3, 7, 2, 4, 6, 3, 7, 5, 2, 20, 6, 4, 2, 6, 1, 2, 1, 4, 3, 4, 3, 5, 5, 5, 16, 2, 14, 3, 3, 2, 2, 5, 5

Offset: 1

Amiram Eldar, Aug 05 2019

Gagola calculated the first 669 terms of this sequence (for all the primes p < 5000) using an HP 9830 in 1981. She found that the largest value of k was only 45 and that 84% of the values of k were less than or equal to 10.

The Generalized Dickson Conjecture implies that the sequence contains each positive integer infinitely many times. - Robert Israel, Aug 05 2019

			a(1) = 1 since 4*1^2*prime(1)^2 + 1 = 4*1*2^2 + 1 = 17 is prime.

Jason Yuen, Table of n, a(n) for n = 1..10000
Gloria Gagola, Progress on primes, News and Letters, Mathematics Magazine, Vol. 54, No. 1 (1981), p. 43.

Cf. A002496, A005574, A035092, A052291.

f:= proc(n) local q,k;
  q:= 4*ithprime(n)^2;
  for k from 1 do
     if isprime(q*k^2+1) then return k fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Aug 05 2019

a[n_] := Module[{k = 1, p = Prime[n]}, While[!PrimeQ[4 * k^2 * p^2 + 1], k++]; k]; Array[a, 100]

a(n) = my(k=1, p=prime(n)); while (!isprime(4*p^2*k^2 + 1), k++); k; \\ Michel Marcus, Aug 05 2019

A245746 Prime numbers P such that Q=4P^2+1, R=4Q^2+1, S=4R^2+1, T=4S^2+1 and Q, R, S, T are all prime numbers.

Original entry on oeis.org

9220303, 16079387, 17232253, 43606237, 66063373, 85403083, 97649917, 104719757, 159685553, 180467533, 197072563, 344777863, 492619373, 517774063, 647320727, 672712637, 715230127, 769494413, 790845563, 909545573, 944196137, 975302173, 1120585597, 1123182763

Offset: 1

Pierre CAMI, Jul 31 2014

nonn

Subsequence of A122429.

Pierre CAMI, Table of n, a(n) for n = 1..30

Cf. A052291, A122424, A122429.

pnQ[n_]:=Module[{q=4n^2+1,r,s,t},r=4q^2+1;s=4r^2+1;t=4s^2+1;AllTrue[ {q,r,s,t},PrimeQ]]; Select[Prime[Range[5678*10^4]],pnQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 08 2019 *)

A333803 Primes p such that 2p+1 and 4p^2+1 are also prime.

Original entry on oeis.org

2, 3, 5, 233, 653, 683, 1013, 1973, 2003, 2393, 2543, 2753, 3023, 3413, 5003, 5333, 7043, 7823, 8663, 9293, 10613, 13463, 13913, 14783, 15233, 15923, 16823, 18233, 20693, 20753, 21713, 21803, 22433, 27743, 27983, 29723, 30323, 30773, 31253, 31793, 32003, 33053, 33623, 33773, 34283, 36083, 37013

Offset: 1

Robert Israel, Apr 05 2020

nonn

The generalized Bunyakovsky conjecture implies there are infinitely many terms.

			a(3)=5 is a member because 5, 2*5+1=11 and 4*5^2+1= 101 are all prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Mathematics StackExchange,Find all p such that p, 2p+1, 4p^2+1 are prime

Intersection of A005384 and A052291.

filter:= proc(n)
  isprime(n) and isprime(2*n+1) and isprime(4*n^2+1)
end proc:
select(filter, [2,3,seq(i,i=5..10^5,6)]);

Select[Prime[Range[4000]],AllTrue[{2#+1,4#^2+1},PrimeQ]&] (* Harvey P. Dale, Sep 15 2021 *)

A363059 Numbers k such that the number of divisors of k^2 equals the number of divisors of phi(k), where phi is the Euler totient function.

Original entry on oeis.org

1, 5, 57, 74, 202, 292, 394, 514, 652, 1354, 2114, 2125, 3145, 3208, 3395, 3723, 3783, 4053, 4401, 5018, 5225, 5298, 5425, 5770, 6039, 6363, 6795, 6918, 7564, 7667, 7676, 7852, 7964, 8585, 9050, 9154, 10178, 10535, 10802, 10818, 10954, 11223, 12411, 13074, 13634

Offset: 1

Amiram Eldar, May 16 2023

nonn

Numbers k such that A048691(k) = A062821(k).
Amroune et al. (2023) characterize solutions to this equation and prove that Dickson's conjecture implies that this sequence is infinite.
They show that the only squarefree semiprime terms are 57, 514 and some of the numbers of the form 2*(4*p^2+1), where p and 4*p^2+1 are both primes (a subsequence of A259021).

			5 is a term since both 5^2 = 25 and phi(5) = 4 have 3 divisors.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Zahra Amroune, Djamel Bellaouar and Abdelmadjid Boudaoud, A class of solutions of the equation d(n^2) = d(phi(n)), Notes on Number Theory and Discrete Mathematics, Vol. 29, No. 2 (2023), pp. 284-309.
Wikipedia, Dickson's conjecture.

Cf. A000005, A000010, A048691, A062821.

Cf. A052291, A259021.

Select[Range[15000], DivisorSigma[0, #^2] == DivisorSigma[0, EulerPhi[#]] &]

is(n) = numdiv(n^2) == numdiv(eulerphi(n));

cp's OEIS Frontend

A378146 Primes p such that 16*p^4 + 1 is prime.

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Author

Keywords

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Magma

Mathematica

PARI

Formula

A060429 a(n) = 4*prime(n)^2+1.

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Maple

Mathematica

PARI

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A121834 Primes p of the form 4*n^2 + 1 such that 4*p^2+1 is also prime.

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Mathematica

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A122429 Primes p such that q = 4p^2 + 1, r = 4q^2 + 1 and s = 4r^2 + 1 are all primes.

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Mathematica

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Extensions

A188583 Primes p such that 6*p^3+1 is also prime.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A306882 Even numbers k such that phi(m) = k^2 has no solution.

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Author

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Comments

Examples

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Crossrefs

Programs

Maple

PARI

A309498 Least number k > 0 such that 4*p^2*k^2 + 1 is prime, where p = prime(n) is the n-th prime.

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Author

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Comments

Examples

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Crossrefs

Programs

Maple

Mathematica

PARI

A245746 Prime numbers P such that Q=4*P^2+1, R=4*Q^2+1, S=4*R^2+1, T=4*S^2+1 and Q, R, S, T are all prime numbers.

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A121834 Primes p of the form 4n^2 + 1 such that 4p^2+1 is also prime.

A309498 Least number k > 0 such that 4p^2k^2 + 1 is prime, where p = prime(n) is the n-th prime.

A245746 Prime numbers P such that Q=4P^2+1, R=4Q^2+1, S=4R^2+1, T=4S^2+1 and Q, R, S, T are all prime numbers.

A333803 Primes p such that 2p+1 and 4p^2+1 are also prime.