A056899 -id:A056899 - cp's OEIS frontend

A216981 Primes of the form n^7+2.

2, 3, 4782971, 1801088543, 1174711139839, 3938980639169, 93206534790701, 425927596977749, 1107984764452583, 2149422977421877, 7416552901015627, 19891027786401119, 307732862434921877, 830512886046548069, 1042842864990234377, 3678954248903875651

Offset: 1

Michel Lagneau, Sep 21 2012

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

Cf. A056899, A144953, A216975, A216977, A216979.

[a: n in [0..500] | IsPrime(a) where a is n^7+2]; // Vincenzo Librandi, Mar 15 2013

lst={}; Do[p=n^7+2; If[PrimeQ[p], AppendTo[lst, p]], {n, 6!}]; lst
Select[Table[n^7 + 2, {n, 0, 400}], PrimeQ] (* Vincenzo Librandi, Mar 15 2013 *)

v=select(n->isprime(n^7+2),vector(2000,n,n-1)); /* A216980 */
vector(#v, n, v[n]^7+2)
/* Joerg Arndt, Sep 21 2012 */

select(isprime, vector(2000,n,(n-1)^7+2)) \\ Charles R Greathouse IV, Sep 21 2012

A241528 Primes p such that p + 1234567890 is also prime where 1234567890 is the first pandigital number with digits in order.

Original entry on oeis.org

17, 23, 37, 59, 131, 139, 157, 199, 241, 311, 353, 359, 397, 433, 479, 547, 673, 691, 769, 877, 937, 947, 953, 1051, 1091, 1097, 1181, 1297, 1301, 1409, 1451, 1471, 1489, 1531, 1609, 1619, 1697, 1709, 1861, 1879, 1889, 1913, 1951, 2053, 2063, 2089, 2099, 2113

Offset: 1

K. D. Bajpai, Apr 25 2014

nonn

			17 is prime and appears in the sequence because 17 + 1234567890 = 1234567907, which is also prime.
23 is prime and appears in the sequence because 23 + 1234567890 = 1234567913, which is also prime.
19 is prime but not included in the sequence since 19 + 1234567890 = 1234567909 = (59107)*(20887), which is not prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A002496, A028871, A050289, A056899, A234812, A235640, A240587.

KD := proc() local a,k; k:=ithprime(n);a:=k+1234567890; if isprime(a) then RETURN (k); fi; end: seq(KD(), n=1..1000);

lst={}; Do[p=Prime[n]; If[PrimeQ[p+1234567890], AppendTo[lst,p]],{n,1,1000}]; lst
(* For the b-file *)  c=0; k=Prime[n]; a=k+1234567890; Do[If[PrimeQ[a], c++; Print[c," ",k]],{n,1,10^5}]
Select[Prime[Range[400]],PrimeQ[#+1234567890]&] (* Harvey P. Dale, Nov 18 2021 *)

s=[]; forprime(p=2, 3000, if(isprime(p+1234567890), s=concat(s, p))); s \\ Colin Barker, Apr 25 2014

A241537 Cubes c such that c + 1234567890 is prime where 1234567890 is the first pandigital number with digits in order.

Original entry on oeis.org

1, 50653, 79507, 456533, 571787, 1295029, 1685159, 1771561, 2248091, 2685619, 3307949, 4173281, 7880599, 9393931, 10218313, 10793861, 11697083, 17373979, 18191447, 22665187, 30664297, 47045881, 70444997, 111284641, 146363183, 151419437, 156590819, 192100033

Offset: 1

K. D. Bajpai, Apr 25 2014

nonn

			50653 = 37^3 and appears in the sequence because 50653 + 1234567890 = 1234618543, which is prime.
79507 = 43^3  and appears in the sequence because 79507 + 1234567890 = 1234647397, which is prime.
64000 = 40^3 but not included in the sequence since 64000 + 1234567890 = 1234631890 = (2)*(5)*(29389)*(4201), which is not prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A002496, A028871, A050289, A056899, A234812, A235640, A240587.

KD := proc() local a,c; c:=n^3;a:=c+1234567890; if isprime(a) then RETURN (c); fi; end: seq(KD(), n=1..1000);

lst={}; Do[c=n^3; If[PrimeQ[c+1234567890], AppendTo[lst,c]], {n,1,1000}]; lst
(*For the b-file*)  m=0; c=n^3; a=c+1234567890; Do[If[PrimeQ[a],m++; Print[m," ",c]], {n,1,4*10^5}]

s=[]; for(n=1, 1000, c=n^3; if(isprime(c+1234567890), s=concat(s, c))); s \\ Colin Barker, Apr 25 2014

A241538 Squares s such that s + 1234567890 is prime.

Original entry on oeis.org

1, 169, 1681, 6889, 8281, 11881, 24649, 27889, 41209, 57121, 58081, 67081, 80089, 101761, 124609, 175561, 185761, 201601, 212521, 332929, 380689, 413449, 461041, 508369, 534361, 609961, 625681, 654481, 683929, 693889, 822649, 829921, 833569, 1014049, 1018081

Offset: 1

K. D. Bajpai, Apr 25 2014

1234567890 is the first pandigital number with digits in order.

			169 = 13^2 and appears in the sequence because 169 + 1234567890 = 1234568059, which is prime.
1681 = 41^2  and appears in the sequence because 1681 + 1234567890 = 1234569571, which is prime.
625 = 25^2 but is not included in the sequence since 625 + 1234567890 = 1234568515 = (5)*(246913703), which is not prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A002496, A028871, A050289, A056899, A234812, A235640, A240587.

KD := proc() local a,s; s:=n^2;a:=s+1234567890; if isprime(a) then RETURN (s); fi; end: seq(KD(), n=1..2000);

A241538 = {}; Do[s = n^2; If[PrimeQ[s + 1234567890], AppendTo[A241538, s]], {n, 2000}]; A241538
(* For the b-file *) c = 0; s = n^2; a = s + 1234567890; Do[If[PrimeQ[a], c++; Print[c, " ", s]], {n, 4*10^5}] (* Bajpai *)
Select[Range[1000]^2, PrimeQ[# + 1234567890] &] (* Alonso del Arte, Apr 25 2014 *)

A248825 a(n) = n^2 + 1 - (-1)^n.

Original entry on oeis.org

0, 3, 4, 11, 16, 27, 36, 51, 64, 83, 100, 123, 144, 171, 196, 227, 256, 291, 324, 363, 400, 443, 484, 531, 576, 627, 676, 731, 784, 843, 900, 963, 1024, 1091, 1156, 1227, 1296, 1371, 1444, 1523, 1600, 1683, 1764, 1851, 1936, 2027, 2116

Offset: 0

Paul Curtz, Oct 15 2014

Also, A016742 and A164897 interleaved.
See the spiral in Example field of A054552: after 0, the sequence is given by the terms of the semidiagonals 4, 16, 36, 64, 100, ... and 3, 11, 27, 51, 83, ... sorted into ascending order.
Primes of the sequence are in A056899.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000290, A008586, A008602, A010673, A016742, A042963, A056899, A164897, A166519, A168277, A248800.

[n^2+1-(-1)^n: n in [0..60]]; // Vincenzo Librandi, Oct 16 2014

Table[n^2 + 1 - (-1)^n, {n, 0, 60}] (* Vincenzo Librandi, Oct 16 2014 *)
LinearRecurrence[{2,0,-2,1},{0,3,4,11},60] (* Harvey P. Dale, Jun 30 2019 *)

vector(100,n,(n-1)^2+1+(-1)^n) \\ Derek Orr, Oct 15 2014

[n^2+1-(-1)^n for n in (0..60)] # Bruno Berselli, Oct 16 2014

a(n) = a(-n) = 2*a(n-1) - 2*(n-3) + a(n-4).
a(n) = n^2 + A010673(n) = (n+1)^2 - A168277(n+1).
a(n+1) = A248800(n) + A042963(n+1) = a(n) + A166519(n).
a(n+2) = a(n) + 4*n.
a(n+5) = a(n-5) + A008602(n).
G.f.: x*(3 - 2*x + 3*x^2)/((1 + x)*(1 - x)^3). - Bruno Berselli, Oct 15 2014
Sum_{n>=1} 1/a(n) = Pi^2/24 + tanh(Pi/sqrt(2))*Pi/(4*sqrt(2)). - Amiram Eldar, Aug 21 2022

Edited by Bruno Berselli, Oct 16 2014

A167675 Least prime p such that p-2 has n divisors, or 0 if no such prime exists.

Original entry on oeis.org

3, 5, 11, 17, 83, 47, 0, 107, 227, 569, 59051, 317, 0, 9479, 2027, 947, 0, 2207, 0, 2837, 88211, 295247, 0, 3467, 50627, 9034499, 11027, 47387, 0, 14177, 0, 15017, 1476227, 215233607, 455627, 17327, 150094635296999123, 15884240051, 89813531, 36857, 0

Offset: 1

T. D. Noe, Nov 09 2009

nonn

This sequence is the idea of Alonso Del Arte. For n>2, a(n) is conjectured to be the smallest number that is orderly (see A167408) for n-1 values of k. For example, 11 is orderly for k=3 and 9. See A056899 for other primes p that are orderly for two k. It is a conjecture because it is not known whether there are composite numbers that are orderly for more than one value of k.

The terms a(n) for prime n are 0 except when 3^(n-1)+2 is prime. Using A051783, we find the exceptional primes to be n=2, 3, 5, 11, 37, 127, 6959.... For these n, a(n) = 3^(n-1)+2. For any n, it is easy to use the factorization of n to find the forms of numbers that have n divisors. For example, for n=38=2*19, we know that the prime must have the form 2+q*r^18 with q and r prime. The smallest such prime is 2+41*3^18.

Cf. A066814 (smallest prime p such that p-1 has n divisors)

nn=25; t=Table[0,{nn}]; Do[p=Prime[n]; k=DivisorSigma[0,p-2]; If[k<=nn && t[[k]]==0, t[[k]]=p], {n,2,10^6}]; t

A178639 Numbers m such that all three values m^2 + 13^k, k = 1, 2, 3, are prime.

Original entry on oeis.org

10, 12, 200, 268, 340, 418, 488, 530, 838, 840, 1102, 1720, 1830, 2240, 2410, 2768, 3148, 3202, 3318, 3322, 3958, 4162, 4610, 5080, 5672, 5700, 5722, 5870, 6178, 6302, 6480, 7490, 8130, 8262, 8888, 9132, 9602, 9618, 10638

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 31 2010

nonn

Subsequence of A176969.

The least-significant digit of all terms is one of 0, 2 or 8, because for odd digits m^2 + 13^k would be even (not prime), and for digits 4 and 6 the number m^2 + 13^2 is a multiple of 5.

			m=10 is in the sequence because 10^2 + 13 = 113 = prime(30), 10^2 + 13^2 = 269 = prime(57), 10^2 + 13^3 = 2297 = prime(342).
m=8888 is in the sequence because 8888^2 + 13 = 78996557 = prime(4614261), 8888^2 + 13^2 = 78996713 = prime(4614269), 8888^2 + 13^3 = 78998741 = prime(4614379).
m=6480 yields a prime 6480^2 + 13^k even for k=0.
m=7490 yields a prime 7490^2 + 13^k even for k=0 and k=4.

B. Bunch: The Kingdom of Infinite Number: A Field Guide, W. H. Freeman, 2001.
R. Courant, H. Robbins: What Is Mathematics? An Elementary Approach to Ideas and Methods, Oxford University Press, 1996.
G. H. Hardy, E. M. Wright, E. M., An Introduction to the Theory of Numbers (5th edition), Oxford University Press, 1980.

Cf. A000040, A000290, A055394, A056899, A086380, A113536, A176371, A176969.

keyword:base removed by R. J. Mathar, Jul 13 2010

A214001 Numbers n such that n^2+2, n^3+2, n^4+2 and n^5+2 are all prime.

Original entry on oeis.org

0, 1, 909, 2055, 11925, 145881, 191079, 254199, 358875, 490215, 614241, 642105, 648261, 689655, 864159, 959595, 1030911, 1047585, 1056981, 1150335, 1366971, 1406571, 1669845, 1746525, 2299485, 2357751, 2491809, 2494329, 2629869, 2876859, 3162159, 3220041, 3257595

Offset: 1

Michel Lagneau, Feb 15 2013

nonn

n^6+2 is also prime for n = 0, 1, 1746525, 2876859, …

Cf. A056899, A144953, A182343, A216977, A216979, A216930.

Select[Range[3500000], And@@PrimeQ/@(Table[n^i+2, {i, 2, 5}]/.n->#)&]
Select[Range[0,33*10^5],AllTrue[#^Range[2,5]+2,PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 21 2018 *)

A283222 Integer area of integer-sided triangle such that the sides are of the form p, p+2, 2(p-1), where p, p+2 and (p-1)/2 are prime numbers.

Original entry on oeis.org

66, 6810, 182430, 105470250, 17356640970, 678676246650, 1879504308930, 4491035717130, 10618004862030, 21136679055030, 23751520478010, 27081671511090, 27596192489190, 31721097756750, 115248550935750, 133303609919430, 140838829659930, 182797297112430, 197799116497230

Offset: 1

Michel Lagneau, Mar 03 2017

nonn

Subsequence of A257049.
The area of a triangle (a,b,c) is given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) where its side lengths are a, b, c and semiperimeter s = (a+b+c)/2.
We observe that the sides of each triangle are of the form (k^2+2, k^2+4, 2k^2+2) and Heron's formula gives immediately the area k(2k^2+4) => a(n)= 2*A086381(n)*A253639(n).
The corresponding primes p are a subsequence of A056899 (primes of the form n^2+2): 11, 227, 2027, 140627, 4223027, 48650627, 95942027, 171479027, ...
We observe that p == 11 mod 72, or p == 11, 83 mod 144. For p>11, p == 27, 227, 627 mod 1000.
An interesting property: the greatest prime divisor of a(n) is equal to p. For instance, the prime divisors of 6810 are {2, 3, 5, 227} => p = 227 is the length of the smallest side of the triangle (227, 229, 452).
The following table gives the first values of A, the sides of the triangles and the primes (p-1)/2.
+-----------+--------+--------+--------+---------+
|         A |      p |    p+2 |  2(p-1)| (p-1)/2 |
+-----------+--------+--------+--------+---------+
|        66 |     11 |     13 |     20 |       5 |
|      6810 |    227 |    229 |    452 |     113 |
|    182430 |   2027 |   2029 |   4052 |    1013 |
| 105470250 | 140627 | 140629 | 281252 |   70313 |
+-----------+--------+--------+--------+---------+

			66 is in the sequence because the area of the triangle (11, 13, 20) is given by Heron's formula with s = 22 and A = sqrt(22(22-11)(22-13)(22-20)) = 66. The numbers 11, 13 and 5 = (11-1)/2 are primes.

Cf. A056899, A085554, A086381, A188158, A253639, A257049.

nn:=100000:
for n from 1 by 2 to nn do:
if isprime(n^2+2) and isprime(n^2+4) and isprime((n^2+1)/2)
then
printf(`%d, `,n*(2*n^2+4)):
else
fi:
od:

nn=10000;lst={};Do[s=(2*Prime[c]-2+Prime[c+1]+Prime[c])/2;If[IntegerQ[s],area2=s (s-2*Prime[c]+2)(s-Prime[c+1])(s-Prime[c]); If[area2>0&&IntegerQ[Sqrt[area2]] &&Prime[c+1] ==Prime[c]+2 && PrimeQ[(Prime[c]-1)/2], AppendTo[lst,Sqrt[area2]]]], {c,nn}];Union[lst]

lista(nn) = {forprime(p=2, nn, if (isprime(p+2) && isprime((p-1)/2), ca = p; cb = p+2; cc = 2*(p-1); sp = (ca+cb+cc)/2; a2 = sp*(sp-ca)*(sp-cb)*(sp-cc); if (issquare(a2), print1(sqrtint(a2), ", "));););} \\ Michel Marcus, Mar 04 2017

a(n) == 6 mod 30.

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A216981 Primes of the form n^7+2.

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A241528 Primes p such that p + 1234567890 is also prime where 1234567890 is the first pandigital number with digits in order.

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A241537 Cubes c such that c + 1234567890 is prime where 1234567890 is the first pandigital number with digits in order.

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A241538 Squares s such that s + 1234567890 is prime.

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A248825 a(n) = n^2 + 1 - (-1)^n.

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A167675 Least prime p such that p-2 has n divisors, or 0 if no such prime exists.

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Mathematica

A178639 Numbers m such that all three values m^2 + 13^k, k = 1, 2, 3, are prime.

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A214001 Numbers n such that n^2+2, n^3+2, n^4+2 and n^5+2 are all prime.

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