A145294 -id:A145294 - cp's OEIS frontend

A145292 Composite numbers generated by the Euler polynomial x^2 + x + 41.

1681, 1763, 2021, 2491, 3233, 4331, 5893, 6683, 6847, 7181, 7697, 8051, 8413, 9353, 10547, 10961, 12031, 13847, 14803, 15047, 15293, 16043, 16297, 17071, 18673, 19223, 19781, 20633, 21797, 24221, 25481, 26123, 26447, 26773, 27101, 29111

Offset: 1

Artur Jasinski, Oct 06 2008

nonn

The Euler polynomial x^2 + x + 41 gives primes for consecutive x from 0 to 39.
For numbers x for which x^2 + x + 41 is not prime see A007634.
Let P(x)=x^2 + x + 41. In view of identity P(x+P(x))=P(x)*P(x+1), all values of P(x+P(x)) are in the sequence. - Vladimir Shevelev, Jul 16 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A005846, A007634, A145293, A145294.

Intersection of A002808 and A202018; A010051.

a145292 n = a145292_list !! (n-1)
a145292_list = filter ((== 0) . a010051) a202018_list
-- Reinhard Zumkeller, Dec 09 2011

a = {}; Do[If[PrimeQ[x^2 + x + 41], null,AppendTo[a, x^2 + x + 41]], {x, 0, 500}]; a
Select[Table[x^2+x+41,{x,200}],CompositeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 21 2018 *)

for(n=1,1e3,if(!isprime(t=n^2+n+41),print1(t", "))) \\ Charles R Greathouse IV, Dec 08 2011

a(n) ~ n^2. [Charles R Greathouse IV, Dec 08 2011]

A145293 a(n) is the smallest nonnegative x such that the Euler polynomial x^2 + x + 41 has exactly n distinct prime proper divisors.

Original entry on oeis.org

0, 41, 420, 2911, 38913, 707864, 6618260, 78776990, 725005500

Offset: 1

Artur Jasinski, Oct 07 2008

The Euler polynomial gives primes for consecutive x from 0 to 39.
For numbers x for which x^2 + x + 41 is not prime, see A007634.
For composite numbers of the form x^2 + x + 41, see A145292.

			a(1)=0 because when x=0 then x^2+x+41=41 (1 distinct prime divisor);
a(2)=41 because when x=41 then x^2+x+41=1763=41*43 (2 distinct prime divisors);
a(3)=420 because when x=420 then x^2+x+41=176861=47*53*71 (3 distinct prime divisors);
a(4)=2911 because when x=2911 then x^2+x+41=8476873=41*47*53*83 (4 distinct prime divisors);
a(5)=38913 because when x=38913 then x^2+x+41=1514260523=43*47*61*71*173 (5 distinct prime divisors);
a(6)=707864 because when x=707864 then x^2+x+41=501072150401=41*43*47*53*71*1607 (6 distinct prime divisors);
a(7)=6618260 because when x=6618260 then x^2+x+41=43801372045901=41*43*47*61*83*131*797 (7 distinct prime divisors);
a(8)=78776990 because when x=78776990 then x^2+x+41=6205814232237131=41*43*61*71*97*131*167*383 (8 distinct prime divisors).
a(9)=725005500: a(9)^2 + a(9) + 41 = 525632975755255541 = 41*43*47*53*61*71*151*397*461. - _Hugo Pfoertner_, Mar 05 2018

Cf. A005846, A007634, A097822, A145292, A145294, A145295.

Cf. A228122. - Zak Seidov, Feb 03 2016

a = {}; Do[x = 1; While[Length[FactorInteger[x^2 + x + 41]] < k - 1, x++ ]; AppendTo[a, x]; Print[x], {k, 2, 10}]; a

Corrected and edited, a(8) added by Zak Seidov, Jan 31 2016
Example for a(8) corrected by Hugo Pfoertner, Mar 02 2018
a(9) from Hugo Pfoertner, Mar 05 2018

A147520 a(n) = Smallest number x such that Euler Polynomial x^2 + x + 41 is divisible by 41^n.

Original entry on oeis.org

0, 40, 1721, 139563, 14268368, 1636255182, 6386359423, 1953929098233, 149759650255065, 1814531956108700, 243422399538851918, 9662500171353620019, 122479951673184550424, 12148820281768361731597, 177497315692809432279207, 14173382150616650630276616, 1225594969529024683212496795

Offset: 1

Artur Jasinski, Nov 06 2008

nonn

For values of x^2 + x + 41, see A147521. For values (x^2 + x + 41)/(41^n), see A147522.

By Hensel's lemma, x^2 + x + 41 has two roots mod 41^n; their sum == -1 mod 41^n. Thus 0 <= a(n) < 41^n/2. - Robert Israel, Apr 09 2018

Robert Israel, Table of n, a(n) for n = 1..620

Cf. A145292, A145293, A145294, A147520, A147521, A147522.

f:= n -> min(map(t -> rhs(op(t)), [msolve(x^2+x+41, 41^n)])):
map(f, [$1..30]); # Robert Israel, Apr 09 2018

a = {}; Do[x = 0; While[Mod[x^2 + x + 41, 41^n] != 0, x++ ]; AppendTo[a,x];Print[{n, x, x^2 + x + 41, (x^2 + x + 41)/41^n}], {n, 1, 6}];a (* Artur Jasinski *)

More terms from Robert Israel, Apr 09 2018

A147521 a(n) is the smallest value of Euler polynomial x^2 + x + 41 which is divisible by 41^n.

Original entry on oeis.org

41, 1681, 2963603, 19477970573, 203586339651833, 2677331022258108347, 40785586686127252393, 3817838920923578492820563, 22427952844519540079208409331, 3292526219739666997778171798741, 59254464597252454704406353071130683, 93363909561408736238900593787191180421

Offset: 1

Artur Jasinski, Nov 06 2008

nonn

For values x see A147520 For values (x^2 + x + 41)/(41^n) see A147522.

Cf. A145292, A145293, A145294, A147520, A147521, A147522.

a = {}; Do[x = 0; While[Mod[x^2 + x + 41, 41^n] != 0, x++ ]; AppendTo[a,x^2 + x + 41];Print[{n, x, x^2 + x + 41, (x^2 + x + 41)/41^n}], {n, 1, 6}];a

a(7)-a(12) from Hugo Pfoertner, Jan 12 2019

A147522 Minimal number such that a(n)*41^n is of the form x^2 + x + 41.

Original entry on oeis.org

1, 1, 43, 6893, 1757233, 563636267, 209420753, 478130829203, 68506995922171, 245296117830341, 107670977147002963, 4137830996824703141, 16215855859012574521, 3891295582697539371727, 20259360308241533953897, 3150687184319350782852593, 574603643148719626677940181

Offset: 1

Artur Jasinski, Nov 06 2008

nonn

For values x see A147520. For x^2 + x + 41 values see A147521.

Cf. A145292, A145293, A145294, A147520, A147521, A147522.

a = {}; Do[x = 0; While[Mod[x^2 + x + 41, 41^n] != 0, x++ ]; AppendTo[a,(x^2 + x + 41)/41^n];Print[{n, x, x^2 + x + 41, (x^2 + x + 41)/41^n}], {n, 1, 6}]; a

a(7)-a(17) from Hugo Pfoertner, Jan 12 2019

A268405 Numbers m such that m^2 + m + 41 is a product of 7 primes.

Original entry on oeis.org

3019035, 3312609, 4005577, 4205871, 4270887, 4502832, 4838229, 4933775, 5086008, 6142338, 6618260, 6932403, 6941996, 7263518, 7375900, 7643466, 7939002, 8268798, 8473961, 8485664, 8499341, 8892530, 8978097, 8991587, 9075462, 9317324, 9469974, 9709914, 9736792, 9745217

Offset: 1

Zak Seidov, Feb 04 2016

nonn

Note that a(1) = 3019035 = A228122(7) and a(11) = 6618260 = A145293(7).

Zak Seidov, Table of n, a(n) for n = 1..140

Cf. A145293, A228122, A268290, A005846, A007634, A145292, A145294, A145295.

cp's OEIS Frontend

A145292 Composite numbers generated by the Euler polynomial x^2 + x + 41.

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A145293 a(n) is the smallest nonnegative x such that the Euler polynomial x^2 + x + 41 has exactly n distinct prime proper divisors.

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A147520 a(n) = Smallest number x such that Euler Polynomial x^2 + x + 41 is divisible by 41^n.

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A147521 a(n) is the smallest value of Euler polynomial x^2 + x + 41 which is divisible by 41^n.

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A147522 Minimal number such that a(n)*41^n is of the form x^2 + x + 41.

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A268405 Numbers m such that m^2 + m + 41 is a product of 7 primes.

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