A145293 -id:A145293 - 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]

A145294 Smallest x >= 0 such that the Euler polynomial x^2 + x + 41 has a prime divisor of multiplicity n.

Original entry on oeis.org

0, 40, 1721, 14144, 2294005, 326924482, 6386359423, 1341160319494, 149759650255065, 1167478867440605, 243422399538851918, 9662500171353620019, 122479951673184550424, 12148820281768361731597, 177497315692809432279207, 11767210525408975519141638

Offset: 1

Artur Jasinski, Oct 07 2008

nonn

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.
For the smallest x such that polynomial x^2 + x + 41 has exactly n distinct prime divisors, see A145293.
Sequence interpreted as a(n)^2 + a(n) + 41 having a prime divisor with multiplicity that is exactly n. - Bert Dobbelaere, Jan 22 2019

			a(2)=40 because when x=40 then x^2 + x + 41 = 1681 = 41^2;
a(3)=1721 because when x=1721 then x^2 + x + 41 = 2963603 = 43*41^3;
a(4)=14144 because when x=14144 then x^2 + x + 41 = 200066921 = 41*47^4;
a(5)=2294005 because when x=2294005 then x^2 + x + 41 = 5262461234071 = 35797*43^5.
a(6)=326924482: a(6)^2 + a(6) + 41 = 106879617257892847 = 9915343 * 47^6. - _Hugo Pfoertner_, Mar 08 2018

Bert Dobbelaere, Table of n, a(n) for n = 1..100
Bert Dobbelaere, Python program

Cf. A005846, A007634, A145292, A145293, A145295.

Title changed, a(1) and a(6) from Hugo Pfoertner, Mar 08 2018

More terms from Bert Dobbelaere, Jan 22 2019

A097822 Numbers n such that n^2+n+41 (Euler's "prime generating polynomial") has more than 2 prime factors.

Original entry on oeis.org

420, 431, 491, 492, 514, 533, 573, 574, 603, 614, 655, 686, 738, 775, 798, 858, 861, 890, 895, 901, 904, 917, 919, 942, 984, 989, 1025, 1059, 1116, 1130, 1162, 1169, 1188, 1215, 1222, 1224, 1245, 1251, 1253, 1268, 1271, 1318, 1321, 1334, 1365, 1374, 1407

Offset: 1

Hugo Pfoertner, Aug 26 2004

nonn

All visible sequence terms give exactly 3 prime factors. The smallest composite of the form p(n)=n^2+n+41 with 4 prime factors occurs for p(1721)=2963603=43*41^3. Smallest n with 4 distinct prime factors: p(2911)=8476873=83*53*47*41, smallest n with 5 prime factors: p(14144)=200066921=47^4*41, smallest n with 5 distinct prime factors: p(38913)=1514260523=173*71*61*47*43.

			a(1)=420 because 420^2+420+41=176861=71*53*47 is the first n for which p(n)=n^2+n+41 has more than 2 prime factors. For all smaller n p(n) is either prime or semiprime.

Harvey P. Dale, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A002837, A007634, A005846, A097823, A145293.

Select[Range[1500],PrimeOmega[#^2+#+41]>2&] (* Harvey P. Dale, Dec 26 2017 *)

isok(n) = #factor(n^2+n+41)~ > 2; \\ Michel Marcus, Sep 07 2017

Corrected a(19) by Hugo Pfoertner, Sep 07 2017

A228122 Smallest nonnegative number x such that x^2 + x + 41 has exactly n prime factors counting multiplicities.

Original entry on oeis.org

0, 40, 420, 1721, 14144, 139563, 3019035, 24304266, 206583092, 3838101265

Offset: 1

Shyam Sunder Gupta, Aug 11 2013

			a(1) = 0 because if x = 0 then x^2 + x + 41 = 41, which has 1 prime factor.
a(2) = 40 because if x = 40 then x^2 + x + 41 = 1681 = 41*41, which has 2 prime factors, counting multiplicities.
a(3) = 420 because if x = 420 then x^2 + x + 41 = 176861 = 47*53*71, which has 3 prime factors.

Cf. A005846, A007634, A145292, A145293, A056561.

a = {}; Do[x = 0; While[PrimeOmega[x^2 + x + 41] != k, x++]; AppendTo[a, x], {k, 9}]; a

a(n) = {my(m=0); while (bigomega(m^2+m+41) != n, m++); m;} \\ Michel Marcus, Jan 31 2016

from sympy import factorint
def A228122(n):
    k = 0
    while sum(factorint(k*(k+1)+41).values()) != n:
        k += 1
    return k # Chai Wah Wu, Sep 07 2018

a(9) from Zak Seidov, Feb 01 2016

a(10) from Giovanni Resta, Sep 08 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

A268290 Numbers m such that m^2 + m + 41 is a product of 8 primes.

Original entry on oeis.org

24304266, 25056402, 41492163, 44215670, 47394564, 50520814, 50979769, 58929176, 66582359, 67293751, 75860168, 77698074, 78776990, 78919547, 83396050, 88914075, 89994056, 97631455, 100382268, 101241663, 102396844, 104964961, 105399150

Offset: 1

Zak Seidov, Feb 01 2016

nonn

Note that a(1)=24304266=A228122(8) and a(13)=78776990=A145293(8).

Zak Seidov, Table of n, a(n) for n = 1..57

Cf. A145293, A228122.

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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A145294 Smallest x >= 0 such that the Euler polynomial x^2 + x + 41 has a prime divisor of multiplicity n.

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A097822 Numbers n such that n^2+n+41 (Euler's "prime generating polynomial") has more than 2 prime factors.

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A228122 Smallest nonnegative number x such that x^2 + x + 41 has exactly n prime factors counting multiplicities.

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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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A268290 Numbers m such that m^2 + m + 41 is a product of 8 primes.

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