Pedja Terzic - cp's OEIS frontend

A323388 a(n) = b(n+1)/b(n) - 1 where b(1)=3 and b(k) = b(k-1) + lcm(floor(sqrt(3)*k), b(k-1)).

1, 5, 1, 1, 5, 1, 13, 5, 17, 19, 1, 11, 1, 5, 1, 29, 31, 1, 17, 1, 19, 13, 41, 43, 1, 23, 1, 1, 17, 53, 1, 19, 29, 1, 31, 1, 13, 67, 23, 71, 1, 37, 1, 1, 79, 1, 83, 1, 43, 1, 1, 13, 31, 1, 1, 1, 1, 1, 103, 1, 107, 109, 1, 1, 1, 29, 1, 1, 1, 61, 1, 1

Offset: 1

Pedja Terzic, Jan 13 2019

nonn

Conjectures:
1. This sequence consists only of 1's and primes.
2. Every odd prime of the form floor(sqrt(3)*m) greater than 3 is a term of this sequence.
3. At the first appearance of each prime of the form floor(sqrt(3)*m), it is larger than any prime that has already appeared.
The 2nd and 3rd conjectures are proved at the Mathematics Stack Exchange link. - Sungjin Kim, Jul 17 2019

Michel Marcus, Table of n, a(n) for n = 1..10000
Mathematics Stack Exchange, Generating primes of the form floor(sqrt(3)*n)

Cf. A184796 (primes of the form floor(sqrt(3)*m)).

Cf. A135506, A323359, A323386.

Generator(n)={b1=3; list=[]; for(k=2, n, b2=b1+lcm(floor(sqrt(3)*k), b1); a=b2/b1-1; list=concat(list,a); b1=b2); print(list)}

lista(nn)={my(b1=3, b2, va=vector(nn)); for(k=2, nn+1, b2=b1+lcm(sqrtint(3*k^2), b1); va[k-1]=b2/b1-1; b1=b2); va}; \\ Michel Marcus, Aug 20 2022

A323386 a(n) = b(n+1)/b(n) - 1 where b(1)=2 and b(k) = b(k-1) + lcm(floor(sqrt(2)*k),b(k-1)).

Original entry on oeis.org

1, 1, 5, 7, 1, 3, 11, 1, 7, 5, 1, 1, 19, 7, 11, 1, 5, 13, 1, 29, 31, 1, 11, 1, 1, 19, 13, 41, 1, 43, 1, 23, 1, 1, 1, 13, 53, 1, 1, 19, 59, 1, 31, 1, 13, 1, 67, 23, 1, 1, 73, 1, 19, 1, 79, 1, 41, 83, 1, 43, 29, 89, 1, 13, 31, 47, 1, 97, 1, 1, 101, 103

Offset: 1

Pedja Terzic, Jan 13 2019

nonn

Conjectures:
This sequence consists only of 1's and primes.
Every odd prime of the form floor(sqrt(2)*m) is a term of this sequence.
At the first appearance of each prime of the form floor(sqrt(2)*m), it is the next prime after the largest prime that has already appeared.

Cf. A135506, A008578, A323359, A323388.

Generator(n)={b1=2; list=[]; for(k=2, n, b2=b1+lcm(floor(sqrt(2)*k), b1); a=b2/b1-1; list=concat(list,a); b1=b2); print(list)}

A323359 a(n) = b(n+1)/b(n) - 1 where b(1)=2 and b(k) = b(k-1) + lcm(floor(sqrt(k^3)), b(k-1)).

Original entry on oeis.org

1, 5, 1, 11, 7, 1, 11, 1, 31, 1, 41, 23, 13, 29, 1, 1, 19, 41, 89, 1, 103, 11, 1, 1, 11, 1, 37, 1, 41, 43, 181, 1, 1, 23, 1, 1, 1, 1, 1, 131, 17, 281, 97, 43, 311, 23, 83, 1, 353, 1, 17, 1, 1, 37, 419, 43, 1, 151, 29, 17, 61, 1, 1, 131, 67, 137, 1, 191, 1, 1, 61, 89

Offset: 1

Pedja Terzic, Jan 12 2019

nonn

Conjectures:
1. This sequence consists only of 1's and primes.
2. Every odd prime of the form floor(sqrt(m^3)) is a term of this sequence.
3. At the first appearance of each prime of the form floor(sqrt(m^3)), it is the next prime after the largest prime that has already appeared.
Record values appear to be A291139(m), m > 1. - Bill McEachen, Jun 23 2023

Cf. A135506, A008578, A323386, A323388, A291139.

Generator(n)={b1=2;list=[]; for(k=2, n, b2=b1+lcm(sqrtint(k^3),b1); a=b2/b1-1; list=concat(list,a);b1=b2); return(list)}

A304822 a(n) = A304821(n+3)/A304821(n) - 1.

Original entry on oeis.org

1, 1, 2, 5, 1, 7, 1, 3, 5, 11, 1, 13, 7, 5, 1, 17, 1, 19, 5, 7, 11, 23, 1, 5, 13, 3, 1, 29, 5, 31, 1, 11, 17, 1, 1, 37, 19, 13, 1, 41, 1, 43, 11, 5, 23, 47, 1, 7, 1, 17, 13, 53, 1, 1, 1, 19, 29, 59, 1, 61, 31, 1, 1, 13, 11, 67, 17, 23, 1, 71, 1, 73

Offset: 1

Pedja Terzic, May 19 2018

nonn

Conjecture: This sequence consists of 1's and primes only.

a(n) divides n+1. - A.H.M. Smeets, Jul 02 2018

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A135504, A217662, A217663, A304821.

Drop[#, 3] &@ Fold[Append[#1, {#2, #2/#1 - 1} & @@ {#1[[-3, 1]], #1[[-3, 1]] + LCM[#1[[-3, 1]], #2 + 1]}] &, {{6, 0}, {6, 0}, {6, 0}}, Range@ 72] [[All, -1]] (* Michael De Vlieger, May 20 2018 *)

Generator(n)={b1=6;b2=6;b3=6;list=[];for(k=4,n,b4=b1+lcm(k-2,b1);a=b4/b1-1;list=concat(list,a);b1=b2;b2=b3;b3=b4);print(list)}

from math import gcd
n,an0,an1,an2 = 3,6,6,6
while n-3 < 200:
    n += 1
    an0,an1,an2,an3 = an2+an2*(n-2)//gcd(an2,n-2),an0,an1,an2
    # an0 = A304821(n), an3 = A304821(n-3)
    print(n-3,an0//an3-1)
# A.H.M. Smeets, Jul 02 2018

A304821 For n > 3, a(n) = a(n-3) + lcm(a(n-3), n-2) with a(1)=6, a(2)=6, a(3)=6.

Original entry on oeis.org

6, 6, 6, 12, 12, 18, 72, 24, 144, 144, 96, 864, 1728, 192, 12096, 13824, 1152, 24192, 248832, 2304, 483840, 1492992, 18432, 5806080, 35831808, 36864, 34836480, 501645312, 147456, 69672960, 15049359360, 884736

Offset: 1

Pedja Terzic, May 19 2018

nonn

Cf. A135504, A217662.

Fold[Append[#1, #1[[-3]] + LCM[#1[[-3]], #2 + 1]] &, {6, 6, 6}, Range@ 29] (* Michael De Vlieger, May 20 2018 *)

Recurrence(n)={b1=6;b2=6;b3=6;list=[6,6,6];for(k=4,n,b4=b1+lcm(k-2,b1);list=concat(list,b4);b1=b2;b2=b3;b3=b4);print(list)}

A304177 Union of sequences b and c defined by: b(1)=8, b(2)=488, b(n)=62b(n-1) - b(n-2) and c(1)=22, c(2)=10582, c(n)=482c(n-1) - c(n-2).

Original entry on oeis.org

8, 22, 488, 10582, 30248, 1874888, 5100502, 116212808, 2458431382, 7203319208, 446489578088, 1184958825622, 27675150522248, 571147695518422, 1715412842801288, 106327921103157608, 275292004281053782, 6590615695552970408, 132690174915772404502, 408511845203181007688

Offset: 1

Pedja Terzic, May 07 2018

Conjecture: Each member of this sequence can be used as an initial value for Inkeri's primality test for Fermat numbers.

Inkeri's primality test for Fermat numbers: Fermat's number F_{m}=2^2^m+1 (m => 2) is prime if and only if F_{m} divides the term v_{2^m-2} of the series v_{0}=8 , v_{i}=(v_{i-1})^2-2 .

K. Inkeri, Tests for primality, Ann. Acad. Sci. Fenn., A I 279, 119 (1960).

Pedja Terzic, Initial values of Inkeri's primality test for Fermat numbers, Math StackExchange, May 2018.

Cf. A018844, A000215.

b=RecurrenceTable[{a[1]==8,a[2]==488,a[n]==62a[n-1]-a[n-2]},a,{n,12}]; c= RecurrenceTable[{a[1]==22,a[2]==10582,a[n]==482a[n-1]-a[n-2]},a,{n,12}]; Join[ b,c]//Union (* Harvey P. Dale, May 05 2022 *)

InitialValues(n)= {l=[8,22,488,10582];b1=8;b2=488;i=3;while(i<=n,b=62*b2-b1;l=concat(l,b);b1=b2;b2=b;i++);c1=22;c2=10582;j=3;while(j<=n,c=482*c2-c1;l=concat(l,c);c1=c2;c2=c;j++);print(vecsort(l))}

A218457 a(n) = 6n^3 - 263n^2 + 3469*n - 12841.

Original entry on oeis.org

-12841, -9629, -6907, -4639, -2789, -1321, -199, 613, 1151, 1451, 1549, 1481, 1283, 991, 641, 269, -89, -397, -619, -719, -661, -409, 73, 821, 1871, 3259, 5021, 7193, 9811, 12911, 16529, 20701, 25463, 30851, 36901, 43649, 51131, 59383, 68441, 78341, 89119

Offset: 0

Pedja Terzic, Oct 29 2012

A prime-producing cubic polynomial. Produces 78 distinct primes if we scan the absolute values of the first 100 terms.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A124363, A076808, A218456.

Mathematica
```
Table[6n^3-263n^2+3469n-12841,{n,0,99}]
```

a(n) = {6*n^3 - 263*n^2 + 3469*n - 12841} \\ Andrew Howroyd, Apr 27 2020

Signs of terms corrected and a(32) and beyond from Andrew Howroyd, Apr 27 2020

A218458 a(n) = 2n^3 - 163n^2 + 2777*n - 11927.

Original entry on oeis.org

-11927, -9311, -7009, -5009, -3299, -1867, -701, 211, 881, 1321, 1543, 1559, 1381, 1021, 491, -197, -1031, -1999, -3089, -4289, -5587, -6971, -8429, -9949, -11519, -13127, -14761, -16409, -18059, -19699, -21317, -22901, -24439, -25919

Offset: 0

Pedja Terzic, Oct 29 2012

A prime-producing cubic polynomial. Produces 78 distinct primes if we scan the absolute values of the first 100 terms.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A124363, A076808, A218456, A218457.

[2*n^3 - 163*n^2 + 2777*n - 11927 : n in [0..60]]; // Wesley Ivan Hurt, Apr 21 2021

Table[2n^3-163n^2+2777n-11927,{n,0,99}]
LinearRecurrence[{4,-6,4,-1},{-11927,-9311,-7009,-5009},40] (* Harvey P. Dale, Jan 31 2017 *)

A218458(n):=2*n^3-163*n^2+2777*n-11927$
makelist(A218458(n),n,0,30); /* Martin Ettl, Nov 08 2012 */

G.f.: (-11927+38397*x-41327*x^2+14869*x^3)/(x-1)^4. - R. J. Mathar, Nov 07 2012

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Wesley Ivan Hurt, Apr 21 2021

A218456 2n^3 - 313n^2 + 6823*n - 13633.

Original entry on oeis.org

-13633, -7121, -1223, 4073, 8779, 12907, 16469, 19477, 21943, 23879, 25297, 26209, 26627, 26563, 26029, 25037, 23599, 21727, 19433, 16729, 13627, 10139, 6277, 2053, -2521, -7433, -12671, -18223, -24077, -30221, -36643, -43331, -50273, -57457

Offset: 0

Pedja Terzic, Oct 29 2012

A prime-producing cubic polynomial. Produces 79 distinct primes if we scan the absolute values of the first 100 terms..

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A124363, A076808.

Table[2n^3-313n^2+6823n-13633,{n,0,99}]
LinearRecurrence[{4,-6,4,-1},{-13633,-7121,-1223,4073},40] (* Harvey P. Dale, May 03 2018 *)

A218456(n):=2*n^3-313*n^2+6823*n-13633$
makelist(A218456(n),n,0,30); /* Martin Ettl, Nov 08 2012 */

G.f.: (20771*x^3-54537*x^2+47411*x-13633)/(x-1)^4. [Colin Barker, Nov 10 2012]

A217606 a(n) is the least unused prime greater than 3 such that (a(n) + a(n-1))/2 is prime, with a(0)=13.

Original entry on oeis.org

13, 61, 73, 181, 37, 97, 109, 193, 229, 157, 241, 313, 349, 277, 337, 397, 421, 373, 541, 433, 409, 457, 757, 661, 577, 709, 613, 601, 853, 769, 733, 1021, 997, 877, 829, 673, 1033, 1009, 1069, 1117, 1129, 937, 1201, 1297, 1549, 1093, 1153, 1249, 1213, 1381

Offset: 0

Pedja Terzic, Oct 08 2012

nonn

Conjecture: every prime of the form 12k+1 is a member.

Cf. A086519.

a:=5:
l:=13:
L:=[l]:
while l < 3400 do
if isprime((l+a)/2) then
if not(a in L) then
if not a mod 12 = 1 then
print(a);
break;
end if;
L:=[op(L),a]:
l:=a:
a:=5:
else
a:=nextprime(a):
end if;
else
a:=nextprime(a):
end if;
end do;
L;

cp's OEIS Frontend

User: Pedja Terzic

A323388 a(n) = b(n+1)/b(n) - 1 where b(1)=3 and b(k) = b(k-1) + lcm(floor(sqrt(3)*k), b(k-1)).

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A323386 a(n) = b(n+1)/b(n) - 1 where b(1)=2 and b(k) = b(k-1) + lcm(floor(sqrt(2)*k),b(k-1)).

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A323359 a(n) = b(n+1)/b(n) - 1 where b(1)=2 and b(k) = b(k-1) + lcm(floor(sqrt(k^3)), b(k-1)).

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A304822 a(n) = A304821(n+3)/A304821(n) - 1.

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A304821 For n > 3, a(n) = a(n-3) + lcm(a(n-3), n-2) with a(1)=6, a(2)=6, a(3)=6.

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A304177 Union of sequences b and c defined by: b(1)=8, b(2)=488, b(n)=62*b(n-1) - b(n-2) and c(1)=22, c(2)=10582, c(n)=482*c(n-1) - c(n-2).

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A218457 a(n) = 6*n^3 - 263*n^2 + 3469*n - 12841.

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Extensions

A218458 a(n) = 2*n^3 - 163*n^2 + 2777*n - 11927.

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A304177 Union of sequences b and c defined by: b(1)=8, b(2)=488, b(n)=62b(n-1) - b(n-2) and c(1)=22, c(2)=10582, c(n)=482c(n-1) - c(n-2).

A218457 a(n) = 6n^3 - 263n^2 + 3469*n - 12841.

A218458 a(n) = 2n^3 - 163n^2 + 2777*n - 11927.

A218456 2n^3 - 313n^2 + 6823*n - 13633.