Matt C. Anderson - cp's OEIS frontend

A350856 Initial members of prime triples (p, p+2, p+14).

3, 5, 17, 29, 59, 137, 149, 179, 197, 227, 269, 419, 599, 617, 659, 809, 1019, 1049, 1277, 1289, 1607, 1787, 1997, 2129, 2237, 2267, 2657, 2789, 3167, 3257, 3299, 3329, 3359, 3527, 3557, 3917, 3929, 4217, 4229, 4259, 4547, 4637, 4649, 4787, 4799, 5009, 5099

Offset: 1

Matt C. Anderson, Jan 19 2022

nonn

According to the k-tuple conjecture this sequence is theoretically infinite.

Chris K. Caldwell prime k-tuple conjecture
Chris K. Caldwell k-tuple

Cf. A022004 (p,p+2,p+6), A046134 (p,p+2,p+8), A046135 (p,p+2,p+12).

for a from 3 to 1000 by 2 do
if isprime(a) and isprime(a+2) and isprime(a+14) then
print(a);
end if
end do
# second Maple program:
q:= p-> andmap(isprime, [p, p+2, p+14]):
select(q, [$1..10000])[];  # Alois P. Heinz, Jan 28 2022

Select[Range[7000], And @@ PrimeQ[# + {0, 2, 14}] &] (* Amiram Eldar, Jan 20 2022 *)

A290124 a(n) = a(n-1) + 12*a(n-2) with a(1) = 1 and a(2) = 2.

Original entry on oeis.org

1, 2, 14, 38, 206, 662, 3134, 11078, 48686, 181622, 765854, 2945318, 12135566, 47479382, 193106174, 762858758, 3080132846, 12234437942, 49196032094, 196009287398, 786361672526, 3138473121302, 12574813191614, 50236490647238, 201134248946606, 803972136713462, 3217583124072734

Offset: 1

Matt C. Anderson, Jul 20 2017

The binomial transform is 1, 3, 19, 87,.... (A015528 shifted). - R. J. Mathar, Apr 07 2022

Index entries for linear recurrences with constant coefficients, signature (1,12).

Cf. A000045, A000079, A133577.

[(5/28)*4^n-(2/21)*(-3)^n: n in [1..30]]; // Vincenzo Librandi, Aug 27 2017

CoefficientList[Series[(1 + x) / ((1 + 3 x) (1 - 4 x)), {x, 0, 33}], x] (* Vincenzo Librandi, Aug 27 2017 *)

a(n) = if (n==1, 1, if (n==2, 2, a(n-1) + 12*a(n-2))); \\ Michel Marcus, Jul 25 2017

a(n) = (5/28)*4^n - (2/21)*(-3)^n.

G.f.: x*(1+x)/((1+3*x)*(1-4*x)). - Vincenzo Librandi, Aug 27 2017

A277338 Reverse and Add! sequence starting with 295.

Original entry on oeis.org

295, 887, 1675, 7436, 13783, 52514, 94039, 187088, 1067869, 10755470, 18211171, 35322452, 60744805, 111589511, 227574622, 454050344, 897100798, 1794102596, 8746117567, 16403234045, 70446464506, 130992928913, 450822227944, 900544455998, 1800098901007, 8801197801088, 17602285712176, 84724043932847, 159547977975595

Offset: 0

Matt C. Anderson, Oct 09 2016

Apart from the initial term in both sequences, the same as A006960.
a(0) = 295; a(n+1) = a(n) + A004086(a(n)).
295 is conjectured to be the second smallest initial term which does not lead to a palindrome. Also, 196 is possibly the smallest initial term which does not lead to a palindrome. a(0) = 196 is described in A006960.

			a(0) = 295
a(1) = 295 + 592 = 887
a(2) = 887 + 788 = 1675
...

Wikipedia, Lychrel number

Cf. A004086.
Almost the same as A006960.
See index entries at A023108.

with(StringTools):
revnum := proc (n)
local a, b, c;
description "to REVerse the digits of a NUMber";
a := convert(n, string);
b := Reverse(a);
c := convert(b, decimal, 10)
end proc;
f := 0;
e := 295;
count := 0;
while f <> e do
e := e+f;
f := revnum(e);
count := count+1
end do;

a[1] = 295; a[n_] := a[n] = FromDigits@ Reverse@ IntegerDigits@ # + # &@ a[n - 1]; Array[a, 29] (* Michael De Vlieger, Oct 14 2016 *)

terms(n) = my(x=295, i=0); while(1, print1(x, ", "); x=x+eval(concat(Vecrev(Str(x)))); i++; if(i==n, break))
/* Print initial 30 terms as follows: */
terms(30) \\ Felix Fröhlich, Nov 15 2016

a(n) = A006960(n) for n >= 1.

a(n) = A243238(295, n+1). - Felix Fröhlich, Nov 20 2016

A247018 Numbers of the form 3*z^2 + z + 3 for some integer z.

Original entry on oeis.org

3, 5, 7, 13, 17, 27, 33, 47, 55, 73, 83, 105, 117, 143, 157, 187, 203, 237, 255, 293, 313, 355, 377, 423, 447, 497, 523, 577, 605, 663, 693, 755, 787, 853, 887, 957, 993, 1067, 1105, 1183, 1223, 1305, 1347, 1433, 1477, 1567, 1613, 1707, 1755, 1853, 1903

Offset: 1

Matt C. Anderson, Sep 09 2014

Note that z is allowed to be negative. - N. J. A. Sloane, Jul 09 2021
Subsequence of A134407.
Numbers k such that 12*k - 35 is a square. - Robert Israel, Sep 18 2014

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A002522, A001105.

select(t -> issqr(12*t-35), [$1..1000]); # Robert Israel, Sep 18 2014

Union[Flatten[Table[3z^2+{z,-z}+3,{z,0,40}]]] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{3,5,7,13,17},60] (* Harvey P. Dale, Jul 10 2021 *)

Vec(x*(3 + 2*x - 4*x^2 + 2*x^3 + 3*x^4) / ((1 - x)^3*(1 + x)^2) + O(x^60)) \\ Colin Barker, Feb 01 2018

From Colin Barker, Feb 01 2018: (Start)
G.f.: x*(3 + 2*x - 4*x^2 + 2*x^3 + 3*x^4) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5. (End)

At some point in the history of this entry the definition was changed from the correct definition to the erroneous "a(n) = 3*n^2 + n + 3". I have restored the original definition, and I deleted some incorrect programs. Thanks to Harvey P. Dale for pointing out that something was wrong. - N. J. A. Sloane, Jul 09 2021.

A248015 Positive numbers n such that n^2 + 1 is composite and there are no positive integers c and z such that n = c*z^2 + z + c.

Original entry on oeis.org

8, 18, 28, 30, 34, 44, 46, 48, 50, 58, 60, 64, 68, 70, 76, 78, 86, 88, 96, 98, 100, 104, 108, 114, 118, 128, 136, 144, 148, 158, 164, 166, 168, 178, 186, 188, 190, 194, 196, 198, 200

Offset: 1

Matt C. Anderson, Sep 29 2014

nonn

Subset of A134407.
If f(x) = x^2 + 1 and g(c,y) = c*y^2 + y + c then the algebraic substitution of g for x gives a factorization: f(g(c,y)) = (y^2 + 1)*(c^2*y^2 + c^2 + 2*c*y + 1).  Since both factors of f(g(c,y)) are integers greater than one, f(g(c,y)) is a composite number.
The numbers are necessarily even terms from A134407 since for odd n = 2c + 1 one has the "forbidden" decomposition with z = 1. - M. F. Hasler, Oct 04 2014

Eric Weisstein's World of Mathematics, Landau's Problems

Cf. A134407.

maxn:=200:
mb:=proc(n::integer)::integer;
  if isprime(n^2+1)=false then return n else return -1 fi;
end proc:
A134407 := Vector(maxn):
for a from 1 to maxn do A134407[a]:= mb(a): end do:
A134407s:=convert(A134407,'set') minus {-1}:
A134407l:=convert(A134407s,'list'):
for c from 1 to 200 do
  for z from 1 to 20 do
    A134407s := A134407s minus {c*z^2 + z + c}:
  end do:
end do:
A134407s;

is(n)={!bittest(n,0)&&!isprime(n^2+1)&&!for(z=2,sqrtint(n),(n-z)%(z^2+1)||return)} \\ M. F. Hasler, Oct 04 2014

A241529 Positive numbers k such that k^2 + k + 41 is composite and there are no positive integers a,c,d such that k = caz^2 + ((((d+2)(1/3))c-2)a/d+1)z + ((((367d^2+d+1)(1/9))c^2-((d+2)(1/3))c+1)a/d^2 - (((d-1)(1/3))c+1)/d)/c for an integer z.

Original entry on oeis.org

2887, 2969, 3056, 3220, 3365, 3464, 3565, 3611, 3719, 3746, 3814, 3836, 3874, 3879, 3955, 4142, 4147, 4211, 4277, 4371, 4403, 4483, 4564, 4572, 4661, 4730, 4813, 4881, 4888, 4902, 4906, 4965, 4982, 5132, 5175, 5208, 5410, 5431, 5509, 5527, 5564, 5624, 5669

Offset: 1

Matt C. Anderson, Apr 27 2014

nonn

This sequence has a restriction involving 4 variables. More composite cases are described with a better restrictive expression.  The expression for k(a,c,d,z) will force k^2 + k + 41 to be either a fraction or a composite number.
The condition on k(a,c,d,z) was determined by quadratic curve fitting.  It has been automated with the Maple Interactive() command.  The ultimate motivation is to try to find a closed-from expression that generates all the composite cases of k^2 + k + 41 for integer k.
What is the smallest value of n where this sequence's a(n) < 2n? (For A194634, this value is 2358.) - J. Lowell, Feb 25 2019

John Stillwell, Elements of Number Theory, Springer, 2003, page 3.

Matt C. Anderson, Graph of composite values for n^2 + n + 41 with a modular symmetry.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A007634, A055390, A201998, and with division, A235381.

# Euler considered the prime values for n^2 + n + 41;
# This is a 76 second calculation on a 2.93 GHz machine
h := n^2+n+41;
y := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c;
y2 := subs(n = y, h);
y3 := factor(y2);
# note that y is an expression in 4 variables.
# After a composition of functions, an algebraic factorization
# can be observed in y3.  As long as y3 is an integer, it will
# be composite.  This is because y3 factors and both factors
# are integers bigger than one.
maxn := 6000;
A := {}:
for n to maxn do
g := n^2+n+41:
if isprime(g) = false then A := `union`(A, {n}) end if :
end do:
# now the A set contains composite values of the form
# n^2 + n + 41 less than maxn.
c := 1: a := 1: d := 1: z := -1: p := 41:
q := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c:
A2 := A:
while q < maxn do
while `and`(q < maxn, d < 100) do
while q < maxn do while
q < maxn do
A2 := `minus`(A2, {q});
A2 := `minus`(A2, {c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c});
z := z+1;
A2 := `minus`(A2, {c*a*z^2-((((d+2)*(1/3))*c-2)*a/d+1)*(1*z)+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c}); q := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c
end do;
a := a+1; z := -1;
q := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c :
end do;
d := d+1: a := 1:
q := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c :
end do:
c := c+1: d := 1:
q := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c :
end do:
A2;
# Matt C. Anderson, May 13 2014

A235381 Positive numbers n such that n^2 + n + 41 is composite and there are no positive integers c or d such that n = cdx^2 + ((d-2)c + 1)x + ((41d^2 - d + 1)c -1)/d for an integer x.

Original entry on oeis.org

611, 622, 630, 663, 679, 734, 758, 835, 867, 966, 978, 995, 1006, 1009, 1060, 1088, 1127, 1142, 1157, 1173, 1175, 1183, 1228, 1280, 1345, 1355, 1368, 1388, 1390, 1426, 1433, 1455, 1457, 1467, 1497, 1538, 1539, 1543, 1554, 1578, 1603, 1612, 1613, 1630, 1661

Offset: 1

Matt C. Anderson, Jan 08 2014

nonn

Restricting c and d so that c is congruent to 1 modulo d, we have that the composition of functions k(x) factors. k(x) = (1/d^2)*((1 + x*d^2 + x^2*d^2 - d - 2*x*d + 41*d^2)*(c^2*d^2*x^2 + x*d^2*c^2 + 41*c^2*d^2 + 2*x*d*c^2 - 2*x*d*c^2 + c*d - c^2*d + 1). So k(x) is the product of two integers greater than one and is thus composite.

			If d = 1 then n = c*n^2 + (1 - c)*x + 41*c  - 1. This is, up to a change of variables, equivalent to A201998.

John Stillwell, Elements of Number Theory, Springer 2003, page 3.

Matt C. Anderson, Table of n, a(n) for n = 1..75

Cf. A007634 (numbers n such that n^2 + n + 41 is composite).

Cf. A201998 and A241529 (similar subsequences of A007634).

maxn := 1000;
A := {};
for n to maxn do
g := n^2+n+41;
if isprime(g) = false then
A := `union`(A, {n}) :
end if :
end do :
A:
# the A list now contains Positive numbers n such that
# n^2 + n + 41 is composite.
# an upper limit for the number of iterations in the
# triple nested while loops is 1000^3 or a billion.
c:=1:
d:=1:
x:=-1:
p:=41:
q:=c*d*x^2+((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d;
A2:=A:
while q < maxn do
while q < maxn do
while q < maxn do
  A2:=A2 minus {q}:
  A2:=A2 minus {c*x^(2)+(c+1)*x+c*p}:
  A2:=A2 minus {c*d*x^2-((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d}:
  x:=x+1:
  q:=c*d*x^2+((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d:
end do:
c:=c+1:
x:=-1:
q:=c*d*x^2+((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d:
end do:
d:=d+1:
c:=1:
x:=-1:
q:=c*d*x^2+((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d:
end do:
A2

Corrected and edited by Matt C. Anderson, Jan 23 2014

A214947 Primes p such that p + (0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48) are all prime.

Original entry on oeis.org

186460616596321, 7582919852522851, 31979851757518501, 49357906247864281, 79287805466244211, 85276506263432551, 89309633704415191, 89374633724310001, 98147762882334001, 136667406812471371, 137803293675931951, 152004604862224951, 157168285586497021, 159054409963103491

Offset: 1

Matt C. Anderson, Jul 30 2012

nonn

These are prime 13-tuplets.
All terms congruent to 991 (modulo 2310). - Matt C. Anderson, May 29 2015
All terms congruent to 14851 or 24091 (modulo 30030). - Matt C. Anderson, May 31 2015

Vladimir Vlesycit and Matt C. Anderson and Dana Jacobsen, Table of n, a(n) for n = 1..854 [first 20 terms from Vladimir Vlesycit, first 82 terms from Matt C. Anderson]
Tony Forbes and Norman Luhn, Smallest Prime k-tuplets
Norman Luhn, Table of n, a(n) for n = 1..5579

Cf. A186702.

use ntheory ":all"; say for sieve_prime_cluster(1,10**15, 6,12,16,18,22,28,30,36,40,42,46,48); # Dana Jacobsen, Oct 07 2015

A213646 Initial members of prime 11-tuplets: primes p such that p + (0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36) are all prime.

Original entry on oeis.org

1418575498573, 2118274828903, 4396774576273, 6368171154193, 6953798916913, 27899359258003, 28138953913303, 34460918582323, 40362095929003, 42023308245613, 44058461657443, 61062361183903, 76075560855373, 80114623697803, 84510447435493, 85160397055813, 90589658803723

Offset: 1

Matt C. Anderson, Jun 17 2012

nonn

All terms are congruent to 1003 (modulo 2310). - Matt C. Anderson, May 29 2015

Dana Jacobsen, Table of n, a(n) for n = 1..6923 [first 40 terms from Vladimir Vlesycit and Joerg Waldvogel, first 100 terms from Matt C. Anderson]
Tony Forbes and Norman Luhn, Smallest Prime k-tuplets
Norman Luhn, Table of n, a(n) for n = 1..100000

use ntheory ":all"; say for sieve_prime_cluster(1,1e14, 4,6,10,16,18,24,28,30,34,36); # Dana Jacobsen, Oct 01 2015

A213647 Initial members of prime 11-tuplets: primes p such that p + (0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36) are all prime.

Original entry on oeis.org

11, 7908189600581, 10527733922591, 12640876669691, 38545620633251, 43564522846961, 60268613366231, 60596839933361, 71431649320301, 79405799458871, 109319665100531, 153467532929981, 171316998238271, 216585060731771, 254583955361621, 259685796605351, 268349524548221

Offset: 1

Matt C. Anderson, Jun 17 2012

nonn

0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 are the first terms of A135311.
All terms are congruent to 11 (modulo 210). - Zak Seidov, Sep 15 2014
Subsequence of A202282. - Zak Seidov, Sep 15 2014
All terms, except the first one, are congruent to 1271 (modulo 2310). - Matt C. Anderson, May 29 2015

Matt C. Anderson and Dana Jacobsen, Table of n, a(n) for n = 1..6800 (first 100 terms from Matt C. Anderson)
Norman Luhn, Table of n, a(n) for n = 1..100000

Cf. A135311, A202282.

use ntheory ":all"; say for sieve_prime_cluster(1,1e14, 2,6,8,12,18,20,26,30,32,36); # Dana Jacobsen, Oct 01 2015

a(89) corrected by Dana Jacobsen, Oct 01 2015

cp's OEIS Frontend

User: Matt C. Anderson

A350856 Initial members of prime triples (p, p+2, p+14).

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A290124 a(n) = a(n-1) + 12*a(n-2) with a(1) = 1 and a(2) = 2.

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Magma

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A277338 Reverse and Add! sequence starting with 295.

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A247018 Numbers of the form 3*z^2 + z + 3 for some integer z.

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A248015 Positive numbers n such that n^2 + 1 is composite and there are no positive integers c and z such that n = c*z^2 + z + c.

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A241529 Positive numbers k such that k^2 + k + 41 is composite and there are no positive integers a,c,d such that k = c*a*z^2 + ((((d+2)*(1/3))*c-2)*a/d+1)*z + ((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2 - (((d-1)*(1/3))*c+1)/d)/c for an integer z.

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A235381 Positive numbers n such that n^2 + n + 41 is composite and there are no positive integers c or d such that n = c*d*x^2 + ((d-2)*c + 1)*x + ((41*d^2 - d + 1)*c -1)/d for an integer x.

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A241529 Positive numbers k such that k^2 + k + 41 is composite and there are no positive integers a,c,d such that k = caz^2 + ((((d+2)(1/3))c-2)a/d+1)z + ((((367d^2+d+1)(1/9))c^2-((d+2)(1/3))c+1)a/d^2 - (((d-1)(1/3))c+1)/d)/c for an integer z.

A235381 Positive numbers n such that n^2 + n + 41 is composite and there are no positive integers c or d such that n = cdx^2 + ((d-2)c + 1)x + ((41d^2 - d + 1)c -1)/d for an integer x.