A017077 -id:A017077 - cp's OEIS frontend

A205171 The lesser of twin primes == 1 (mod 8).

17, 41, 137, 281, 521, 569, 617, 641, 809, 857, 881, 1049, 1289, 1481, 1697, 1721, 2081, 2129, 2657, 2729, 2801, 2969, 3257, 3329, 3929, 4001, 4049, 4217, 4241, 4337, 4481, 4649, 4721, 5009, 5417, 5441, 5657, 5849, 6089, 6449, 6569, 6689, 6761, 7457

Offset: 1

Robert G. Wilson v, Jan 22 2012

Cf. A071695, A205172.

Intersection of A017077 and A001359.

Select[ Prime@ Range@ 1000, Mod[#, 8] == 1 && PrimeQ[# + 2] &]

A218028 a(n) is the smallest positive integer k such that k^4 + 1 == 0 mod p, where p is the n-th prime of the form p = 1 + 8*b (see A007519).

Original entry on oeis.org

2, 3, 10, 12, 33, 18, 10, 9, 12, 8, 4, 60, 5, 85, 70, 45, 31, 79, 92, 170, 43, 76, 152, 59, 59, 139, 256, 64, 62, 40, 44, 188, 177, 18, 14, 156, 227, 192, 231, 223, 79, 31, 75, 362, 7, 239, 338, 402, 6, 235, 114, 72, 342, 511, 15, 483, 310, 355, 104, 292, 232

Offset: 1

Michel Lagneau, Oct 22 2012

nonn

A007519(n) : primes of form 8n+1.

			a(5) = 33 because 33^4+1 = 1185922 = 2 * 97 * 6113 with A007519(5) = 97.

Robert Israel, Table of n, a(n) for n = 1..10000
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 521.

Cf. A017077, A007519.

V:= Vector(100): count:= 0:
for p from 9 by 8 while count < 100 do
  if isprime(p) then
      count:= count+1; V[count]:=min(map(rhs@op,[msolve(k^4+1,p)]))
    fi
od:
convert(V,list); # Robert Israel, Mar 13 2018

aa = {}; Do[p = Prime[n]; If[Mod[p, 8] == 1, k = 1; While[ ! Mod[k^4 + 1, p] == 0, k++ ]; AppendTo[aa, k]], {n, 300}]; aa

A231233 Triangle T(n,k) = greatest prime factor of n*k+1.

Original entry on oeis.org

2, 3, 5, 2, 7, 5, 5, 3, 13, 17, 3, 11, 2, 7, 13, 7, 13, 19, 5, 31, 37, 2, 5, 11, 29, 3, 43, 5, 3, 17, 5, 11, 41, 7, 19, 13, 5, 19, 7, 37, 23, 11, 2, 73, 41, 11, 7, 31, 41, 17, 61, 71, 3, 13, 101, 3, 23, 17, 5, 7, 67, 13, 89, 5, 37, 61, 13, 5, 37, 7, 61, 73, 17, 97, 109, 11, 19, 29

Offset: 1

Michel Marcus, Nov 06 2013

			Triangle begins:
  2;
  3,  5;
  2,  7,  5;
  5,  3, 13, 17;
  3, 11,  2,  7, 13;
  7, 13, 19,  5, 31, 37;
  2,  5, 11, 29,  3, 43,  5;

Michel Marcus, Rows n = 1..100 of triangle, flattened
Étienne Fouvry, On the greatest prime factor of ab+1, arXiv:1311.1161 [math.NT], 2013.

Cf. A005408, A016777, A016813, A016861, A016921, A016993, A017077, A017173.

Flat(List([1..12],n->List([1..n],k->Maximum(FactorsInt(n*k+1))))); # Muniru A Asiru, Sep 23 2018

T[n_, k_] := FactorInteger[n k + 1][[-1, 1]];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 23 2018 *)

tabl(nn) = {for (n=1, nn, for (k=1, n, f = factor(n*k+1); print1(f[#f~, 1], ", ");); print(););}

T(n, k) = A006530(n*k+1).

A341488 a(0) = 1, and for any n > 0, a(n) = A341458(a(n-1), n).

Original entry on oeis.org

1, 1, 2, 5, 8, 4, 2, 8, 1, 9, 25, 116, 8, 117, 25, 16, 1, 17, 25, 108, 8, 109, 25, 24, 1, 25, 2, 29, 8, 28, 2, 32, 1, 33, 97, 92, 8, 93, 97, 40, 1, 41, 122, 45, 8, 44, 122, 48, 1, 49, 122, 53, 8, 52, 122, 56, 1, 57, 97, 68, 8, 69, 97, 64, 1, 65, 97, 60, 8, 61

Offset: 0

Rémy Sigrist, Feb 13 2021

This sequence is an analog of the factorial function for the group described in A341458, and has interesting graphical features.

Rémy Sigrist, Table of n, a(n) for n = 0..32769
Rémy Sigrist, PARI program for A341488

Cf. A000142, A008590, A017077, A341458.

PARI
```
See Links section.
```

Apparently:
- a(n) = 1 iff n = 1 or n belongs to A008590,
- a(n) = n iff n belongs to A017077.

A375090 Omit the trailing 4 from the terms of the Lucas-Lehmer sequence A003010.

Original entry on oeis.org

0, 1, 19, 3763, 141631795, 200595654682274611, 402386166774103602282563565610210099, 1619146272111567178177755907012051366495859012549915851432930874097578803

Offset: 0

Stefano Spezia, Jul 29 2024

The next term a(8) has 146 digits.
Proposition: A003010(n) has only a trailing 4.
Proof: the case n = 0 is trivial since a(0) = 4 has only a digit. For n > 0, we need to prove that the terms have at least a trailing 4 and that this is unique. A003010(n) == 4 (mod 10) because A003010(0) = 4 and 4^2 - 2 = 14 == 4 (mod 10). To prove that this is unique, we prove that the tenth digit of A003010(n) is never equal to 4 for n > 0, or equivalently, that it is odd. Considering that (10*h + 4)^2 = 100*h^2 + 80*h + 16 and 80*h + 16 has the tenth digit of the form 8*k + 1 mod 10, it follows that the tenth digit of A003010(n-1)^2 is odd, and therefore, also that of A003010(n-1)^2 - 2 = A003010(n). QED.

Stefano Spezia, Table of n, a(n) for n = 0..10

Cf. A003010, A017077.

A003010[0]=4; A003010[n_]:=A003010[n-1]^2-2; Array[A003010, 9, 0]; a[n_]:=(A003010[n]-4)/10; Array[a,8,0]

a(n) = (A003010(n) - 4)/10.

a(n) = 10*a(n-1)^2 + 8*a(n-1) + 1 for n > 0.

A376102 Array read by ascending antidiagonals: A(n,k) = k*2^(n+1) + 1.

Original entry on oeis.org

1, 1, 3, 1, 5, 5, 1, 9, 9, 7, 1, 17, 17, 13, 9, 1, 33, 33, 25, 17, 11, 1, 65, 65, 49, 33, 21, 13, 1, 129, 129, 97, 65, 41, 25, 15, 1, 257, 257, 193, 129, 81, 49, 29, 17, 1, 513, 513, 385, 257, 161, 97, 57, 33, 19, 1, 1025, 1025, 769, 513, 321, 193, 113, 65, 37, 21

Offset: 0

Stefano Spezia, Sep 14 2024

In 1747, Euler showed that any factor of a Fermat number A000215(n) is of the form k*2^(n+1) + 1. See Wells at p. 148.

			The array begins as:
  1,   3,   5,   7,   9,  11,  13, ...
  1,   5,   9,  13,  17,  21,  25, ...
  1,   9,  17,  25,  33,  41,  49, ...
  1,  17,  33,  49,  65,  81,  97, ...
  1,  33,  65,  97, 129, 161, 193, ...
  1,  65, 129, 193, 257, 321, 385, ...
  1, 129, 257, 385, 513, 641, 769, ...
  ...

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 70-71, 237-242.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 136.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987.

Cf. A000079, A000215, A000295.

Cf. A000012 (k=0), A000051, A000337, A004119, A005408 (n=0), A016813 (n=1), A017077 (n=2), A158057 (n=3).

A[n_,k_]:=k*2^(n+1)+1; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

G.f.: (1 - 2*x + y)/((1 - x)*(1 - 2*x)*(1 - y)^2).
E.g.f.: exp(x+y)*(1 + 2*exp(x)*y).
Sum_{0<=k<=n} A(n-k,k) = A000295(n+2).
A(n,1) = A000051(n+1).
A(n,3) = A004119(n+2).
A(n,n) = A000337(n+1).

cp's OEIS Frontend

A205171 The lesser of twin primes == 1 (mod 8).

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A218028 a(n) is the smallest positive integer k such that k^4 + 1 == 0 mod p, where p is the n-th prime of the form p = 1 + 8*b (see A007519).

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A231233 Triangle T(n,k) = greatest prime factor of n*k+1.

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A341488 a(0) = 1, and for any n > 0, a(n) = A341458(a(n-1), n).

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A375090 Omit the trailing 4 from the terms of the Lucas-Lehmer sequence A003010.

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A376102 Array read by ascending antidiagonals: A(n,k) = k*2^(n+1) + 1.

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