Warut Roonguthai - cp's OEIS frontend

A130230 Primes p == 5 (mod 8) such that the Diophantine equation x^2 - p*y^2 = -4 has a solution in odd integers x, y.

5, 13, 29, 53, 61, 109, 149, 157, 173, 181, 229, 277, 293, 317, 397, 421, 461, 509, 541, 613, 653, 661, 733, 773, 797, 821, 853, 941, 1013, 1021, 1061, 1069, 1093, 1109, 1117, 1181, 1229, 1237, 1277, 1373, 1381, 1429, 1453, 1493, 1549, 1597

Offset: 1

Warut Roonguthai, Aug 06 2007

nonn

For the Diophantine equation x^2 - p*y^2 = -4 to have a solution in odd integers x, y we must have p == 5 (mod 8)
Calculated using Dario Alpern's quadratic Diophantine solver, see link.
Suggested by a discussion on the Number Theory Mailing List, circa Aug 01 2007.

Robin Visser, Table of n, a(n) for n = 1..10000
Dario Alpern, Generic two integer variable equation solver.

Cf. A130229.

A130229 Primes p == 5 (mod 8) such that the Diophantine equation x^2 - p*y^2 = -4 has no solution in odd integers x, y.

Original entry on oeis.org

37, 101, 197, 269, 349, 373, 389, 557, 677, 701, 709, 757, 829, 877, 997, 1213, 1301, 1613, 1861, 1901, 1949, 1973, 2069, 2221, 2269, 2341, 2357, 2621, 2797, 2837, 2917, 3109, 3181, 3301, 3413, 3709, 3797, 3821, 3853, 3877, 4013, 4021, 4093

Offset: 1

Warut Roonguthai, Aug 06 2007

nonn

For the Diophantine equation x^2 - p*y^2 = -4 to have a solution in odd integers x, y we must have p == 5 (mod 8).
Calculated using Dario Alpern's quadratic Diophantine solver, see link.
Suggested by a discussion on the Number Theory Mailing List, circa Aug 01 2007.

Robin Visser, Table of n, a(n) for n = 1..10000
Dario Alpern, Generic two integer variable equation solver.
Florian Breuer, Periods of Ducci sequences and odd solutions to a Pellian equation, University of Newcastle, Australia, 2018.
Florian Breuer and Cameron Shaw-Carmody, Parity bias in fundamental units of real quadratic fields, Univ. Newcastle (Australia), Comp.-Assisted Res. Math. Appl. (2024). See pp. 1-4.
J. Xue, T.-C. Yang, C.-F. Yu, Supersingular abelian surfaces and Eichler class number formula, arXiv preprint arXiv:1404.2978, 2014
Jiangwei Xue, TC Yang, CF Yu, Numerical Invariants of Totally Imaginary Quadratic Z[sqrt{p}]-orders, arXiv preprint arXiv:1603.02789, 2016

Cf. A130230.

A117804 Natural position of n in the string 12345678910111213....

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120

Offset: 1

Warut Roonguthai, Jul 23 2007

The number of digits necessary to write down all the numbers 0, 1, 2, ..., n-1. Thus, the partial sums of A055642 are given by a(n+1). - Hieronymus Fischer, Jun 08 2012
From Daniel Forgues, Mar 21 2013: (Start)
From n = 10^0 + 1 to 10^1: a(n) - a(n-1) = 1 (9 * 10^0 terms);
From n = 10^1 + 1 to 10^2: a(n) - a(n-1) = 2 (9 * 10^1 terms);
From n = 10^2 + 1 to 10^3: a(n) - a(n-1) = 3 (9 * 10^2 terms);
...
From n = 10^k + 1 to 10^(k+1): a(n) - a(n-1) = k+1 (9 * 10^k terms). (End)
By the "number of digits" definition, a(n) = 1 + A058183(n-1) for n > 1. - David Fifield, Jun 02 2019

			12 begins at the 14th place in 12345678910111213... (we are ignoring "early bird" occurrences here, cf. A116700), so a(12) = 14.
From _Daniel Forgues_, Mar 21 2013: (Start)
a(10^1) = 10. (1*10^1 - 0)
a(10^2) = 190. (2*10^2 - 10)
a(10^3) = 2890. (3*10^3 - 110)
a(10^4) = 38890. (4*10^4 - 1110)
a(10^5) = 488890. (5*10^5 - 11110)
a(10^6) = 5888890. (6*10^6 - 111110)
...
a(10^k) = k*10^k - R_k + 1, R_k := k-th repunit (cf. A002275)
(the number of digits necessary to write down the numbers 0..10^k-1). (End)

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A116700.

Cf. A055640, A055641, A055642, A102669-A102685.

a(n) = d*n + 1 - (10^d - 1)/9 where d is the number of decimal digits in n, i.e., d = floor(log_10(n)) + 1.
From Hieronymus Fischer, Jun 08 2012: (Start)
a(n) = Sum_{j=0..n-1} A055642(j).
a(n) = 1 + A055642(n-1)*n - (10^A055642(n-1)-1)/9.
a(n) = 1 + A055642(n)*n - (10^A055642(n)-1)/9.
a(10^n) = (9*n-1)*(10^n-1)/9 + n + 1. (This is the total number of digits necessary to write down all the numbers with <= n places.)
G.f.: g(x) = x/(1-x) + (x/(1-x)^2)*Sum_{j>=0} x^10^j; corrected by Ilya Gutkovskiy, Jan 09 2017 (End)

A118537 Number of functions f: {1, 2, ..., n} --> {1, 2, ..., n} such that f(1) != f(2), f(2) != f(3), ..., f(n-1) != f(n), f(n) != f(1).

Original entry on oeis.org

2, 6, 84, 1020, 15630, 279930, 5764808, 134217720, 3486784410, 99999999990, 3138428376732, 106993205379060, 3937376385699302, 155568095557812210, 6568408355712890640, 295147905179352825840, 14063084452067724991026

Offset: 2

Warut Roonguthai, May 06 2006

nonn

a(n) is also the number of circuits of length n in the complete graph on n vertices. - Thibaut Lienart (syncthib(AT)gmail.com), Jan 29 2010
Circuits are allowed to be self-intersecting and are directional with a designated start node. The number of (self-avoiding) directed cycles is given by A124355. - Andrew Howroyd, Sep 05 2018
a(n) is also the number of graph colorings of the cycle graph C_n with n colors. - Orson R. L. Peters, Jul 27 2020

Andrew Howroyd, Table of n, a(n) for n = 2..100

Cf. A055897, A124355.

[(n-1)^n + (-1)^n*(n-1) : n in [2..20]]; // Wesley Ivan Hurt, Jul 27 2020

a[n_]:=(n-1)^n + (-1)^n*(n-1); Array[a, 50, {2, 51}] (* Stefano Spezia, Sep 07 2018 *)

a(n) = (n-1)^n + (-1)^n*(n-1); \\ Andrew Howroyd, Sep 05 2018

a(n) = (n-1)^n + (-1)^n*(n-1).

A051835 Palindromic Sophie Germain primes.

Original entry on oeis.org

2, 3, 5, 11, 131, 191, 12821, 14741, 19391, 19991, 36563, 38183, 93239, 96269, 1028201, 1074701, 1150511, 1178711, 1243421, 1281821, 1317131, 1333331, 1407041, 1456541, 1508051, 1532351, 1557551, 1598951, 1600061, 1609061

Offset: 1

Warut Roonguthai Dec 11 1999

p and 2p+1 are primes (cf. A005384) and p is a palindrome.

Robert Israel, Table of n, a(n) for n = 1..10000
H. Dubner, Palindromic Sophie Germain primes, Journal of Recreational Mathematics, Vol. 26, No. 1, pp. 38-41, 1994.

Cf. A002385, A005384.

makepali:= proc(n, d) local L; # case with d odd
  L:= convert(n, base, 10);
  10^((d-1)/2)*n + add(L[i]*10^((d+1)/2-i), i=2..(d+1)/2)
end proc:
N:= 100: # for a(1)..a(N)
R:= 2,3,5,11: count:= 4:
for d from 3 by 2  while count < N do
  for i in [1,3,7,9] while count < N do
    for x from 0 to 10^((d-1)/2)-1 while count < N do
      y:= makepali(i*10^((d-1)/2)+x,d);
      if isprime(y) and isprime(2*y+1) then
        R:= R, y;
        count:= count+1;
       fi
od od od:
R; # Robert Israel, Nov 22 2020

Select[Prime[Range[125000]],PrimeQ[2#+1]&&PalindromeQ[#]&] (* Harvey P. Dale, Nov 21 2021 *)

A030997 Smallest prime which is a concatenation of n consecutive primes.

Original entry on oeis.org

2, 23, 5711, 2357, 711131719, 113127131137139149, 29313741434753, 107109113127131137139149, 211223227229233239241251257, 691701709719727733739743751757, 2329313741434753596167

Offset: 1

Patrick De Geest and Warut Roonguthai

			a(5) = 711131719 is the smallest prime which is the concatenation of five consecutive primes 7, 11, 13, 17 and 19.

Hans Havermann, Table of n, a(n) for n = 1..100

Cf. A030996, A068655, A030473, A086041, A099727, A167517.

Cf. A030461 (primes that are concatenations of two primes), A030469 (three primes), A030473 (four primes), A086041 (five primes).

for(k=1,19, for(i=0,1e9, isprime( eval( p=concat( vector( k,j,Str( prime( i+j )))))) & break); print1(p,", ")) \\ M. F. Hasler, Nov 10 2009

A028916 Friedlander-Iwaniec primes: Primes of form a^2 + b^4.

Original entry on oeis.org

2, 5, 17, 37, 41, 97, 101, 137, 181, 197, 241, 257, 277, 281, 337, 401, 457, 577, 617, 641, 661, 677, 757, 769, 821, 857, 881, 977, 1097, 1109, 1201, 1217, 1237, 1297, 1301, 1321, 1409, 1481, 1601, 1657, 1697, 1777, 2017, 2069, 2137, 2281, 2389, 2417, 2437

Offset: 1

Warut Roonguthai

nonn

John Friedlander and Henryk Iwaniec proved that there are infinitely many such primes.
A256852(A049084(a(n))) > 0. - Reinhard Zumkeller, Apr 11 2015
Primes in A111925. - Robert Israel, Oct 02 2015
Its intersection with A185086 is A262340, by the uniqueness part of Fermat's two-squares theorem. - Jonathan Sondow, Oct 05 2015
Cunningham calls these semi-quartan primes. - Charles R Greathouse IV, Aug 21 2017
Primes of the form (x^2 + y^2)/2, where x > y > 0, such that (x-y)/2 or (x+y)/2 is square. - Thomas Ordowski, Dec 04 2017
Named after the Canadian mathematician John Benjamin Friedlander (b. 1941) and the Polish-American mathematician Henryk Iwaniec (b. 1947). - Amiram Eldar, Jun 19 2021

			2 = 1^2 + 1^4.
5 = 2^2 + 1^4.
17 = 4^2 + 1^4 = 1^2 + 2^4.

T. D. Noe, Table of n, a(n) for n = 1..10000
Art of Problem Solving, Fermat's Two Squares Theorem.
A. J. C. Cunningham, High quartan factorisations and primes, Messenger of Mathematics, Vol. 36 (1907), pp. 145-174.
John Friedlander and Henryk Iwaniec, Using a parity-sensitive sieve to count prime values of a polynomial, Proc. Nat. Acad. Sci., Vol. 94 (1997), pp. 1054-1058.
J. Friedlander and H. Iwaniec, The polynomial x^2 + y^4 captures its primes, arXiv:math/9811185 [math.NT], 1998; Ann. of Math. 148 (1998), 945-1040.
Charles R Greathouse IV, Tables of special primes.
Wikipedia, Friedlander-Iwaniec theorem.

Cf. A078523, A111925.
Cf. A000290,  A000583, A000040, A256852, A256863 (complement), A002645 (subsequence), subsequence of A247857.
Primes of form n^2 + b^4, b fixed: A002496 (b = 1), A243451 (b = 2), A256775 (b = 3), A256776 (b = 4), A256777 (b = 5), A256834 (b = 6), A256835 (b = 7), A256836 (b = 8), A256837 (b = 9), A256838 (b = 10), A256839 (b = 11), A256840 (b = 12), A256841 (b = 13).

a028916 n = a028916_list !! (n-1)
a028916_list = map a000040 $ filter ((> 0) . a256852) [1..]
-- Reinhard Zumkeller, Apr 11 2015

N:= 10^5: # to get all terms <= N
S:= {seq(seq(a^2+b^4, a = 1 .. floor((N-b^4)^(1/2))),b=1..floor(N^(1/4)))}:
sort(convert(select(isprime,S),list)); # Robert Israel, Oct 02 2015

nn = 10000; t = {}; Do[n = a^2 + b^4; If[n <= nn && PrimeQ[n], AppendTo[t, n]], {a, Sqrt[nn]}, {b, nn^(1/4)}]; Union[t] (* T. D. Noe, Aug 06 2012 *)

list(lim)=my(v=List([2]),t);for(a=1,sqrt(lim\=1),forstep(b=a%2+1, sqrtint(sqrtint(lim-a^2)), 2, t=a^2+b^4;if(isprime(t),listput(v,t)))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jun 12 2013

Title expanded by Jonathan Sondow, Oct 02 2015

A027569 Initial members of prime decaplets (p, p+2, p+6, p+8, p+12, p+18, p+20, p+26, p+30, p+32).

Original entry on oeis.org

11, 33081664151, 83122625471, 294920291201, 573459229151, 663903555851, 688697679401, 730121110331, 1044815397161, 1089869189021, 1108671297731, 1235039237891, 1291458592421, 1738278660731

Offset: 1

Warut Roonguthai

nonn

All terms are congruent to 11 (modulo 210). - Matt C. Anderson, May 28 2015

Matt C. Anderson and Dana Jacobsen, Table of n, a(n) for n = 1..10000 [first 100 terms from Matt C. Anderson]
Tony Forbes and Norman Luhn, Prime k-tuplets
Norman Luhn, 1 million terms of A027569, zip compressed (7.93 MB) (2021)

Cf. A027570, A202281, A202282.

composite_small := proc (n::integer)
description "procedure to determine if n has a prime factor less than 100";
if igcd(2305567963945518424753102147331756070, n) = 1 then
return false else return true end if;
end proc;
# begin initialization section
p := [0, 2, 6, 8, 12, 18, 20, 26, 30, 32]; o := [1271, 1691]; m := 2310;
# end initialization section
with(ArrayTools); os := Size(o, 2); ps := Size(p, 2);
loopstop := 10^11; loopstart := 0;
print(11);
for n from loopstart to loopstop do
for a to os do
counter := 0; wc := 0; wd := 0;
while `and`(wd > -10, wd < ps) do
wd := wd+1;
if composite_small(m*n+o[a]+p[wd]) = false then wd := wd+1 else wd := -10 end if;
end do;
if wd >= 9 then while `and`(counter >= 0, wc < ps) do
wc := wc+1;
if isprime(m*n+o[a]+p[wc]) then counter := counter+1 else counter := -1 end if end do;
end if;
if counter = ps then print(m*n+o[a]); end if;
end do;
end do;
# Matt C. Anderson, Apr 30 2015

is(n)=isprime(n) && isprime(n+2) && isprime(n+6) && isprime(n+8) && isprime(n+12) && isprime(n+18) && isprime(n+20) && isprime(n+26) && isprime(n+30) && isprime(n+32)
v=primes(10); t=1; forprime(p=31,1e11, v[t]=p; t=(t%10)+1; if(p-v[t]==32 && is(v[t]), print1(v[t]", "))) \\ Charles R Greathouse IV, May 20 2015

use ntheory ":all"; say for sieve_prime_cluster(1,1e13, 2,6,8,12,18,20,26,30,32); # Dana Jacobsen, Sep 30 2015

A027570 Initial members of prime decaplets (p, p+2, p+6, p+12, p+14, p+20, p+24, p+26, p+30, p+32).

Original entry on oeis.org

9853497737, 21956291867, 22741837817, 164444511587, 179590045487, 217999764107, 231255798857, 242360943257, 666413245007, 696391309697, 867132039857, 974275568237, 976136848847, 1002263588297, 1086344116367

Offset: 1

Warut Roonguthai

nonn

All terms are congruent to 167 (modulo 210). - Matt C. Anderson, May 29 2015

Matt C. Anderson and Dana Jacobsen, Table of n, a(n) for n = 1..10000 [first 100 terms from Matt C. Anderson]
Tony Forbes and Norman Luhn, Prime k-tuplets
Norman Luhn, 1 million terms of A027570, zip compressed (7.94 MB) (2021).

Cf. A027569, A202361, A202362.

a := 1:
for b to 25 do
a := a*ithprime(b):
end do:
a;
# now 'a' is the product of the primes less than 100.
composite_small := proc (n::integer)
description "procedure to determine if n has a prime factor less than 100";
if igcd(2305567963945518424753102147331756070, n) = 1 then return false
else return true;
end if;
end proc:
# so composite_small tests if there are any factors 2 through 97.
#begin initialization section
p := [0, 2, 6, 12, 14, 20, 24, 26, 30, 32];
o := [7517, 10247, 12137, 14447, 14867, 17177, 21377, 24107, 25997, 28727];
m := 30030;
#end initialization section
# implement isprime(m*n+o+p)
with(ArrayTools):
os:=Size(o,2):
ps:=Size(p,2):
#here ps is 10 so a prime constellation of length 10.
loopstop := 10^11:
loopstart := 0:
for n from loopstart to loopstop do
for a to os do
counter := 0; wc := 0; wd := 0;
  while `and`(wd > -10, wd < ps) do
  wd := wd+1;
  if composite_small(m*n+o[a]+p[wd]) = false then wd := wd+1
  else wd := -10 end if;
  end do;
if wd >= 9 then
while `and`(counter >= 0, wc < ps) do
  wc := wc+1;
  if isprime(m*n+o[a]+p[wc]) then counter := counter+1;
  else counter := -1
  end if;
end do;
end if;
if counter = ps then print(m*n+o[a]) end if;
end do:
end do:
# Matt C. Anderson, Apr 15 2015

use ntheory ":all"; say for sieve_prime_cluster(1,1e13, 2,6,12,14,20,24,26,30,32); # Dana Jacobsen, Sep 30 2015

A030431 Primes of form 10n+3.

Original entry on oeis.org

3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263, 283, 293, 313, 353, 373, 383, 433, 443, 463, 503, 523, 563, 593, 613, 643, 653, 673, 683, 733, 743, 773, 823, 853, 863, 883, 953, 983, 1013, 1033, 1063, 1093, 1103, 1123, 1153, 1163, 1193

Offset: 1

Warut Roonguthai

nonn

Also primes of form 5n+3.
Union of A132233, A132235, {3}. - Ray Chandler, Apr 07 2009
Primes p such that arithmetic mean of divisors of p^4 is an integer. There are 2 such sequences of primes, this one and A030430. - Ctibor O. Zizka, Oct 20 2009
5 is not quadratic residue of primes of this form. - Vincenzo Librandi, Jun 25 2014
Intersection of A000040 and A017305. - Iain Fox, Dec 30 2017

Michael B. Porter, Table of n, a(n) for n = 1..100000
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.

Cf. A045414, A049508, A053179, A102338.

Select[Prime@Range[200], Mod[ #, 10] == 3 &] (* Ray Chandler, Nov 07 2006 *)
Select[10 Range[0, 150] + 3, PrimeQ]  (* Harvey P. Dale, Apr 06 2011 *)

select(n->n%10==3, primes(500)) \\ Charles R Greathouse IV, Apr 29 2015

a(n) = 10*A102338(n) + 3.

Extended by Ray Chandler, Nov 07 2006

cp's OEIS Frontend

User: Warut Roonguthai

A130230 Primes p == 5 (mod 8) such that the Diophantine equation x^2 - p*y^2 = -4 has a solution in odd integers x, y.

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A130229 Primes p == 5 (mod 8) such that the Diophantine equation x^2 - p*y^2 = -4 has no solution in odd integers x, y.

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A117804 Natural position of n in the string 12345678910111213....

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A118537 Number of functions f: {1, 2, ..., n} --> {1, 2, ..., n} such that f(1) != f(2), f(2) != f(3), ..., f(n-1) != f(n), f(n) != f(1).

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Magma

Mathematica

PARI

Formula

A051835 Palindromic Sophie Germain primes.

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Maple

Mathematica

A030997 Smallest prime which is a concatenation of n consecutive primes.

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PARI

A028916 Friedlander-Iwaniec primes: Primes of form a^2 + b^4.

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Haskell

Maple

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Extensions

A027569 Initial members of prime decaplets (p, p+2, p+6, p+8, p+12, p+18, p+20, p+26, p+30, p+32).

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Maple

PARI