A005098 -id:A005098 - cp's OEIS frontend

A175600 Primes of form 4k+1 where k is a Pythagorean prime.

53, 149, 293, 389, 773, 1109, 1493, 1637, 1733, 2309, 2693, 2837, 3413, 3989, 4133, 4373, 4517, 5189, 5717, 5813, 6197, 6389, 7013, 7109, 8069, 8117, 9173, 9749, 10709, 10853, 11813, 12149, 12197, 12437, 12917, 13829, 13877, 14549, 15077, 15173

Offset: 1

Zak Seidov, Jul 22 2010

nonn

"Double-Pythagorean" primes: primes of form 4k+1 with k prime of form 4m+1.

All terms are congruent to 5 modulo 48. - Zak Seidov, Jun 05 2014

			53 = A002144(7) = 4*13 + 1, 13 = A002144(2);
149 = A002144(16) = 4*37 + 1, 37 = A002144(5).

Cf. A002144 (Pythagorean primes: primes of form 4n+1), A005098 (Numbers n such that 4n+1 is prime).

se=Select[Range[5,100000,4],PrimeQ]; (* se=A002144 *)
se2=Select[se,MemberQ[se,(#-1)/4]&]
(* (se2-1)/4 = intersection (A005098, A002144) *)

A227541 a(n) = floor(13*n^2/4).

Original entry on oeis.org

0, 3, 13, 29, 52, 81, 117, 159, 208, 263, 325, 393, 468, 549, 637, 731, 832, 939, 1053, 1173, 1300, 1433, 1573, 1719, 1872, 2031, 2197, 2369, 2548, 2733, 2925, 3123, 3328, 3539, 3757, 3981, 4212, 4449, 4693, 4943, 5200, 5463, 5733, 6009

Offset: 0

Wolfdieter Lang, Aug 07 2013

This generalizes A032527, which uses 5, to 13 (the next prime 1 (mod 4)). The figures in A032527 use n/2 concentric dotted pentagons for even n, and (n-1)/2 concentric dotted pentagons plus an extra dot in the middle if n is odd. In the present case one can take n/2 concentric dotted 13-gons (the dot numbers of each side for these 13-gons are 2, 4, 6, ..., n) for even n>=2. There is no figure for n = 0. For odd n one has (n-1)/2 concentric dotted 13-gons (the dot numbers  of each side for these 13-gons are  3, 5, 7, ..., n) and an extra dotted 3-gon in the middle. See the example section below for the counting.
a(n) = -N(-floor(n/2),n) with the N(a,b) = ((2*a+b)^2 - b^2*13)/4, the norm for integers a + b*omega(13), a, b rational integers, in the quadratic number field Q(sqrt(13)), where omega(13) = (1 + sqrt(13))/2.
a(n) = max({|N(a,n)|,a = -n..+n}) = |N(-floor(n/2),n)| = 3*n^2 + n*floor(n/2) - floor(n/2)^2 = floor(13*n^2/4) (the last eq. checks for even and odd n).
In the general case one has for primes 1 (mod 4), p(k) = A002144(k), k >= 1, a(p(k);n) = floor(p(k)*n^2/4) with o.g.f. G(p(k);x) = x*(A(k)*(1+x^2) + B(k)*x)/((1-x)^3*(1+x)), where A(k) = A005098(k) = (p(k)-1)/4 and B(k) = A119681(k) = (p(k)+1)/2. This follows from the alternative formula a(p(k),n) = p(k)*n^2/4 + ((-1)^n-1)/8, n >= 0 (which checks for even and odd n). Because the denominator of the o.g.f. is 1-2*x+2*x^3-x^4 the recurrence given by Bruno Berselli below holds for all a(k;n) sequences with inputs for n = -1, 0, 1, 2 given by (p(k)-1)/4, 0, (p(k)-1)/4, p(k), respectively.
The dot counting in the concentric p(k)-gons is similar to the one described for p = 5 in A032527 and for p=13 here. For odd n one puts an additional dotted A(k)-gon into the center. - Wolfdieter Lang, Aug 08 2013

			Counting dots in the concentric dotted 13-gons described above in a comment:
a(2*k), k >= 1: (2-1)*13 = 13, (1+(4-1))*13 = 52, (1+3+(6-1))*13 = 117, (1+3+5+7)*13 = 208, ... a(2*k+1), k >= 0: 3, 3+(3-1)*13 = 29, 3+(2+(5-1))*13 = 81, 3+2*(1+2+3)*13 = 159, ... (a dotted triangle is put into the middle of the k concentric 13-gons).

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A032527 (case for prime 5).

Cf. A002144, A005098, A119681.

[ Floor(13*n^2/4) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

A227541:=n->floor(13*n^2/4); seq(A227541(n), n=0..50); # Wesley Ivan Hurt, Jun 09 2014

Table[Floor[13*n^2/4], {n, 0, 50}] (* Wesley Ivan Hurt, Jun 09 2014 *)

a(n) = 13*n^2/4+((-1)^n-1)/8, n >= 0 (use even or odd n to prove it).
G.f.: x*(3+7*x+3*x^2)/((1-x)^3*(1+x)).
a(2*k) = k^2*13, k >= 0.
a(2*k+1) = 3 + k*(k+1)*13, k >= 0.
a(n) = a(-n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Bruno Berselli, Aug 08 2013
a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n+5)/2). - Wesley Ivan Hurt, Mar 12 2015
Sum_{n>=1} 1/a(n) = Pi^2/78 + tan(Pi/(2*sqrt(13)))*Pi/sqrt(13). - Amiram Eldar, Jul 30 2024

A320599 Numbers k such that 4k + 1 and 8k + 1 are both primes.

Original entry on oeis.org

9, 24, 39, 57, 84, 144, 150, 165, 207, 219, 234, 249, 252, 267, 309, 324, 357, 402, 414, 507, 522, 534, 555, 570, 639, 654, 759, 765, 777, 792, 795, 882, 924, 927, 942, 969, 1044, 1065, 1089, 1155, 1200, 1215, 1227, 1389, 1395, 1437, 1509, 1530, 1554, 1557

Offset: 1

Amiram Eldar, Nov 20 2018

nonn

Rotkiewicz proved that if k is in this sequence then (4k + 1)*(8k + 1) is a triangular Fermat pseudoprime to base 2 (A293622), and thus under Schinzel's Hypothesis H there are infinitely many triangular Fermat pseudoprimes to base 2.

The corresponding pseudoprimes are 2701, 18721, 49141, 104653, 226801, 665281, 721801, ...

			9 is in the sequence since 4*9 + 1 = 37 and 8*9 + 1 = 73 are both primes.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.
Wikipedia, Schinzel's Hypothesis H.

Cf. A000217, A001567, A293622.

Intersection of A005098 and A005123.

Select[Range[1000], PrimeQ[4#+1] && PrimeQ[8#+1] &]

isok(n) = isprime(4*n+1) && isprime(8*n+1); \\ Michel Marcus, Nov 20 2018

from sympy import isprime
def ok(n): return isprime(4*n + 1) and isprime(8*n + 1)
print(list(filter(ok, range(1558)))) # Michael S. Branicky, Sep 24 2021

A199427 Numbers n such that 4n+1 and 8n+3 are prime.

Original entry on oeis.org

1, 7, 10, 13, 22, 28, 43, 58, 70, 73, 127, 148, 160, 163, 190, 202, 238, 253, 262, 307, 322, 352, 370, 400, 433, 472, 475, 493, 517, 532, 535, 568, 598, 637, 673, 685, 688, 742, 832, 847, 853, 862, 898, 940, 955, 1018, 1087, 1093, 1102, 1120, 1183, 1198, 1270

Offset: 1

Martin Renner, Nov 06 2011

nonn

According to Beiler: the integer 2 is a primitive root of all primes of the form 8n+3 provided 4n+1 is a prime.

			For n = 1, both 11 and 5 are primes, hence 2 is a primitive root of 11.

Albert H. Beiler: Recreations in the theory of numbers. New York: Dover, (2nd ed.) 1966, p. 102, nr. 4.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A001122, A005098, A005124.

Select[Range[1270], PrimeQ[4*# + 1] && PrimeQ[8*# + 3] &] (* T. D. Noe, Nov 07 2011 *)

a(n) = intersection(A005098, A005124).

A254010 Numbers k such that 4k+1 and 4(k+1)+1 are primes.

Original entry on oeis.org

3, 9, 24, 27, 48, 57, 69, 78, 87, 99, 114, 153, 168, 189, 192, 213, 219, 234, 252, 273, 303, 324, 357, 372, 387, 399, 402, 423, 468, 498, 534, 567, 573, 594, 597, 609, 618, 654, 672, 687, 699, 708, 714, 738, 759, 804, 813, 864, 882, 903, 918, 924, 948, 969, 1032, 1038, 1128, 1182, 1197, 1203, 1233, 1242, 1269, 1308, 1353

Offset: 1

Zak Seidov, Jan 22 2015

nonn

Both k and k+1 are terms in A005098. All terms are multiples of 3.

4k+1 and 4(k+1)+1 are pairs of consecutive primes. Notice that in all cases, the numbers 4(k-1)+1 and 4(k+2)+1 are not prime as they are multiples of 3.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A005098.

[n: n in [0..1000] | IsPrime(4*n+1) and IsPrime(4*(n+1)+1)]; // Vincenzo Librandi, Apr 24 2015

A254010:=n->`if`(isprime(4*n+1) and isprime(4*(n+1)+1), n, NULL): seq(A254010(n), n=1..2000); # Wesley Ivan Hurt, Apr 23 2015

Select[Range[1000], PrimeQ[4 # + 1] && PrimeQ[4 (# + 1) + 1] &] (* Vincenzo Librandi, Apr 24 2015 *)

A254288 Numbers k such that 4*k + {1, 3, 7, 9, 13, 19} are all prime.

Original entry on oeis.org

1, 370, 41425, 81535, 255625, 267175, 311590, 365350, 1054570, 1381750, 2533600, 2975125, 3266080, 3930205, 4684210, 4782385, 4802860, 5940850, 6414610, 7986565, 8429245, 8570470, 8636305, 8810080, 9270715, 9857980, 10459525, 13708225, 13917490, 15127720, 15252460

Offset: 1

K. D. Bajpai, Jan 27 2015

nonn

All terms in this sequence are congruent to 1 mod 3.

Subsequence of A123986.

			a(2) = 370;
4*370 +  1 = 1481;
4*370 +  3 = 1483;
4*370 +  7 = 1487;
4*370 +  9 = 1489;
4*370 + 13 = 1493;
4*370 + 19 = 1499;
All six are prime.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000 (first 726 terms from K. D. Bajpai)

Cf. A000040, A005098, A095278, A123986.

[n: n in [0..10^8] | forall{4*n+i: i in [1, 3, 7, 9, 13, 19] |  IsPrime(4*n+i)}]; // Vincenzo Librandi, Mar 12 2015

Select[Range[5*10^7], PrimeQ[4*# + 1] && PrimeQ[4*# + 3] && PrimeQ[4*# + 7] && PrimeQ[4*# + 9] && PrimeQ[4*# + 13] && PrimeQ[4*# + 19] &]
Select[Range[5*10^6], And @@ PrimeQ /@ ({1, 3, 7, 9, 13, 19} + 4 #) &]

for(n=1,10^7, if( isprime(4*n + 1) && isprime(4*n + 3) &&isprime(4*n + 7) &&isprime(4*n + 9) &&isprime(4*n + 13) &&isprime(4*n + 19) , print1(n,", ")))

A254376 Numbers n such that 4n+1, 4n+3, 4n+7, 4n+9 and 4n+13 are prime.

Original entry on oeis.org

1, 25, 370, 4015, 4855, 10945, 36040, 41425, 41710, 50455, 56335, 61900, 81535, 86995, 116290, 129700, 134110, 158365, 207430, 239635, 255625, 265990, 267175, 272815, 293395, 311590, 335080, 337810, 339700, 342115, 365350, 393385, 403960, 481345, 488590, 550990

Offset: 1

K. D. Bajpai, Jan 29 2015

nonn

All terms in this sequence are 1 mod 3.
Each term yields a set of five consecutive primes.
Alternatively: Numbers n such that 4n+k forms a set of five consecutive primes for k = {1,3,7,9,13}.
Subsequence of A123986.

			25 is in the list because 4*25 + 1 = 101, 4*25 + 3 = 103, 4*25 + 7 = 107, 4*25 + 9 = 109 and 4*25 + 13 = 113 are all prime.

K. D. Bajpai, Table of n, a(n) for n = 1..8458

Cf. A000040, A005098, A095278, A123986.

[n: n in [0..10^6] | forall{4*n+r: r in [1,3,7,9,13] | IsPrime(4*n+r)}]; // Vincenzo Librandi, Feb 16 2015

Select[Range[1, 500000], PrimeQ[4*# + 1] && PrimeQ[4*# + 3] && PrimeQ[4*# + 7] && PrimeQ[4*# + 9] && PrimeQ[4*# + 13] &]
Select[Range[5*10^6], And @@ PrimeQ /@ ({1, 3, 7, 9, 13} + 4 #) &]

for(n=1,10^7, if( isprime(4*n + 1) && isprime(4*n + 3) &&isprime(4*n + 7) &&isprime(4*n + 9) &&isprime(4*n + 13), print1(n,", ")))

A260821 Least positive integer k for which n*2^(2^k) + 1 is composite.

Original entry on oeis.org

5, 1, 2, 2, 1, 1, 3, 1, 2, 2, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 3, 1, 2, 3, 1, 3, 3, 1, 1, 1, 1, 1, 2, 1, 1, 5, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 4, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 2

Offset: 1

Marco Ripà, Jul 31 2015

nonn

a(n) = 1 for nonzero n in A045751. - Michel Marcus, Aug 01 2015

			a(7)=3 because 7*2^2 + 1 = 29 is prime and 7*2^(2^2) + 1 = 113 is also prime, while 7*2^(2^3) + 1 = 11*163.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A045751, A005098.

A260821[n_] := Module[{k = 0}, While[PrimeQ[n*2^(2^++k) + 1]]; k];
Array[A260821, 100] (* Paolo Xausa, Jan 31 2024 *)

a(n) = {k = 1; while (isprime(n*2^2^k+1), k++); k;} \\ Michel Marcus, Aug 01 2015

More terms from Michel Marcus, Aug 01 2015

A333721 Numbers k such that k + 1, 2k + 1, 3k + 1, 4k + 1, and 6k + 1 are all prime.

Original entry on oeis.org

1530, 4260, 25410, 26040, 78540, 111720, 174990, 211050, 214830, 395430, 403260, 409290, 459690, 487830, 512820, 711120, 779790, 910560, 1023750, 1135950, 1280370, 1312350, 1451520, 1464810, 1487070, 1563510, 1623360, 1698060, 1824330, 1933680, 2006340, 2097480

Offset: 1

Pedro Caceres, May 04 2020

nonn

All terms are multiples of 6.

All terms are multiples of 30. - Robert Israel, Jun 17 2020

			25410 is in the sequence because 25411, 50821, 76231, 101641, 152461 are all prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Pedro Caceres, 30 Ideas about Prime Numbers

Cf. A006093, A005097, A024892, A087370, A005098, A005099, A024898, A024899.

select(t -> andmap(isprime, [t+1,2*t+1,3*t+1,4*t+1,6*t+1]), [seq(i,i=30..3*10^6,30)]); # Robert Israel, Jun 17 2020

isok(m)={for(i=1, 6, if(i<>5&&!isprime(i*m+1), return(0))); 1}
{ forstep(n=0, 3*10^6, 6, if(isok(n), print1(n, ", "))) } \\ Andrew Howroyd, May 04 2020

cp's OEIS Frontend

A175600 Primes of form 4k+1 where k is a Pythagorean prime.

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A227541 a(n) = floor(13*n^2/4).

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A320599 Numbers k such that 4k + 1 and 8k + 1 are both primes.

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A199427 Numbers n such that 4n+1 and 8n+3 are prime.

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A254010 Numbers k such that 4k+1 and 4(k+1)+1 are primes.

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A254288 Numbers k such that 4*k + {1, 3, 7, 9, 13, 19} are all prime.

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A254376 Numbers n such that 4n+1, 4n+3, 4n+7, 4n+9 and 4n+13 are prime.

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