A078586 -id:A078586 - cp's OEIS frontend

A006278 a(n) is the product of the first n primes congruent to 1 (mod 4).

5, 65, 1105, 32045, 1185665, 48612265, 2576450045, 157163452745, 11472932050385, 1021090952484265, 99045822390973705, 10003628061488344205, 1090395458702229518345, 123214686833351935572985

Offset: 1

Gene_Salamin(AT)cohr.com

nonn

a(n)+2 is prime for n=1,2. No others are prime for n <= 200. Compare A002110 and A078586. - T. D. Noe, Dec 01 2002
Also, a(n) is least hypotenuse of exactly A003462(n) Pythagorean triangles of which 2^(n-1) are primitive. - Lekraj Beedassy, Dec 06 2003
Also, a(n) are the record setting values of m, for the number of solutions to "m*k-1 is a square", for some k, 1 <= k < m. There is one solution for m=2, and for a given m = a(n) there are 2^n solutions. For a given m there also 2^(n-1) solutions for primitively representing m as x^2 + y^2. See A008782. Also compare with A102476, which applies to "m*k+1 is a square". - Richard R. Forberg, Mar 18 2016
a(n) is the smallest m such that A000089(m) = 2^n. Also, numbers k for which A000089(k) sets a new record. - Jianing Song, Apr 27 2019

T. D. Noe, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A002110, A002144, A038346, A078586, A185952.

maxN=15; pLst={}; k=0; While[Length[pLst]Harvey P. Dale, Jun 16 2013 *)

tree(v)=my(t=#v); if(t<4, factorback(v), tree(v[1..t\2])*tree(v[t\2+1..t]));
a(n,x=9*n\4+2)=my(P=select(p->p%4==1, primes(x))); if(#PCharles R Greathouse IV, Jan 08 2018

a(n) = Product_{i=1..n} A002144(i). - Alois P. Heinz, Mar 01 2021

A121940 Product of the first n primes of the form 6k+1.

Original entry on oeis.org

7, 91, 1729, 53599, 1983163, 85276009, 5201836549, 348523048783, 25442182561159, 2009932422331561, 194963444966161417, 20081234831514625951, 2188854596635094228659, 277984533772656967039693, 38639850194399318418517327, 5834617379354297081196116377

Offset: 1

Jonathan Vos Post, Sep 03 2006

For n>1, a(n) is the least positive integer that can be primitively represented as m^2+mn+n^2 with 0<=m<=n and gcd(m,n)=1 in exactly 2^(n-1) ways. - Ray Chandler, Oct 01 2007
From Bernard Schott, Mar 30 2021: (Start)
Also, for n >= 1, a(n) is the smallest positive integer m such that m^2 can be primitively represented as k^2-k*q+q^2 with 1 <= k < q and gcd(k,q)= 1 in exactly 2^n ways. For example (a(1))^2 = 7^2 = 3^2 - 3*8 + 8^2 = 5^2 - 5*8 + 8^2.
It follows that a(n) is the smallest middle side b that appears exactly 2^n times consecutively in the data of A335895, for integer-sided triangles whose angles A < B < C are in arithmetic progression. (End)
Also, a(n) is the smallest largest side c that appears exactly 2^(n-1) times consecutively in the data of A357277 for integer-sided triangles with angles A < B < C = 2*Pi/3 = 120 degrees. - Bernard Schott, Oct 21 2022

			a(4) = 53599 = 7 * 13 * 19 * 31.

Ray Chandler, Table of n, a(n) for n=1..100

Cf. A002476, A002110, A006278, A078586, A335895.

Greedy inverse of A005088.

Rest@FoldList[Times, 1, Select[6 Range[100] + 1, PrimeQ]] (* Ray Chandler, Oct 01 2007 *)

lista(nn) = {my(pr=1, list=List()); forprime(p=1, nn, if ((p%3) == 1, listput(list,pr *= p));); Vec(list);} \\ Michel Marcus, Jul 17 2020

a(n) = Product_{i=1..n} A002476(i).

Extended by Ray Chandler, Oct 01 2007

A323739 a(n) is the number of residues modulo (4*primorial(n)) of the squares of primes greater than or equal to prime(n+1).

Original entry on oeis.org

2, 1, 1, 2, 6, 30, 180, 1440, 12960, 142560, 1995840, 29937600, 538876800, 10777536000, 226328256000, 5205549888000, 135344297088000, 3924984615552000, 117749538466560000, 3885734769396480000, 136000716928876800000, 4896025809439564800000, 190945006568143027200000

Offset: 0

Jon E. Schoenfield, Feb 20 2019

Here, "primorial(n)" is A002110(n) = Product_{k=1..n} prime(k).

For n >= 1, a(n) is the number of coprime squares modulo 4*primorial(n). Note that 4*primorial(n) = A102476(n+1) is the smallest k such that rank((Z/kZ)*) = n+1 for n >= 1. (The rank of a finitely generated group rank(G) is defined to be the size of the minimal generating sets of G. In particular, rank((Z/kZ)*) = 0 if k <= 2 and A046072(k) otherwise.) - Jianing Song, Oct 18 2021

			a(3) = 2 because, for every prime p >= prime(3+1) = 7, p^2 mod (4*2*3*5 = 120) is one of the 2 values {1, 49}:
   7^2 mod 120 =  49 mod 120 = 49
  11^2 mod 120 = 121 mod 120 =  1
  13^2 mod 120 = 169 mod 120 = 49
  17^2 mod 120 = 289 mod 120 = 49
  19^2 mod 120 = 361 mod 120 =  1
  23^2 mod 120 = 529 mod 120 = 49
  29^2 mod 120 = 841 mod 120 =  1
  ...
.
   q=(n+1)st        b =          residues p^2 mod b
n    prime    4*primorial(n)         for p >= q         a(n)
=  =========  ===============  =======================  ====
0      2      4         =   4           {0,1}             2
1      3      4*2       =   8            {1}              1
2      5      4*2*3     =  24            {1}              1
3      7      4*2*3*5   = 120           {1,49}            2
4     11      4*2*3*5*7 = 840  {1,121,169,289,361,529}    6

Jianing Song, Table of n, a(n) for n = 0..200
Jianing Song, Further terms: table of n, a(n) for n = 0..1000

Cf. A002110, A005867, A240775, A046072, A102476.

a(n) = if(n==0, 2, my(t=1); forprime(p=3, , t*=(p-1)/2; if(n--<2, return(t)))) \\ Jianing Song, Oct 18 2021, following Charles R Greathouse IV's program for A078586

Conjecture: a(n) = 2^(1-n)*Product_{j=1..n} (prime(j)-1) for n >= 0, so a(n) = a(n-1)*(prime(n)-1)/2 for n >= 1.
From Charlie Neder, Feb 28 2019: (Start)
Conjecture is true. Since there exists a prime congruent to r modulo 4*primorial(n) for any r coprime to primorial(n), this set is precisely the set of coprime quadratic residues of 4*primorial(n). If n >= 1, each residue can be broken down into congruences modulo 8 and the first n-1 odd primes, each odd prime p has (p-1)/2 residue classes, and every combination eventually occurs, giving the formula. (End)

More terms from Jianing Song, Oct 18 2021

A038346 Sum of first n primes of form 4k+1.

Original entry on oeis.org

5, 18, 35, 64, 101, 142, 195, 256, 329, 418, 515, 616, 725, 838, 975, 1124, 1281, 1454, 1635, 1828, 2025, 2254, 2487, 2728, 2985, 3254, 3531, 3812, 4105, 4418, 4735, 5072, 5421, 5774, 6147, 6536, 6933

Offset: 1

Den Roussel (DenRoussel(AT)webtv.net)

Jianing Song, Table of n, a(n) for n = 1..10001

Cf. A002144, A002313, A006278, A038347.

Accumulate[Select[Prime[Range[100]],IntegerQ[(#-1)/4]&]] (* Harvey P. Dale, Jul 25 2011 *)

a(n) = my(s=0); forprime(p=2, , if(p%4==1, s+=p; if(n--<1, return(s)))) \\ following Charles R Greathouse IV's program for A078586

a(n) = Sum_{k=1..n} A002144(k). - Michel Marcus, Oct 18 2021

Offset corrected by Jianing Song, Oct 18 2021

A038347 Sum of first n primes of form 4k-1.

Original entry on oeis.org

3, 10, 21, 40, 63, 94, 137, 184, 243, 310, 381, 460, 543, 646, 753, 880, 1011, 1150, 1301, 1464, 1631, 1810, 2001, 2200, 2411, 2634, 2861, 3100, 3351, 3614, 3885, 4168, 4475, 4786, 5117, 5464, 5823, 6190, 6569, 6952, 7371, 7802, 8241, 8684, 9147, 9614

Offset: 1

Den Roussel (DenRoussel(AT)webtv.net)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002145, A038346, A078586.

ListTools:-PartialSums(select(isprime, [seq(i,i=3..1000,4)])); # Robert Israel, Feb 27 2017

Accumulate[Select[Prime[Range[250]],Mod[#,4]==3&]] (* Harvey P. Dale, Jul 04 2013 *)

A057130 Product of first n primes of form 6k-1.

Original entry on oeis.org

5, 55, 935, 21505, 623645, 25569445, 1201763915, 63693487495, 3757915762205, 266812019116555, 22145397586674065, 1970940385213991785, 199064978906613170285, 21299952743007609220495, 2406894659959859841915935, 315303200454741639290987485

Offset: 1

Henry Bottomley, Aug 11 2000

			a(3) = 5*11*17 = 935.
a(4) = 21505 = 5 * 11 * 17 * 23.

Cf. A000040, A002110, A002476, A006278, A007528, A074040, A078586, A121940.

Cf. A008543, A057131.

lista(nn) = {pp = 1; for (n = 1, nn, p = prime(n); if (Mod(p, 6) == -1, pp *= p; print1(pp, ", ")););} \\ Michel Marcus, Sep 08 2013

a(n) = a(n-1)*A007528(n) = (A057131(n)+1)/6.

More terms from Larry Reeves (larryr(AT)acm.org), Oct 06 2000

More terms from Michel Marcus, Sep 08 2013

A348418 a(n) is the smallest k with rank((Z/kZ)*) = n such that there are an odd number of coprime squares modulo k.

Original entry on oeis.org

1, 3, 8, 24, 168, 1848, 35112, 807576, 25034856, 1076498808, 50595443976, 2985131194584, 200003790037128, 14200269092636088, 1121821258318250952, 93111164440414829016, 9590449937362727388648, 1026178143297811830585336, 130324624198822102484337672

Offset: 0

Jianing Song, Oct 18 2021

The rank of a finitely generated group rank(G) is defined to be the size of the minimal generating sets of G. In particular, rank((Z/kZ)*) = 0 if k <= 2 and A046072(k) otherwise.
The number of coprime squares modulo a(n) is given by A046073(a(n)) = A348420(n-2) for n >= 2.
a(n) is the least k such that the Sylow 2-subgroup of (Z/kZ)* is (C_2)^n. - Jianing Song, Aug 13 2023

			a(2) = 8;
a(3) = 8 * 3 = 24;
a(4) = 8 * 3 * 7 = 168;
a(5) = 8 * 3 * 7 * 11 = 1848;
a(6) = 8 * 3 * 7 * 11 * 19 = 35112.

Jianing Song, Table of n, a(n) for n = 0..200

Cf. A046072, A046073, A348420, A002145, A102476.

a(n) = if(n<=2, [1, 3, 8][n+1], my(t=8); forprime(p=2, , if(p%4==3, t*=p; if(n--<3, return(t))))) \\ following Charles R Greathouse IV's program for A078586

a(n) = 8 * A078586(n-2) = 8 * (Product_{k=1..n-2} A002145(k)) for n > 2.

A348420 a(n) = Product_{k=1..n} (p_k - 1)/2 where p_1, p_2, ..., p_n are the first n primes congruent to 3 modulo 4.

Original entry on oeis.org

1, 1, 3, 15, 135, 1485, 22275, 467775, 10758825, 312005925, 10296195525, 360366843375, 14054306891625, 576226582556625, 29387555710387875, 1557540452650557375, 98125048516985114625, 6378128153604032450625, 440090842598678239093125

Offset: 0

Jianing Song, Oct 18 2021

a(n) is the number of coprime squares modulo A348418(n+2), where A348418(n) is the smallest k with rank((Z/kZ)*) = n such that there are an odd number of coprime squares modulo k. (The rank of a finitely generated group rank(G) is defined to be the size of the minimal generating sets of G. In particular, rank((Z/kZ)*) = 0 if k <= 2 and A046072(k) otherwise.)

			A348418(2) = 8, and the number of coprime squares modulo 8 is a(0) = 1;
A348418(3) = 8 * 3 = 24, and the number of coprime squares modulo 24 is a(1) = (3-1)/2 = 1;
A348418(4) = 8 * 3 * 7 = 168, and the number of coprime squares modulo 168 is a(2) = ((3-1)/2) * ((7-1)/2) = 3;
A348418(5) = 8 * 3 * 7 * 11 = 1848, and the number of coprime squares modulo 1848 is a(3) = ((3-1)/2) * ((7-1)/2) * ((11-1)/2) = 15;
A348418(6) = 8 * 3 * 7 * 11 * 19 = 35112, and the number of coprime squares modulo 35112 is a(4) = ((3-1)/2) * ((7-1)/2) * ((11-1)/2) * ((19-1)/2) = 135.

Jianing Song, Table of n, a(n) for n = 0..200

Cf. A002145, A046073, A348418, A046072, A323739.

a(n) = my(t=1); forprime(p=2, , if(p%4==3, t*=(p-1)/2; if(n--<1, return(t)))) \\ following Charles R Greathouse IV's program for A078586

a(n) = Product_{k=1..n} (A002145(k) - 1)/2.

a(n) = A046073(A348418(n+2)).

cp's OEIS Frontend

A006278 a(n) is the product of the first n primes congruent to 1 (mod 4).

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A121940 Product of the first n primes of the form 6k+1.

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A323739 a(n) is the number of residues modulo (4*primorial(n)) of the squares of primes greater than or equal to prime(n+1).

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A038346 Sum of first n primes of form 4k+1.

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A038347 Sum of first n primes of form 4k-1.

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A057130 Product of first n primes of form 6k-1.

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A348418 a(n) is the smallest k with rank((Z/kZ)*) = n such that there are an odd number of coprime squares modulo k.

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A348420 a(n) = Product_{k=1..n} (p_k - 1)/2 where p_1, p_2, ..., p_n are the first n primes congruent to 3 modulo 4.

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