A102476 -id:A102476 - 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

A112086 a(n) = the period of the first differences of the n-th row of A112060 (or A112070), or 0 if that row does not have a periodic first difference.

Original entry on oeis.org

2, 4, 6, 16, 72, 420, 3240

Offset: 1

Antti Karttunen, Aug 28 2005

These values have been computed empirically. An independent recomputation or a mathematical proof would be welcome. The initial terms factored: 2, 2*2, 2*3, 2*2*2*3*3, 2*2*7*3*5, 2*2*2*3*3*3*3*5, ...

These are the periods of A010684, A112132, A112133, A112134, A112135, A112136, A112137, etc. (Periods of A112138 & A112139 not computed yet.) If we sum the period length prefixes of these sequences, as Sum_{i=1..a(1)} A010684(i), Sum_{i=1..a(2)} A112132(i), Sum_{i=1..a(3)} A112133(i), etc., we get the sequence 4, 12, 60, 420, 4620, 60060, 1021020, ... (cf. A097250) and when doubled, it yields: 8, 24, 120, 840, 9240, 120120, 2042040, ... (cf. A066631 and A102476).

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

A303557 a(0) = 1; a(n) = 2^(n-1)*prime(n)#, where prime(n)# is the product of first n primes.

Original entry on oeis.org

1, 2, 12, 120, 1680, 36960, 960960, 32672640, 1241560320, 57111774720, 3312482933760, 205373941893120, 15197671700090880, 1246209079407452160, 107173980829040885760, 10074354197929843261440, 1067881544980563385712640, 126010022307706479514091520, 15373222721540190500719165440

Offset: 0

Ilya Gutkovskiy, Apr 26 2018

nonn

For n > 0, a(n) is the smallest number m having exactly n distinct prime divisors and exactly 2*n - 1 prime divisors counted with multiplicity.

			a(1) = 2^1;
a(2) = 2^2*3;
a(3) = 2^3*3*5;
a(4) = 2^4*3*5*7;
a(5) = 2^5*3*5*7*11, etc.

Central terms of triangle A303555 (for n > 0).

Cf. A000079, A002110, A003680, A005179, A011782, A038547, A061283, A070175, A088860, A102476.

Join[{1}, Table[2^(n - 1) Product[Prime[j], {j, n}], {n, 18}]]

a(n) = A011782(n)*A002110(n).

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.

A378133 Irregular triangle T(n,k) = P(n)*2^k, n >= 0, k = 0..floor(log_2 prime(k+1)), where P = A002110.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 30, 60, 120, 210, 420, 840, 1680, 2310, 4620, 9240, 18480, 30030, 60060, 120120, 240240, 480480, 510510, 1021020, 2042040, 4084080, 8168160, 9699690, 19399380, 38798760, 77597520, 155195040, 223092870, 446185740, 892371480, 1784742960, 3569485920

Offset: 0

Michael De Vlieger, Nov 17 2024

Subset of A060735.

a(n) = min(2*a(n-1), f(a(n-1))), where f(k) is the smallest primorial (A002110) greater than k, for n > 0. - Hal M. Switkay, Mar 19 2025

			Rows n = 0..9:
n\k |         0          1          2           3           4
-------------------------------------------------------------
  0 |         1          .          .           .           .
  1 |         2          4          .           .           .
  2 |         6         12         24           .           .
  3 |        30         60        120           .           .
  4 |       210        420        840        1680           .
  5 |      2310       4620       9240       18480           .
  6 |     30030      60060     120120      240240      480480
  7 |    510510    1021020    2042040     4084080     8168160
  8 |   9699690   19399380   38798760    77597520   155195040
  9 | 223092870  446185740  892371480  1784742960  3569485920

Michael De Vlieger, Table of n, a(n) for n = 0..3494 (rows n = 0..349, flattened).

Cf. A000079, A002110, A060735, A088860, A098388, A102476, A378144.

nn = 16;
MapIndexed[Set[P[First[#2] - 1], #1] &,
  FoldList[Times, 1, Prime@ Range[nn + 1] ] ];
Union@ Flatten@
  Table[P[i]*2^Range[0, Floor[Log2[Prime[i + 1] ] ] ], {i, 0, nn}]

T(n,k) = A002110(n)*A000079(k), n >= 0, k = 0..A098388(k+1).
T(n,0) = A002110(n).
T(n,1) = A088860(n), n >= 1.
T(n,2) = A102476(n), n >= 2.
T(n,A098388(k+1)) = A378144(n).
Let S(n,j) = A002110(n)*j, n >= 0, j = 0..A006093(n+1) = P(n)*j, n >= 0, j = 0..prime(n+1)-1. Then T(n,k) = S(n, 2^k).

A379423 Least modulus k such that the multiplicative group modulo k is the direct product of n nontrivial cyclic groups.

Original entry on oeis.org

1, 3, 7, 21, 56, 168, 504, 1736, 5208, 15624, 57288, 171864, 671832, 2234232, 7390152, 32023992, 96071976, 450799272, 1559322072, 5860390536, 20271186936, 95118646392, 385152551784, 1236542403096, 6182712015480, 23494305658824, 82848341007432, 409295535424776

Offset: 0

Asher Gray, Dec 22 2024

nonn

Compare with A102476. That sequence also measures the least modulus k with n nontrivial cyclic groups, but only using the rank, the minimal representation for each such k. For example, A102476(3) = 24 as (Z/24Z)* ≅ C2 x C2 x C2. However with this sequence a(3) = 21 as (Z/21Z)* ≅ C2 x C2 x C3.

			a(2) = 7 because (Z/7Z)* ≅ C2 x C3.

Asher Gray, Table of n, a(n) for n = 0..1000
Asher Gray, Least modulus with n cycles, Github repository.
Asher Gray, Sequences from Group Theory, YouTube Video.

Cf. A102476.

A272590 a(n) is the smallest number m such that the multiplicative group modulo m is the direct product of n cyclic groups.

Original entry on oeis.org

2, 8, 24, 120, 840, 9240, 120120, 2042040, 38798760, 892371480, 25878772920, 802241960520, 29682952539240, 1217001054108840, 52331045326680120, 2459559130353965640, 130356633908760178920, 7691041400616850556280, 469153525437627883933080, 31433286204321068223516360

Offset: 1

Joerg Arndt, May 05 2016

nonn

Arguably a(1)=3, as the multiplicative group mod 2 has only one element, hence its factorization is the empty product. - Joerg Arndt, May 18 2018

For n >= 2, positions of records of A046072. - Joerg Arndt, May 18 2018

Cf. A002322, A102476, A058254.

Numbers n such that the multiplicative group modulo n is the direct product of k cyclic groups: A033948 (k=1), A272593 (k=2), A272593 (k=3), A272594 (k=4), A272595 (k=5), A272596 (k=6), A272597 (k=7), A272598 (k=8), A272599 (k=9).

a(n)=if(n==1,2,4*prod(k=1,n-1,prime(k)));

a(1) = 2, a(n) = 4 * prod(k=1..n-1, prime(k) ) for n >= 2.
a(n) = A102476(n) for n >= 2.
A002322(a(n)) = A058254(n).

A332802 a(n) is the smallest q such that the number of nonnegative k <= q, possessing the property that k + k*q - q is a square, is equal to 2^n.

Original entry on oeis.org

0, 2, 7, 23, 119, 839, 9239, 120119, 2042039, 38798759, 892371479, 25878772919, 802241960519, 29682952539239, 1217001054108839, 52331045326680119, 2459559130353965639, 130356633908760178919, 7691041400616850556279, 469153525437627883933079, 31433286204321068223516379

Offset: 0

Juri-Stepan Gerasimov, Feb 24 2020

nonn

			a(0) = 0 because 2^0 = 1 solution is 0 (where k=0).
a(1) = 2 because 2^1 = 2 solutions are 1 (1) and 4 (2).
a(2) = 7 because 2^2 = 4 solutions are 1 (1), 9 (2), 25 (4), 49 (7).
a(3) = 23 because 2^3 = 8 solutions are 1 (1), 25 (2), 49 (3), 121 (6), 169 (8), 289 (13), 361 (16), 529 (23).
a(4) = 119 because 2^4 = 16 solutions are 1 (1), 121 (2), 361 (4), 841 (8), 961 (9), 1681 (15), 2401 (21), 3481 (30), 3721 (32), 5041 (43), 6241 (53), 7921 (67), 8281 (70), 10201 (86), 11881 (100), 14161 (119).

Cf. A000290, A060594, A102476, A332761.

a(n) = {my(q=0); while (sum(k=0, q, issquare(k + k*q - q)) != 2^n, q++); q;} \\ Michel Marcus, Feb 25 2020

a(n) = A102476(n) - 1. - Jinyuan Wang, Feb 25 2020

a(7) from Michel Marcus, Feb 25 2020

More terms from Jinyuan Wang, Feb 25 2020

A379424 Least modulus k such that the multiplicative group modulo k has a difference of n nontrivial cycles between its minimal and maximal representation.

Original entry on oeis.org

1, 7, 31, 211, 1333, 6541, 45787, 281263, 1968841, 13781887, 93098053, 649998793, 4549991551, 31849940857, 215149600483, 1506047203381, 10542330423667, 86982188480467, 587573558919073, 4113014912433511, 28791104387034577, 247368468304929733

Offset: 0

Asher Gray, Dec 22 2024

nonn

This is equal to the least modulus k such that (Z/kZ)* has a representation as a direct product of cyclic groups, of which n are odd cycles. The number of even cycles in the maximal representation is equal to the total cycles in the minimal representation.

			a(4) = 1333 because (Z/1333Z) ≅ C210 x C6 ≅ C2 x C3 x C5 x C2 x C3 x C7. The first representation has 2 cycles and the second has 6, a difference of 4.

Asher Gray, Table of n, a(n) for n = 0..500
Asher Gray, Least modulus with n cycles, Github repository.
Asher Gray, Sequences from Group Theory, YouTube Video.

Cf. A102476, A379423.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A112086 a(n) = the period of the first differences of the n-th row of A112060 (or A112070), or 0 if that row does not have a periodic first difference.

Views

Author

Keywords

Comments

Crossrefs

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A303557 a(0) = 1; a(n) = 2^(n-1)*prime(n)#, where prime(n)# is the product of first n primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A378133 Irregular triangle T(n,k) = P(n)*2^k, n >= 0, k = 0..floor(log_2 prime(k+1)), where P = A002110.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A379423 Least modulus k such that the multiplicative group modulo k is the direct product of n nontrivial cyclic groups.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A272590 a(n) is the smallest number m such that the multiplicative group modulo m is the direct product of n cyclic groups.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula