A121940 -id:A121940 - cp's OEIS frontend

A329920 Smallest k such that 6kA121940(n)-1 and 6kA121940(n)+1 are twin primes.

1, 2, 2, 15, 36, 10, 13, 26, 30, 228, 24, 138, 520, 59, 110, 456, 700, 670, 146, 300, 390, 53, 2335, 340, 159, 340, 65, 475, 785, 1145, 759, 3557, 490, 169, 990, 1527, 704, 3379, 1426, 1927, 2397, 600, 1603, 4809, 9815, 58, 35, 364, 361, 123, 2197, 4054, 1867, 1827, 5048

Offset: 1

Pierre CAMI, Nov 24 2019

nonn

			A121940(1)=7, 6*1*7-1=41, 41 and 43 are twin primes so a(1)=1.
A121940(2)=91, 6*2*91-1=1091, 1091 and 1093 are twin primes so a(2)=2.

Cf. A001359, A002476, A006512, A121940, A173937, A329336, A329916.

lista(nn) = {my(pp = 1); forprime (p = 1, nn, if (Mod(p, 6) == +1, pp *= p; my(k=1); while (!isprime(6*k*pp-1) || !isprime(6*k*pp+1), k++); print1(k, ", ");););} \\ Michel Marcus, Nov 25 2019

A160498 Number of cubic primitive Dirichlet characters modulo n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0

Offset: 1

Steven Finch, May 15 2009

Also called primitive Dirichlet characters of order 3.
Mobius transform of A060839.
C. David, J. Fearnley & H. Kisilevsky prove that Sum_{k=1..n} a(k) ~ C*n, with C = (11*sqrt(3)/(18*Pi)) * Product_{primes p == 1 (mod 3)} (1 - 2/(p*(p+1))) = 0.3170565167922841205670156...; they credit Cohen, F. Diaz y Diaz, & M. Olivier 2002 (see Proposition 5.2. and Corollary 5.3.). - Charles R Greathouse IV, Aug 26 2009 [corrected by Vaclav Kotesovec, Sep 16 2020]
a(n) is the number of primitive Dirichlet characters modulo n such that all entries are 0 or a cubic root of unity: 1, w = (-1 + sqrt(3)*i)/2 or w^2 = (-1 - sqrt(3)*i)/2. - Jianing Song, Feb 27 2019
Every term is 0 or a power of 2. - Jianing Song, Mar 02 2019
From Jianing Song, Apr 03 2021: (Start)
For n >= 2, a(n) is the number of cyclic cubic fields with discriminant n^2. See A343023 for detailed information.
The first occurrence of 2^t is 9*A121940(t-1) for t >= 2. (End)

			From _Jianing Song_, Mar 02 2019: (Start)
Let w = (-1 + sqrt(3)*i)/2 be one of the primitive 3rd root of unity.
For n = 7, the 2 cubic primitive Dirichlet characters modulo n are [0, 1, w, w^2, w^2, w, 1] and [0, 1, w^2, w, w, w^2, 1], so a(7) = 2.
For n = 9, the 2 cubic primitive Dirichlet characters modulo n are [0, 1, w, 0, w^2, w^2, 0, w, 1] and [0, 1, w^2, 0, w, w, 0, w^2, 1], so a(9) = 2. (End)

Jianing Song, Table of n, a(n) for n = 1..10000
C. David, J. Fearnley and H. Kisilevsky,On the vanishing of twisted L-functions of elliptic curves, Experim. Math. 13 (2004) 185-198.
Steven R. Finch, Cubic and quartic characters.
Steven R. Finch, Cubic and quartic characters.
Steven R. Finch, Quartic and Octic Characters Modulo n, arXiv:0907.4894 [math.NT], 2016.
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n for n = 1..10000000

Cf. A114643 (number of quadratic primitive Dirichlet characters modulo n), A160499 (number of quartic primitive Dirichlet characters modulo n).
Cf. A060839 (number of solutions to x^3 == 1 (mod n)).
Cf. A121940, A343023.

A060839[n_] := Sum[If[Mod[k^3 - 1, n] == 0, 1, 0], {k, 1, n}]; a[n_] := Sum[ MoebiusMu[n/d]*A060839[d], {d, Divisors[n]}]; Table[a[n], {n, 2, 81}] (* Jean-François Alcover, Jun 19 2013 *)
f[3, 2] = 2; f[p_, e_] := If[Mod[p, 3] == 1 && e == 1, 2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

a(n)=sum(d=1, n, if(n%d==0, moebius(n/d)*sum(i=1, d, if((i^3-1)%d, 0, 1)), 0)) \\ Steven Finch, Jun 09 2009

A005088(n)=my(f=factor(n)[,1]); sum(i=1,#f,f[i]%3==1)
A060839(n)=3^((n%9==0)+A005088(n))
a(n)=sumdiv(n,d,moebius(n/d)*A060839(d)) \\ Charles R Greathouse IV, Aug 26 2009

a(n) = my(L=factor(n), w=omega(n)); for(i=1, w, if(!((L[i, 1]%3==1 && L[i, 2]==1) || L[i, 1]^L[i, 2] == 9), return(0))); 2^w \\ Jianing Song, Apr 03 2021

Multiplicative with a(p^e) = 2 if p^e = 9 or p == 1 (mod 3) and e = 1, otherwise 0. - Jianing Song, Mar 02 2019

a(n) = 2*A343023(n) for n >= 2. - Jianing Song, Apr 03 2021

a(1) = 1 prepended by Jianing Song, Feb 27 2019

A005088 Number of primes = 1 mod 3 dividing n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2

Offset: 1

N. J. A. Sloane

nonn

The first instance of a(n)=2 is for n=91; the first instance of a(n)=3 is for n=1729. 1729 is famously Ramanujan's taxi cab number -- see A001235. - Harvey P. Dale, Jun 25 2013

Antti Karttunen, Table of n, a(n) for n = 1..10000
S. R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.

Cf. A001221, A005070, A005090, A020639, A028234, A079978.

Cf. A121940 (first number having n such factors).

A005088 := proc(n)
    local a,pe;
    a := 0 ;
    for pe in ifactors(n)[2] do
        if modp(op(1,pe),3)= 1 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, May 19 2020

Join[{0},Table[Count[Transpose[FactorInteger[n]][[1]],?(Mod[#-1,3] == 0&)],{n,2,100}]] (* _Harvey P. Dale, Sep 22 2021 *)
Array[DivisorSum[#, 1 &, And[PrimeQ@ #, Mod[#, 3] == 1] &] &, 91] (* Michael De Vlieger, Jul 11 2017 *)

a(n)=my(f=factor(n)[,1]); sum(i=1,#f,f[i]%3==1) \\ Charles R Greathouse IV, Jan 16 2017

(define (A005088 n) (if (= 1 n) 0 (+ (modulo (modulo (A020639 n) 3) 2) (A005088 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = 1 if p = 1 (mod 3), 0 otherwise.
From Antti Karttunen, Jul 10 2017: (Start)
a(1) = 0; for n > 1, ((A020639(n) mod 3) mod 2) + a(A028234(n)).
a(n) = A001221(n) - A005090(n) - A079978(n).
(End)

A335895 Middle side of primitive triples, in nondecreasing order, for integer-sided triangles whose angles A < B < C are in arithmetic progression.

Original entry on oeis.org

7, 7, 13, 13, 19, 19, 31, 31, 37, 37, 43, 43, 49, 49, 61, 61, 67, 67, 73, 73, 79, 79, 91, 91, 91, 91, 97, 97, 103, 103, 109, 109, 127, 127, 133, 133, 133, 133, 139, 139, 151, 151, 157, 157, 163, 163, 169, 169, 181, 181, 193, 193, 199, 199, 211, 211, 217, 217, 217, 217

Offset: 1

Bernard Schott, Jul 04 2020

nonn

Equivalently, lengths of the middle side b of primitive non-equilateral triangles that have an angle of Pi/3; indeed, this side is opposite to angle B = Pi/3.
Also solutions b of the Diophantine equation b^2 = a^2 - a*c + c^2 with a < b and gcd(a,b) = 1.
For the corresponding primitive triples and miscellaneous properties and references, see A335893.
When (a, b, c) is a triple or a solution, then (c-a, b, c) is another solution, so every b in the data is present an even number of times (see examples).
From Bernard Schott, Apr 02 2021: (Start)
Terms are primes of the form 6k+1, or products of primes of the form 6k+1. Three observations:
-> The lengths b are in A004611 \ {1} without repetition, 1 corresponds to the equilateral triangle (1, 1, 1).
-> Every term appears 2^k (k>0) times consecutively and the smallest term that appears 2^k times is precisely A121940(k); see examples.
-> The terms that appear precisely twice consecutively are in A133290. (End)

			b = 7 appears twice because A121940(1) = 7 and:
  7^2 = 3^2 - 3*8 + 8^2, with triple (3, 7, 8),
  7^2 = 5^2 - 5*8 + 8^2, with triple (5, 7, 8).
b = 91 appears four times because A121940(2) = 91 and:
  91^2 = 11^2 - 11*96 + 96^2, with triple (11, 91, 96),
  91^2 = 85^2 - 85*96 + 96^2, with triple (85, 91, 96),
  91^2 = 19^2 - 19*99 + 99^2, with triple (19, 91, 99),
  91^2 = 80^2 - 80*99 + 99^2, with triple (80, 91, 99).
b = 1729 appears eight times because A121940(3) = 1729 and the triples of these 2^3 = 8 triangles are (96, 1729, 1775), (1679, 1729, 1775), (249, 1729, 1840), (1591, 1729, 1840), (656, 1729, 1961), (1305, 1729, 1961), (799, 1729, 1984), (1185, 1729, 1984).

Cf. A004611, A121940, A133290.

Cf. A335893 (triples), A335894 (smallest side), this sequence (middle side), A335896 (largest side), A335897 (perimeter).

for b from 3 to 100 by 2 do
for a from 1 to b-1 do
c := (a+ sqrt(4*b^2-3*a^2))/2;
if igcd(a, b) = 1 and issqr(4*b^2-3*a^2) then print(b); end if;
end do;
end do;

lista(nn) = {forstep(b=1, nn, 2, for(a=1, b-1, if (gcd(a, b) == 1, my(d = 4*b^2 - 3*a^2); if (issquare(d), my(c = (a + sqrtint(d))/2); if (denominator(c)==1, print1(b, ", "));););););} \\ Michel Marcus, Jul 05 2020

a(n) = A335893(n, 2).

b is such that b^2 = a^2 - a*c + c^2 with gcd(a,b) = 1 and a < b.

A343023 Number of cyclic cubic fields with discriminant n^2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Jianing Song, Apr 02 2021

Equivalently, number of cubic fields with discriminant n^2. That is to say, it makes no difference if the word "cyclic" is omitted from the title.
Let D be a discriminant of a cubic field F, then F is a cyclic cubic field if and only if D is a square. For D = k^2, k must be of the form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3), in which case there are exactly 2^(t-1) = 2^(omega(k)-1) (cyclic) cubic fields with discriminant D. See Page 17, Theorem 2.7 of the Ka Lun Wong link.
Each term is 0 or a power of 2.
The first occurrence of 2^t is 9*A121940(t) for t >= 1.

			a(7) = 1 since there is only 1 (cyclic) cubic field with discriminant 7^2 = 49 is Q[x]/(x^3 - x^2 + x + 1).
a(63) = 2 since there are 2 (cyclic) cubic fields with discriminant 63^2 = 3969: Q[x]/(x^3 - 21x - 28) and Q[x]/(x^3 - 21x - 35).
a(819) = 4 since there are 4 (cyclic) cubic fields with discriminant 819^2 = 670761: Q[x]/(x^3 - 273x - 91), Q[x]/(x^3 - 273x - 728), Q[x]/(x^3 - 273x - 1547) and Q[x]/(x^3 - 273x - 1729).
a(35) = 0 since it is not of form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3). Indeed, there are no (cyclic) cubic fields with discriminant 35^2 = 1225.

Jianing Song, Table of n, a(n) for n = 1..16000
LMFDB, Cubic fields
Wikipedia, Cubic field
Ka Lun Wong, Maximal Unramified Extensions of Cyclic Cubic Fields, (2011), Theses and Dissertations, 2781.

Cf. A160498, A121940, A343000 (discriminants of cyclic cubic fields), A343001 (indices of positive terms).

a(n) = if(n<=1, 0, my(L=factor(n), w=omega(n)); for(i=1, w, if(!((L[i, 1]%3==1 && L[i, 2]==1) || L[i, 1]^L[i, 2] == 9), return(0))); 2^(w-1))

a(n) = A160498(n)/2 for n > 1.

A319443 Number of distinct Eisenstein primes in the factorization of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 3, 2, 1, 1, 2, 2, 2, 3, 2, 1, 2, 1, 3, 1, 3, 1, 3, 2, 1, 2, 2, 3, 2, 2, 3, 3, 2, 1, 4, 2, 2, 2, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 3, 2, 1, 3, 2, 3, 3, 1, 3, 3, 2, 2, 2, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 2, 1, 2, 1, 4, 2, 3, 2

Offset: 1

Jianing Song, Sep 19 2018

nonn

Equivalent of omega (A001221) in the ring of Eisenstein integers.
z is an Eisenstein prime iff z has prime norm or z is the product of a rational prime congruent to 2 modulo 3 and an Eisenstein unit (one of +-1 or (+-1 +- sqrt(3)*i)/2).
Associated Eisenstein prime divisors are counted only once.
Let s(n) be the smallest k with a(k) = n, then we have: s(0) = 1, s(1) = 2, s(2) = 6, s(2n-1) = 2*A121940(n-1), s(2n) = 6*A121940(n-1).

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
Over the Gaussian integers, 5187 = 3*7*13*19 is factored as w'*(1 + w)^2*(2 + w)*(2 + w')*(3 + w)*(3 + w')*(3 + 2w)*(3 + 2w'), the distinct Eisenstein prime factors are 1 + w, 2 + w, 2 + w', 3 + w, 3 + w', 3 + 2w and 3 + 2w', so a(5187) = 7.
Over the Gaussian integers, 1006655265000 = 2^3*3^2*5^4*7^5*11^3 is factored as w'^2*(1 + w)^4*2^3*(2 + w)*(2 + w')*5^4*11^3, the distinct Eisenstein prime factors are 1 + w, 2, 2 + w, 2 + w', 5 and 11, so a(1006655265000) = 6.

Jianing Song, Table of n, a(n) for n = 1..10000
Wikipedia, Eisenstein integer

Cf. A121940.
Equivalent of arithmetic functions in the ring of Eisenstein integers (the corresponding functions in the ring of integers are in the parentheses): A319442 ("d", A000005), A319449 ("sigma", A000203), A319445 ("phi", A000010), A319446 ("psi", A002322), this sequence ("omega", A001221), A319444 ("Omega", A001222), A319448 ("mu", A008683).
Equivalent in the ring of Gaussian integers: A086275.

f[p_, e_] := If[Mod[p, 3] == 1, 2, 1]; eisOmega[1] = 0; eisOmega[n_] := Plus @@ f @@@ FactorInteger[n]; Array[eisOmega, 100] (* Amiram Eldar, Feb 10 2020 *)

a(n)=my(f=factor(n)[, 1]); sum(i=1, #f, if(f[i]%3==1, 2, 1))

Additive with a(p^e) = 2 if p == 1 (mod 3), 1 otherwise.

A193869 Smallest product of n distinct primes of the form n*k + 1.

Original entry on oeis.org

2, 15, 1729, 32045, 60551711, 85276009, 52814801041129, 1312422595226609, 1130308388231798179, 4182230628909121261, 100166053986652515419641469, 1898732717895963155960377, 1011844196551535741726366525322443

Offset: 1

Omar E. Pol, Sep 01 2011

nonn

Also the row products of triangle A077316.

Note that a(3) = 1729 is known as the Hardy-Ramanujan number.

			a(1) = 2
a(2) = 3*5 = 15
a(3) = 7*13*19 = 1729
a(4) = 5*13*17*29 = 32045
a(5) = 11*31*41*61*71 = 60551711
a(6) = 7*13*19*31*37*43 = 85276009

Nathaniel Johnston, Table of n, a(n) for n = 1..100

Cf. A001235, A077316, A092609, A121940, A154729.

Tj := proc(n,k) option remember: local j,p: if(k=0)then return 0:fi: for j from procname(n,k-1)+1 do if(isprime(n*j+1))then return j: fi: od: end: A193869 := proc(n) return mul(n*Tj(n,k)+1,k=1..n): end: seq(A193869(n),n=1..15); # Nathaniel Johnston, Sep 02 2011

a(7)-a(14) from Nathaniel Johnston, Sep 02 2011

A357277 Largest side c of primitive triples, in nondecreasing order, for integer-sided triangles with angles A < B < C = 2*Pi/3 = 120 degrees.

Original entry on oeis.org

7, 13, 19, 31, 37, 43, 49, 61, 67, 73, 79, 91, 91, 97, 103, 109, 127, 133, 133, 139, 151, 157, 163, 169, 181, 193, 199, 211, 217, 217, 223, 229, 241, 247, 247, 259, 259, 271, 277, 283, 301, 301, 307, 313, 331, 337, 343, 349, 361, 367, 373, 379, 397, 403, 403, 409, 421, 427, 427, 433, 439, 457

Offset: 1

Bernard Schott, Oct 01 2022

nonn

For the corresponding primitive triples and miscellaneous properties and references, see A357274.
Solutions c of the Diophantine equation c^2 = a^2 + a*b + b^2 with gcd(a,b) = 1 and a < b.
Also, side c can be generated with integers u, v such that gcd(u,v) = 1 and 0 < v < u, c = u^2 + u*v + v^2.
Some properties:
-> Terms are primes of the form 6k+1, or products of primes of the form 6k+1.
-> The lengths c are in A004611 \ {1} without repetition, in increasing order.
-> Every term appears 2^(k-1) (k>=1) times consecutively.
-> The smallest term that appears 2^(k-1) times is precisely A121940(k): see examples.
-> The terms that appear only once in this sequence are in A133290.
-> The terms are the same as in A335895 but frequency is not the same: when a term appears m times consecutively here, it appears 2m times consecutively in A335895. This is because if (a,b,c) is a primitive 120-triple, then both (a,a+b,c) and (a+b,b,c) are 60-triples in A335893 (see Emrys Read link, lemma 2 p. 302).
Differs from A088513, the first 20 terms are the same then a(21) = 151 while A088513(21) = 157.
A050931 gives all the possible values of the largest side c, in increasing order without repetition, for all triangles with an angle of 120 degrees, but not necessarily primitive.

			c = 7 appears once because A121940(1) = 7 with triple (3,5,7) and 7^2 = 3^2 + 3*5 + 5^2.
c = 91 is the smallest term to appear twice because A121940(2) = 91 with primitive 120-triples (11, 85, 91) and (19, 80, 91).
c = 1729 is the smallest term to appear four times because A121940(3) = 1729 with triples (96, 1679, 1729), (249, 1591, 1729), (656, 1305, 1729), (799, 1185, 1729).

Emrys Read, On integer-sided triangles containing angles of 120° or 60°, Mathematical Gazette 90, July 2006, 299-305.

Cf. A357274 (triples), A357275(smallest side), A357276 (middle side), A357278 (perimeter).

Cf. Also A004611, A050931, A088513, A121940, A133290, A335895.

for c from 5 to 500 by 2 do
for a from 3 to c-2 do
b := (-a + sqrt(4*c^2-3*a^2))/2;
if b=floor(b) and gcd(a, b)=1 and a

Formula

a(n) = A357274(n, 3).

cp's OEIS Frontend

A329920 Smallest k such that 6*k*A121940(n)-1 and 6*k*A121940(n)+1 are twin primes.

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A160498 Number of cubic primitive Dirichlet characters modulo n.

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A005088 Number of primes = 1 mod 3 dividing n.

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A335895 Middle side of primitive triples, in nondecreasing order, for integer-sided triangles whose angles A < B < C are in arithmetic progression.

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A343023 Number of cyclic cubic fields with discriminant n^2.

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A319443 Number of distinct Eisenstein primes in the factorization of n.

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A193869 Smallest product of n distinct primes of the form n*k + 1.

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A329920 Smallest k such that 6kA121940(n)-1 and 6kA121940(n)+1 are twin primes.