A121940 -id:A121940 - cp's OEIS frontend

A307162 a(n) is the smallest k such that A319100(k) = A025610(n).

1, 3, 8, 7, 24, 21, 120, 56, 1320, 63, 168, 22440, 252, 840, 516120, 504, 9240, 819, 14967480, 2184, 157080, 3276, 613666680, 10920, 3612840, 6552, 28842333960, 120120, 15561, 104772360, 32760, 1528643699880, 2042040, 62244, 4295666760, 207480, 90189978292920, 46966920, 124488

Offset: 1

Jianing Song, Mar 27 2019

nonn

A025610 is the range of A319100.
Let b = A319100. Note that:
- if k is an odd number, then b(2*k) = b(k), b(4*k) = 2*b(k), b(2^e*k) = 4*b(k) for e >= 3;
- if k is not divisible by 3, then b(3*k) = 2*b(k), b(3^e*k) = 6*b(k) for e >= 2;
- for all primes p > 3, if k is not divisible by p, then b(p^e*k) = b(p*k).
As a result, it is easy to see that for every n, a(n) is not congruent to 2 modulo 4 and is not divisible by 16 or 27 or p^2 for any prime p > 3.

Jianing Song, Table of n, a(n) for n = 1..1000

Cf. A025610, A319100, A002476, A007528, A121940, A057130.

isA025610(n) = omega(6*n)==2&&valuation(n,2)>=valuation(n,3)
b(n) = if(isA025610(n), i=1; while(A319100(i)!=n, i++); i)
for(n=1, 216, if(isA025610(n), print1(b(n), ", "))) \\ See A319100 for its program

p(j) = my(t=0,v=vector(j)); for(k=1, oo, if(prime(k)%6==1, t++; v[t]=prime(k)); if(t==j, return(v)))
q(i) = my(t=0,v=vector(i)); for(k=1, oo, if(prime(k)%6==5, t++; v[t]=prime(k)); if(t==i, return(v)))
b(i,j) = {
if(j<=1 && i<=2, my(M=[1,3,8;7,21,56]); return(M[j+1,i+1]));
if(j==0 && i>=3, my(Q=q(i-3)); return(24*prod(k=1, i-3, Q[k])));
if(j>=2 && i<=2, my(P=p(j-1), w=[9,36,72]); return(w[i+1]*prod(k=1, j-1, P[k])));
if(j>=1 && i>=3, my(P=p(j), Q=q(i-2)); return(prod(k=1, j-1, P[k])*8*prod(k=1, i-3, Q[k])*min(9*Q[i-2], 3*P[j])));
}
list(lim) = my(v=A025610(lim), u=vector(#v)); for(k=1, #v, my(i=valuation(v[k],2)-valuation(v[k],3), j=valuation(v[k],3)); u[k]=b(i,j)); u \\ Jianing Song, Jun 04 2019, See A025610 for its program

Let p(j) = A002476(j), q(i) = A007528(i), P(j) = Product_{k=1..j} p(k) = A121940(j) if j > 0, Q(i) = Product_{k=1..i} q(k) = A057130(i) if i > 0. If A025610(n) = 2^i*6^j, then:
(a) if i = 0, then a(n) = 1 if j = 0, 7 if j = 1 and 9*P(j-1) if j >= 2;
(b) if i = 1, then a(n) = 3 if j = 0, 21 if j = 1 and 36*P(j-1) if j >= 2;
(c) if i = 2, then a(n) = 8 if j = 0, 56 if j = 1 and 72*P(j-1) if j >= 2;
(d) if i >= 3, then a(n) = 24*Q(i-3) if j = 0 and P(j-1)*8*Q(i-3)*min{9*q(i-2), 3*p(j)} if j >= 1. [Rewritten by Jianing Song, Jun 04 2019]

A220171 An ordered subset of primitive values of x^2 + x*y + y^2 where at least two ordered pairs (x1,y1) and (x2,y2) with x1 != x2, y1 != y2 and gcd(x1,y1) = gcd(x2,y2) = 1 yield identical primitive values.

Original entry on oeis.org

91, 133, 217, 247, 259, 273, 301, 399, 403, 427, 469, 481, 511, 553, 559, 589, 637, 651, 679, 703, 721, 741, 763, 777, 793, 817, 871, 889, 903, 931, 949, 973, 1027, 1057, 1099, 1141, 1147, 1159, 1183, 1209, 1261, 1267, 1273, 1281, 1333, 1339, 1351

Offset: 1

Frank M Jackson, Dec 06 2012

nonn

The primitive values of x^2 + x*y + y^2 where x >= y >= 0 and gcd(x, y) = 1 are given by A034017. However there are incidents in the sequence A034017 where different values of (x, y) yield the same primitive value. Furthermore, the number of solutions for a given primitive value equates to a power of 2. See A121940.

			a(3) = 217 because it is the 3rd incident in ascending order of the primitive x^2 + x*y + y^2 that yields multiple solutions. This happens when (x, y) = (9, 8) and (13, 3).

Cf. A034017, A121940.

maxLen = 100; sol[k_] := Solve[m^2 + m*n + n^2 == k && m > n > 0 && GCD[m, n] == 1, Integers]; getlist[l_] := Which[Length[sol[l]] == 0, {}, True, {m, n} /. sol[l]]; list = {}; p = 1; While[Length[list] < maxLen, (While[Length[getlist[p]] < 2, p++]; list = Append[list, p]; p++)]; list

n such that n = x1^2 + x1*y1 + y1^2 = x2^2 + x2*y2 + y2^2 with x1 != x2, y1 != y2 and gcd(x1,y1) = gcd(x2,y2) = 1.

cp's OEIS Frontend

A307162 a(n) is the smallest k such that A319100(k) = A025610(n).

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