Alexander Riasanovsky - cp's OEIS frontend

A308612 Moments of the ternary Cantor measure (numerators).

1, 1, 3, 5, 87, 31, 10215, 2675, 2030721, 3791353, 21043039755, 3617048975, 456510966890031, 66989072882759, 4380916942893971361, 8410761713598485675, 4355410489470724905492213, 2471070675390828342358441, 8410576873515817323688553597445

Offset: 0

Alexander Riasanovsky, Jun 10 2019

The ternary Cantor measure, defined many ways, is the unique Borel measure mu on the unit interval [0,1] satisfying the following recurrence relation for any measurable set E: mu(E) = mu(phi_0(E))/2 + mu(phi_2(E))/2.  Here, for j in {0,1,2}, phi_j: [0,1] to [0,1] is the linear function which sends x in [0,1] to (x+j)/3.  For any nonnegative integer k, we define the k-th moment to be I_k to be the integral of x^k with respect to mu.  The described sequence I(0), I(1), I(2), ... is rational and this sequence a(0), a(1), a(2), ... is the sequence of numerators of I(0), I(1), I(2), ....
For the purpose of computing I_k, we note the following recurrence relation: I(0) = 1 and for all positive k, I(k) = (1/(3^k-1))*((1/2) * Sum_{j=0..k-1} binomial(k, j) + (1/2) * Sum_{j=0..k-1} binomial(k, j) * 2^(k-j) * I(j)).
More generally, for any N-dimensional nonnegative vector alpha = (alpha_0, ..., alpha_{N-1}) whose entries sum to 1, there exists a unique Borel measure mu = mu^{alpha} on [0,1] so that for any measurable set E, the following identity holds: mu(E) = Sum_{k=0..N-1} alpha_k * mu(phi_k(E)).  Here, for j in {0, 1, ..., N-1}, phi_j: [0,1] to [0,1] is the linear function which sends x in [0,1] to (x+j)/N.  Defining I(k) to be the integral of x^k with respect to mu, the following recurrence relation holds: I(0) = 1 and for all positive k, I(k) = (1/(N^k-1)) * Sum_{n=0..N-1} alpha_n * Sum_{j=0..k-1} binomial(k, j) * n^{k-j}*I(j).

Michael De Vlieger, Table of n, a(n) for n = 0..97
David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, and Michael G. Rose, Expectations on fractal sets, Applied Mathematics and Computation, Vol. 220 (2013), pp. 695-721, alternative link.
Steven N. Harding, Alexander W. N. Riasanovsky, Moments of the weighted Cantor measures, arXiv:1908.05358 [math.FA], 2019.
Math Stack Exchange, Integrating f(x) = x for x in C, the Cantor set, with respect to a certain measure

Cf. A308613 (denominators).

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, j]*2^(n - j - 1)*a[j], {j, 0, n - 1}]/(3^n - 1); Table[Numerator[a[i]], {i, 0, 19}] (* Amiram Eldar, Aug 03 2019 *)

def am(m, alpha):
    N = len(alpha)
    am = [1]
    for a in [1..m]:
        mm = 0
        for k in [0..N-1]:
            for r in [0..a-1]:
                mm += alpha[k]*binomial(a, r)*k^(a-r)*am[r]
        mm /= (N^a-1)
        am.append(mm)
    return am
[p.numerator() for p in am(15, (1/2, 0, 1/2))]

More terms from Amiram Eldar, Aug 03 2019

A308613 Moments of the ternary Cantor measure (denominators).

Original entry on oeis.org

1, 2, 8, 16, 320, 128, 46592, 13312, 10915840, 21831680, 128911704064, 23438491648, 3114038000353280, 479082769285120, 32734822212030169088, 65469644424060338176, 35228168150276083007094784, 20722451853103578239467520, 72984567358962659964369885986816

Offset: 0

Alexander Riasanovsky, Jun 10 2019

The ternary Cantor measure, defined many ways, is the unique Borel measure mu on the unit interval [0,1] satisfying the following recurrence relation for any measurable set E: mu(E) = mu(phi_0(E))/2 + mu(phi_2(E))/2.  Here, for j in {0,1,2}, phi_j: [0,1] to [0,1] is the linear function which sends x in [0,1] to (x+j)/3.  For any nonnegative integer k, we define the k-th moment to be I(k) to be the integral of x^k with respect to mu.  The described sequence I(0), I(1), I(2), ... is rational and this sequence a(0), a(1), a(2), ... is the sequence of denominators of I(0), I(1), I(2), ....
For the purpose of computing I(k), we note the following recurrence relation: I(0) = 1 and for all positive k, I(k) = (1/(3^k-1))*((1/2) * Sum_{j=0..k-1} binomial(k, j) + (1/2) * Sum_{j=0..k-1} binomial(k, j) * 2^(k-j) * I(j)).
More generally, for any N-dimensional nonnegative vector alpha = (alpha_0, ..., alpha_{N-1}) whose entries sum to 1, there exists a unique Borel measure mu = mu^{alpha} on [0,1] so that for any measurable set E, the following identity holds: mu(E) = Sum_{k=0..N-1} alpha_k * mu(phi_k(E)).  Here, for j in {0, 1, ..., N-1}, phi_j: [0,1] to [0,1] is the linear function which sends x in [0,1] to (x+j)/N.  Defining I(k) to be the integral of x^k with respect to mu, the following recurrence relation holds: I(0) = 1 and for all positive k, I(k) = (1/(N^k-1)) * Sum_{n=0..N-1} alpha_n * Sum_{j=0..k-1} binomial(k, j) * n^(k-j)*I(j).

Michael De Vlieger, Table of n, a(n) for n = 0..97
David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, and Michael G. Rose, Expectations on fractal sets, Applied Mathematics and Computation, Vol. 220 (2013), pp. 695-721, alternative link.
Steven N. Harding, Alexander W. N. Riasanovsky, Moments of the weighted Cantor measures, arXiv:1908.05358 [math.FA], 2019.
Math Stack Exchange, Integrating f(x) = x for x in C, the Cantor set, with respect to a certain measure

Cf. A308612 (numerators).

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, j]*2^(n - j - 1)*a[j], {j, 0, n - 1}]/(3^n - 1); Table[Denominator[a[i]], {i, 0, 19}] (* Amiram Eldar, Aug 03 2019 *)

def am(m, alpha):
    N = len(alpha)
    am = [1]
    for a in [1..m]:
        mm = 0
        for k in [0..N-1]:
            for r in [0..a-1]:
                mm += alpha[k]*binomial(a, r)*k^(a-r)*am[r]
        mm /= (N^a-1)
        am.append(mm)
    return am
[p.denominator() for p in am(15, (1/2, 0, 1/2))]

More terms from Amiram Eldar, Aug 03 2019

A308614 Numerators of the even shifted moments of the ternary Cantor measure.

Original entry on oeis.org

1, 1, 7, 205, 10241, 26601785, 144273569311, 8432005793267, 85813777224887042933, 41391682933691854767291415, 279988393358814530594186727509023, 4597481350195941947735138659876438945979, 137236498421201646022141003769649699705393990756253

Offset: 0

Alexander Riasanovsky, Jun 10 2019

Due to the symmetry of the measure mu with respect to x=1/2 and the parity of the polynomial (x-1/2)^k about the line x=1/2, every odd entry is 0 and is thus omitted.
The ternary Cantor measure, defined many ways, is the unique Borel measure mu on the unit interval [0,1] satisfying the following recurrence relation for any measurable set E: mu(E) = mu(phi_0(E))/2 + mu(phi_2(E))/2.  Here, for j in {0,1,2}, phi_j:[0,1] to [0,1] is the linear function which sends x in [0,1] to (x+j)/3.  For any nonnegative integer k, we define the k-th shifted moment J(k) to be the integral of (x-1/2)^k with respect to mu.  The described sequence J(0), J(1), J(2), ... is rational and this sequence a(0), a(1), a(2), ... is the sequence of numerators of J(0), J(2), J(4), ....
For the purpose of computing J(k), we first compute the (unshifted) moments (see A308612 and A308613) which are the integrals of x^k rather than (x-1/2)^k, expand the polynomial (x-1/2)^k, replace each x^m term with the corresponding moment I(m), and simplify.

Michael De Vlieger, Table of n, a(n) for n = 0..48
Steven N. Harding, Alexander W. N. Riasanovsky, Moments of the weighted Cantor measures, arXiv:1908.05358 [math.FA], 2019.

Matching denominators are A308615. Shifted version of A308612 and A308613.

f[0] = 1; f[n_] := f[n] = Sum[Binomial[n, j]*2^(n - j - 1)*f[j], {j, 0, n - 1}]/(3^n - 1); a[n_] := Sum[Binomial[n, j]*f[j]*(-1/2)^(n - j), {j, 0, n}]; Table[Numerator[a[i]], {i, 0, 24, 2}] (* Amiram Eldar, Aug 03 2019 *)

moms = [1]
for k in [1..15]:
    s = 0
    for j in [0..k-1]:
        s += binomial(k, j)*2^(k-j)*moms[j]/2
    s /= (3^k-1)
    moms.append(s)
var('x')
shmoms = []
for k in [0..15]:
    p = (x-1/2)^k
    p = p.expand()
    s = 0
    for m in [0..k]:
        s += moms[m]*p.coefficient(x, m)
    shmoms.append(s)
[p.numerator() for p in shmoms if p]

More terms from Amiram Eldar, Aug 03 2019

A308615 Denominators of the even shifted moments of the ternary Cantor measure.

Original entry on oeis.org

1, 8, 320, 46592, 10915840, 128911704064, 3114038000353280, 798410297854394368, 35228168150276083007094784, 72984567358962659964369885986816, 2104733804502091904066890388853154119680, 146449616359318768962787815768964807513279037440

Offset: 0

Alexander Riasanovsky, Jun 10 2019

Due to the symmetry of the measure mu with respect to x=1/2 and the parity of the polynomial (x-1/2)^k about the line x=1/2, every odd entry is 0 and is thus omitted.
The ternary Cantor measure, defined many ways, is the unique Borel measure mu on the unit interval [0,1] satisfying the following recurrence relation for any measurable set E: mu(E) = mu(phi_0(E))/2 + mu(phi_2(E))/2.  Here, for j in {0,1,2}, phi_j:[0,1] to [0,1] is the linear function which sends x in [0,1] to (x+j)/3.  For any nonnegative integer k, we define the k-th shifted moment J(k) to be the integral of (x-1/2)^k with respect to mu.  The described sequence J(0), J(1), J(2), ... is rational and this sequence a(0), a(1), a(2), ... is the sequence of denominators of J(0), J(2), J(4), ....
For the purpose of computing J(k), we first compute the (unshifted) moments (see A308612 and A308613) which are the integrals of x^k rather than (x-1/2)^k, expand the polynomial (x-1/2)^k, replace each x^m term with the corresponding moment I(m), and simplify.

Michael De Vlieger, Table of n, a(n) for n = 0..48
Steven N. Harding, Alexander W. N. Riasanovsky, Moments of the weighted Cantor measures, arXiv:1908.05358 [math.FA], 2019.

Matching numerators are A308614. Shifted version of A308612 and A308613.

f[0] = 1; f[n_] := f[n] = Sum[Binomial[n, j]*2^(n - j - 1)*f[j], {j, 0, n - 1}]/(3^n - 1); a[n_] := Sum[Binomial[n, j]*f[j]*(-1/2)^(n - j), {j, 0, n}]; Table[Denominator[a[i]], {i, 0, 24, 2}] (* Amiram Eldar, Aug 03 2019 *)

moms = [1]
for k in [1..15]:
    s = 0
    for j in [0..k-1]:
        s += binomial(k, j)*2^(k-j)*moms[j]/2
    s /= (3^k-1)
    moms.append(s)
x = var('x')
shmoms = []
for k in [0..15]:
    p = (x-1/2)^k
    p = p.expand()
    s = 0
    for m in [0..k]:
        s += moms[m]*p.coefficient(x, m)
    shmoms.append(s)
[p.denominator() for p in shmoms[::2]]

More terms from Amiram Eldar, Aug 03 2019

A236441 Möbius inversion of A235342.

Original entry on oeis.org

0, 1, 0, 1, -2, 0, -1, 1, 0, 0, -2, 0, -2, 0, 0, 1, -2, 0, -1, 0, 0, 0, 2, 0, -2, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -2, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 7, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 5, 0, 1, 0, 0, 0, -2, 0, -2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -2

Offset: 1

Alexander Riasanovsky, Jan 25 2014

sign

Möbius inversion of A235342. Since b(xy) = b(x)+b(y) where b = A235342, it follows that a(n) is zero on nonprime powers and b(p) if n=p^k.

			a(1)=0 since 1 is not a prime power.
a(2)=b(2)=1 since 2=2! and b(2!)=1.
a(3)=b(3)=0 since 3=3!/2! and b(3!/2!)=b(3!)-b(2!)=1-1=0.
a(4)=b(2)=1 (above).
a(5)=b(5)=-2 since 5=5!/(3!2!2!) and b(5!/(3!2!2!))=1-3=-2.
a(6)=0 since 6 is not a prime power.

Antti Karttunen, Table of n, a(n) for n = 1..5041
Alexander Riasanovsky, Sage program

Möbius inversion of A235342.

For n > 0, a(n) = Sum_{d|n} b(d)*mu(n/d) where b(n) = A235342(n).

Data section extended and b-file computed with Riasanovsky's Sage program by Antti Karttunen, Mar 28 2017

A235342 Sum of exponents in the (unique) factorization of n as a ratio of p! terms, p prime.

Original entry on oeis.org

0, 1, 0, 2, -2, 1, -1, 3, 0, -1, -2, 2, -2, 0, -2, 4, -2, 1, -1, 0, -1, -1, 2, 3, -4, -1, 0, 1, 1, -1, 1, 5, -2, -1, -3, 2, -1, 0, -2, 1, 1, 0, 0, 0, -2, 3, -1, 4, -2, -3, -2, 0, 3, 1, -4, 2, -1, 2, 0, 0, 0, 2, -1, 6, -4, -1, -2, 0, 2, -2, 0, 3, -3, 0, -4, 1, -3, -1, 7, 2, 0, 2, -4, 1, -4, 1, 1, 1, 0, -1, -3, 4, 1, 0, -3, 5, -3, -1, -2, -2, 5, -1, 1

Offset: 1

Alexander Riasanovsky, Jan 06 2014

sign

With n > 0, write n = p_1!p_2!...p_k!/(q_1!q2!...q_l!) where p_i,q_j are primes and the fraction is simplified (i.e., no p_i is a q_j). This representation is unique for positive integers (and positive rational numbers), so we let a(n):=#p_i! terms on top-#q_j! terms on bottom.

			a(1)=0 (by convention).
a(2)=1 since 2=2!.
a(3)=0 since 3=3!/2!.
a(4)=2 since 4=2!*2!.
a(5)=-2 since 5=5!/(3!*2!*2!).

Antti Karttunen, Table of n, a(n) for n = 1..5040
Seventieth Annual William Lowell Putnam Mathematical Competition, Problem B1, (2009).

Cf. A001222, A236441.

def plus(c, d, mult):
  for elt in d:
    if elt in c:
      c[elt]+=mult*d[elt]
    else:
      c[elt]=mult*d[elt]
def rep(m):
  if m==1:
    return {}
  if m==2:
    return {2:1}
  f=factor(Integer(m))
  #print f
  if len(f)==1 and f[0][1]==1:
    #print "prime", m
    p=prime_range(m)[-1]
    new={m:1, p:-1}
    r=range(p+1, m)
    #print "range", r
    for k in r:
      plus(new, rep(k), -1)
  else:
    new={}
    #print "not prime", m, f
    for (p, mult) in f:
      #print (p, mult)
      plus(new, rep(p), mult)
  for elt in [elt for elt in new if new[elt]==0]:
    new.pop(elt)
  return new
def weight(m):
  w=0
  r=rep(m)
  for p in r:
    w+=r[p]
  return w
A235342=[weight(m) for m in range(1, 5041)]
# Above code "de-periodicized" by Antti Karttunen, Mar 28 2017
# This is just for outputting a b-file:
i=0
outfp = open('b235342.txt','w')
for an in A235342:
    i = i+1
    outfp.write(str(i) + " " + str(an) + "\n")
outfp.close()

a(1)=0; a(p!)=1, p prime; a(xy)=a(x)+a(y); (group homomorphism from Q^+ to Z).

More terms from Antti Karttunen, Mar 28 2017

A226356 Number of representations of the n-th factorial group as a (nondecreasing) product of (nontrivial) cyclic groups.

Original entry on oeis.org

0, 0, 1, 2, 3, 10, 20, 91, 207, 792, 2589, 17749, 52997

Offset: 0

Alexander Riasanovsky, Jun 04 2013

The algorithm given which generates and counts a(n) goes as follows:
Consider the product [1,2,3,...,n] as Z_1 x Z_2 x ... x Z_n and "refine" wherever possible to create a multiset. For example, Z_6 ~= Z_2 x Z_3, so [1,2,3,4,5,6] refines as the multiset [2,2,3,3,4,5] or {2:2,3:2,4:1,5:1}.
Place the above multiset at the top of a (ranked) poset, row 0.
Set i=0.
While there exists an element on the i-th row which, generate the (i+1)-th row of elements: those which are coarsed from elements on the i-th row.
Find the number of representations generated.

			Note: in the following ~= denotes isomorphism.
For example, G_0=Z_1 which cannot be represented as a product of nontrivial cyclic groups. Hence, a(0)=0. Likewise, G_1=G_0 x Z_1~=Z_1, so a(1)=0.
However G_2~=Z_2 is the only such representation of G_2.
For G_5=Z_1 x Z_2 x Z_3 x Z_4 x Z_5, we have exactly the following representations, sorted by the number of terms:
*Z_2 x Z_3 x Z_4 x Z_5,
*Z_4 x Z_5 x Z_6, Z_3 x Z_4 x Z_10, Z_2 x Z_5 x Z_12, Z_2 x Z_4 x Z_15, Z_2 x Z_3 x Z_20, and
*Z_6 x Z_20, Z_4 x Z_30, Z_10 x Z_12, Z_2 x Z_60.
Hence, a(5)=10.

#NOTE: by uncommenting the second return argument, the reader is given the array of representations.
def d_split(prod):
    p_counts={}
    for term in prod:
        for p, m in term.factor():
            pm = p^m
            if pm in p_counts:
                p_counts[pm]+=1
            else:
                p_counts[pm]=1
    return p_counts
def factorial_group_reps(m):
    if m<2:
        return 0
    i=0
    widest_rep=d_split([Integer(n) for n in range(1,m+1)])
    w_max=sum([widest_rep[p] for p in widest_rep])
    rep_poset=[[widest_rep]]
    r_count=1
    while w_max-i>m//2:
        row_new=[]
        for rep in rep_poset[i]:
            for [a,b] in Combinations(rep,2):
                if gcd([a,b])==1:
                    rep_new=rep.copy()
                    if rep_new[a]==1:
                        rep_new.pop(a)
                    else:
                        rep_new[a]-=1
                    if rep_new[b]==1:
                        rep_new.pop(b)
                    else:
                        rep_new[b]-=1
                    if a*b in rep_new:
                        rep_new[a*b]+=1
                    else:
                        rep_new[a*b]=1
                    if not rep_new in row_new:
                        r_count+=1
                        row_new.append(rep_new)
        rep_poset.append(row_new)
        i+=1
    return r_count#,rep_poset
for i in range(11):
    # for i>10, a(i) is a very tedious computation for this algorithm
    print(i,factorial_group_reps(i))

With Z_k denoting the cyclic group on k letters, let G_0:=Z_1 and for all positive integers i, set G_i:=G_(i-1) x Z_i. Then a(n) is the number of (isomorphic) representations of G_n as a (nondecreasing) product of (nontrivial) cyclic groups.

a(11) and a(12) added by Alexander Riasanovsky, Jun 06 2013

A210450 Numbers n such that 16n + 7 is in A192628.

Original entry on oeis.org

0, 3, 4, 5, 6, 7, 11, 16, 17, 21, 23, 24, 27, 28, 32, 34, 35, 36, 38, 39, 40, 43, 44, 45, 47, 48, 49, 51, 53, 54, 55, 56, 59, 60, 63, 65, 67, 68, 69, 70, 72, 73, 74, 76, 77, 79, 81, 82, 85, 86, 89, 93, 96, 97, 98, 100, 102, 103, 105, 106, 107, 109, 110

Offset: 1

Alexander Riasanovsky, Jan 20 2013

nonn

Reduce the elements of A192718 (which are the elements of A192628 congruent to 7 (mod 16)) by subtracting 7 and dividing by 16. In "On the reciprocal of the binary generating function for the sum of divisors", this sequence is precisely the set T.

J. N. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, arXiv:math/0506496 [math.NT], 2005.
J. N. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, International Journal of Number Theory, 2 no. 4 (2006), 499-522.
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012.
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, Journal of Integer Sequences, Vol. 16 (2013), #13.1.8.

Cf. A192718, A192628.

prec = 2^12
R = PowerSeriesRing(GF(2), 'q', default_prec = prec)
q = R.gen()
sigma = lambda x : 1 if x == 0 else sum(Integer(x).divisors())
SigmaSeries = sum([sigma(m)*q^m for m in range(prec)])
SigmaBarSeries = 1/SigmaSeries
SigmaBarList = SigmaBarSeries.exponents()
reduced = [(m-7)/16 for m in SigmaBarList if mod(m, 8) == 7]
print(reduced[:128])

A192718 Elements of A192628 which are congruent to 7 (mod 8) (equivalently, 7 (mod 16)).

Original entry on oeis.org

7, 55, 71, 87, 103, 119, 183, 263, 279, 343, 375, 391, 439, 455, 519, 551, 567, 583, 615, 631, 647, 695, 711, 727, 759, 775, 791, 823, 855, 871, 887, 903, 951, 967, 1015, 1047, 1079, 1095, 1111, 1127, 1159, 1175, 1191, 1223, 1239, 1271, 1303, 1319, 1367

Offset: 1

Alexander Riasanovsky, Dec 31 2012

nonn

This is the subsequence/subset of A192628 which contains elements congruent to 7 modulo 8.  Equivalently, these elements are also congruent to 7 modulo 16.
By partitioning A192628 into congruence classes k modulo 8, it turns out that it contains only elements congruent to 0, 1, 3, and 7 modulo 8.  Further, the congruence classes 0, 1, and 3 modulo 8 are vanishing--having a density asymptotic to 0.
However, the 7 modulo 8 congruence classes appears to have nonzero density, conjectured 1/32.  A current upper bound on its density (thus the entire density of A192628) is 1/16.

J. Cooper and A. Riasanovsky, On the reciprocal of the binary generating function for the sum-of-divisors, Journal of Integer Sequences (accepted).
J. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, International Journal of Number Theory, 2 no. 4 (2006), 499-522.

J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012; J. Int. Seq. 16 (2013) #13.1.8

prec = 2^12
R = PowerSeriesRing(GF(2), 'q', default_prec = prec)
q = R.gen()
sigma = lambda x : 1 if x == 0 else sum(Integer(x).divisors())
SigmaSeries = sum([sigma(m)*q^m for m in range(prec)])
SigmaBarSeries = 1/SigmaSeries
SigmaBarList = SigmaBarSeries.exponents()
SigmaBar7Mod8 = [m for m in SigmaBarList if mod(m, 8) == 7]
print(SigmaBar7Mod8)

A192717 Positive integers of the form (p^e)(k^2) for p prime congruent to 3 (mod 8), e congruent to 1 (mod 4), and k an odd integer coprime to p.

Original entry on oeis.org

3, 11, 19, 43, 59, 67, 75, 83, 99, 107, 131, 139, 147, 163, 171, 179, 211, 227, 243, 251, 275, 283, 307, 331, 347, 363, 379, 387, 419, 443, 467, 475, 491, 499, 507, 523, 531, 539, 547, 563, 571, 587, 603, 619, 643, 659, 683, 691, 739, 747, 787, 811, 827

Offset: 1

Alexander Riasanovsky, Dec 31 2012

nonn

This sequence is equivalent to all of the following sets (written in increasing order):
- all integers the form (p^e)(k^2) for p prime congruent to 3 (mod 8), e congruent to 1 (mod 4), and k an odd number coprime to p;
- all integers with an odd number of representations as x^2 + 2y^2 for odd x and y; and
- elements of A192628 which are congruent to 3 (mod 8).

			3 is in the sequence since 3 = (3^1)(1^2); 3 is prime and congruent to 3 (mod 8), 1 is congruent to 1 (mod 4), and 1 is an odd integer coprime to 3.
6 is not in the sequence: since it is squarefree, k must be 1, but 6 cannot be written as p^e.
27 is not in the sequence: the only possible values for k are 1 and 3. In the k=1 case, 27 = (3^3)(1^2) does not work since e = 3 is not congruent to 1 (mod 4), and in the k=3 case, 27 = (3^1)(3^2), k=3 and p=3 are not coprime.
243 is in the sequence since 243 = (3^5)(1^2); 3 is prime and congruent to 3 (mod 8), 5 is congruent to 1 (mod 4), and 1 is an odd integer coprime to 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Joshua N. Cooper, Dennis Eichhorn and Kevin O'Bryant, Reciprocals of binary power series, International Journal of Number Theory, Vol. 2, No. 4 (2006), pp. 499-522; arXiv preprint, arXiv:math/0506496 [math.NT], 2005.
Joshua N. Cooper and Alexander W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, J. Int. Seq., Vol. 16 (2013), Article #13.1.8; alternative link.

Cf. A192628.

ofTheFormQ[n_] := If[Length[fin = FactorInteger[n]] == 1 && Mod[fin[[1, 1]], 8] == 3 && Mod[fin[[1, 2]], 4] == 1, True, pe = Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ fin); k = Sqrt[n/pe]; fip = FactorInteger[pe]; Length[fip] == 1 && Mod[p = fip[[1, 1]], 8] == 3 && Mod[e = fip[[1, 2]], 4] == 1 && OddQ[k] && CoprimeQ[k, p]]; Select[Range[1, 999, 2], ofTheFormQ] (* Jean-François Alcover, Jan 22 2013 *)

prec = 2^10
L = []
for n in range(1, prec, 2):
    n = Integer(n)
    sfp = n.squarefree_part()
    if mod(sfp, 8) == 3 and sfp.is_prime() and mod(n.ord(sfp), 4) == 1:
        L.append(n)
print(L)

def BPS(n): #binary power series
    return sum([q^s for s in n])
prec = 2^14
R = PowerSeriesRing(GF(2), 'q', default_prec = prec)
q = R.gen()
dList = [(2*n+1)^2 for n in range(0, (sqrt(prec)-1)/2)]
dSeries = BPS(dList)
print((dSeries^3).exponents()[:128])

cp's OEIS Frontend

User: Alexander Riasanovsky

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