A233998 -id:A233998 - cp's OEIS frontend

A055047 Numbers of the form 9^i(3j+1).

1, 4, 7, 9, 10, 13, 16, 19, 22, 25, 28, 31, 34, 36, 37, 40, 43, 46, 49, 52, 55, 58, 61, 63, 64, 67, 70, 73, 76, 79, 81, 82, 85, 88, 90, 91, 94, 97, 100, 103, 106, 109, 112, 115, 117, 118, 121, 124, 127, 130, 133, 136, 139, 142, 144, 145, 148, 151, 154, 157, 160, 163

Offset: 1

N. J. A. Sloane, Jun 01 2000

nonn

The numbers not of the form 2x^2+3y^2+3z^2.
Also values of n such that numbers of the form x^2+n*y^2 for some integers x, y cannot have prime factor of 3 raised to an odd power. - V. Raman, Dec 18 2013
Numbers whose squarefree part is congruent to 1 modulo 3. - Peter Munn, May 17 2020

Paolo Xausa, Table of n, a(n) for n = 1..10000
L. J. Mordell, A new Waring's problem with squares of linear forms, Quart. J. Math., 1 (1930), 276-288 (see p. 283).

Intersection of A007417 and A189715.
Complement of A055048 with respect to A007417.
Complement of A055040 with respect to A189715.
Cf. A007913, A055041, A055046, A233998, A233999.

A055047Q[k_] := Mod[k/9^IntegerExponent[k, 9], 3] == 1;
Select[Range[300], A055047Q] (* Paolo Xausa, Mar 20 2025 *)

is(n)=n/=9^valuation(n,9); n%3==1 \\ Charles R Greathouse IV and V. Raman, Dec 19 2013

from sympy import integer_log
def A055047(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//9**i-1)//3+1 for i in range(integer_log(x,9)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

a(n) = 8n/3 + O(log n). - Charles R Greathouse IV, Dec 19 2013

a(n) = A055041(n)/3. - Peter Munn, May 17 2020

A055046 Numbers of the form 4^i(8j+3).

Original entry on oeis.org

3, 11, 12, 19, 27, 35, 43, 44, 48, 51, 59, 67, 75, 76, 83, 91, 99, 107, 108, 115, 123, 131, 139, 140, 147, 155, 163, 171, 172, 176, 179, 187, 192, 195, 203, 204, 211, 219, 227, 235, 236, 243, 251, 259, 267, 268, 275, 283, 291, 299, 300, 304, 307, 315, 323, 331, 332

Offset: 1

N. J. A. Sloane, Jun 01 2000

Numbers not of the form x^2+y^2+5z^2.

Also values of n such that numbers of the form x^2+n*y^2 for some integers x, y cannot have prime factor of 2 raised to an odd power. - V. Raman, Dec 18 2013

Paolo Xausa, Table of n, a(n) for n = 1..10000
L. J. Mordell, A new Waring's problem with squares of linear forms, Quart. J. Math., 1 (1930), 276-288 (see p. 283).

Cf. A004215, A055043, A055045, A055047, A233998, A233999, A234000.

A055046Q[k_] := Mod[k/4^IntegerExponent[k, 4], 8] == 3;
Select[Range[500], A055046Q] (* Paolo Xausa, Mar 20 2025 *)

is(n)=n/=4^valuation(n,4); n%8==3 \\ Charles R Greathouse IV and V. Raman, Dec 19 2013

from itertools import count, islice
def A055046_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not (m:=(~n&n-1).bit_length())&1 and (n>>m)&7==3,count(max(startvalue,1)))
A055046_list = list(islice(A055046_gen(),30)) # Chai Wah Wu, Jul 09 2022

def A055046(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(((x>>(i<<1))-3>>3)+1 for i in range(x.bit_length()>>1))
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

a(n) = 6n + O(log n). - Charles R Greathouse IV, Dec 19 2013

a(n) = A055043(n)/2. - Chai Wah Wu, Mar 19 2025

A233999 Values of n such that numbers of the form x^2+n*y^2 for some integers x, y cannot have prime factor of 7 raised to an odd power.

Original entry on oeis.org

1, 2, 4, 8, 9, 11, 15, 16, 18, 22, 23, 25, 29, 30, 32, 36, 37, 39, 43, 44, 46, 49, 50, 51, 53, 57, 58, 60, 64, 65, 67, 71, 72, 74, 78, 79, 81, 85, 86, 88, 92, 93, 95, 98, 99, 100, 102, 106, 107, 109, 113, 114, 116, 120, 121, 123, 127, 128, 130, 134, 135, 137, 141, 142, 144, 148, 149

Offset: 1

V. Raman, Dec 18 2013

Equivalently, numbers of the form 49^n*(7m+1), 49^n*(7m+2), or 49^n*(7m+4). [Corrected by Charles R Greathouse IV, Jan 12 2017]
From Peter Munn, Feb 08 2024: (Start)
Numbers whose squarefree part is congruent to a (nonzero) quadratic residue modulo 7.
The integers in a subgroup of the positive rationals under multiplication.  As such the sequence is closed under multiplication and - where the result is an integer - under division. The subgroup has index 4 and is generated by the primes congruent to a quadratic residue (1, 2 or 4) modulo 7, the square of 7, and 3 times the other primes; that is a generator corresponding to each prime: 2, 3*3, 3*5, 7^2, 11, 3*13, 3*17, 3*19, 23, 29, 3*31, ... .
(End)

Eric Weisstein's World of Mathematics, Squarefree Part.

Cf. A055046, A233998.
Numbers whose squarefree part is congruent to a coprime quadratic residue modulo k: A003159 (k=2), A055047 (k=3), A277549 (k=4), A352272 (k=6), A234000 (k=8), A334832 (k=24).
First differs from A047350 by including 49.

is(n)=n/=49^valuation(n, 49); n%7==1||n%7==2||n%7==4 \\ Charles R Greathouse IV and V. Raman, Dec 19 2013

is_A233999(n)=bittest(22,n/49^valuation(n, 49)%7) \\ - M. F. Hasler, Jan 02 2014

list(lim)=my(v=List(),t,u); forstep(k=1,lim\=1,[1,2,4], listput(v,k)); for(e=1,logint(lim,49), u=49^e; for(i=1,#v, t=u*v[i]; if(t>lim, break); listput(v,t))); Set(v) \\ Charles R Greathouse IV, Jan 12 2017

from sympy import integer_log
def A233999(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in range(integer_log(x,49)[0]+1):
            m = x//49**i
            c -= (m-1)//7+(m-2)//7+(m-4)//7+3
        return c
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

a(n) = 16n/7 + O(log n). - Charles R Greathouse IV, Jan 12 2017

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A055047 Numbers of the form 9^i(3j+1).

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A055046 Numbers of the form 4^i(8j+3).

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A233999 Values of n such that numbers of the form x^2+n*y^2 for some integers x, y cannot have prime factor of 7 raised to an odd power.

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cp's OEIS Frontend

A055047 Numbers of the form 9^i*(3*j+1).

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A055046 Numbers of the form 4^i*(8*j+3).

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Author

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Programs

Mathematica

PARI

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Python

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A233999 Values of n such that numbers of the form x^2+n*y^2 for some integers x, y cannot have prime factor of 7 raised to an odd power.

Views

Author

Keywords

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PARI

PARI

PARI

Python

Formula

A055047 Numbers of the form 9^i(3j+1).

A055046 Numbers of the form 4^i(8j+3).