A047350 -id:A047350 - cp's OEIS frontend

A047235 Numbers that are congruent to {2, 4} mod 6.

2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50, 52, 56, 58, 62, 64, 68, 70, 74, 76, 80, 82, 86, 88, 92, 94, 98, 100, 104, 106, 110, 112, 116, 118, 122, 124, 128, 130, 134, 136, 140, 142, 146, 148, 152, 154, 158, 160, 164, 166, 170, 172, 176, 178, 182, 184, 188, 190, 194, 196, 200, 202, 206

Offset: 1

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 19 ).
Complement of A047273; A093719(a(n)) = 0. - Reinhard Zumkeller, Oct 01 2008
One could prefix an initial term "1" (or not) and define this sequence through a(n+1) = a(n) + (a(n) mod 6). See A001651 for the analog with 3, A235700 (with 5), A047350 (with 7), A007612 (with 9) and A102039 (with 10). Using 4 or 8 yields a constant sequence from that term on. - M. F. Hasler, Jan 14 2014
Nonnegative m such that m^2/6 + 1/3 is an integer. - Bruno Berselli, Apr 13 2017
Numbers divisible by 2 but not by 3. - David James Sycamore, Apr 04 2018
Numbers k for which A276086(k) is of the form 6m+3. - Antti Karttunen, Dec 03 2022

David A. Corneth, Table of n, a(n) for n = 1..10000
Chunhui Lai, A note on potentially K_4-e graphical sequences, arXiv:math/0308105 [math.CO], 2003.
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A020760, A020832, A093719, A047273 (complement), A120325 (characteristic function).
Equals 2*A001651.
Cf. A007310 ((6*n+(-1)^n-3)/2). - Bruno Berselli, Jun 24 2010
Positions of 3's in A053669 and in A358840.

[ n eq 1 select 2 else Self(n-1)+2*(1+n mod 2): n in [1..70] ]; // Klaus Brockhaus, Dec 13 2008

seq(6*floor((n+1)/2) + 3 + (-1)^n, n=1..67); # Gary Detlefs, Mar 02 2010

Flatten[Table[{6n - 4, 6n - 2}, {n, 40}]] (* Alonso del Arte, Oct 27 2014 *)

a(n)=(n-1)\2*6+3+(-1)^n \\ Charles R Greathouse IV, Jul 01 2013

first(n) = my(v = vector(n, i, 3*i - 1)); forstep(i = 2, n, 2, v[i]--); v \\ David A. Corneth, Oct 20 2017

a(n) = 2*A001651(n).
n such that phi(3*n) = phi(2*n). - Benoit Cloitre, Aug 06 2003
G.f.: 2*x*(1 + x + x^2)/((1 + x)*(1 - x)^2). a(n) = 3*n - 3/2 - (-1)^n/2. - R. J. Mathar, Nov 22 2008
a(n) = 3*n + 5..n odd, 3*n + 4..n even a(n) = 6*floor((n+1)/2) + 3 + (-1)^n. - Gary Detlefs, Mar 02 2010
a(n) = 6*n - a(n-1) - 6 (with a(1) = 2). - Vincenzo Librandi, Aug 05 2010
a(n+1) = a(n) + (a(n) mod 6). - M. F. Hasler, Jan 14 2014
Sum_{n>=1} 1/a(n)^2 = Pi^2/27. - Dimitris Valianatos, Oct 10 2017
a(n) = (6*n - (-1)^n - 3)/2. - Ammar Khatab, Aug 23 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)). - Amiram Eldar, Dec 11 2021
E.g.f.: 2 + ((6*x - 3)*exp(x) - exp(-x))/2. - David Lovler, Aug 25 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2/sqrt(3) (10 * A020832).
Product_{n>=1} (1 + (-1)^n/a(n)) = 1/sqrt(3) (A020760). (End)

A102039 a(n) = a(n-1) + last digit of a(n-1), starting at 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 22, 24, 28, 36, 42, 44, 48, 56, 62, 64, 68, 76, 82, 84, 88, 96, 102, 104, 108, 116, 122, 124, 128, 136, 142, 144, 148, 156, 162, 164, 168, 176, 182, 184, 188, 196, 202, 204, 208, 216, 222, 224, 228, 236, 242, 244, 248, 256, 262, 264, 268, 276, 282

Offset: 1

Samantha Stones (devilsdaughter2000(AT)hotmail.com), Dec 25 2004

Sequence A001651 is the "base 3" version. In base 4 this rule leads to (1,2,4,4,4...), in base 5 to (1,2,4,8,11,12,14,18,21,22,24,28...) = A235700. - M. F. Hasler, Jan 14 2014

This and the following sequences (none of which is "base"!) could all be defined by a(1) = 1 and a(n+1) = a(n) + (a(n) mod b) with different values of b: A001651 (b=3), A235700 (b=5), A047235 (b=6), A047350 (b=7), A007612 (b=9). Using b=4 or b=8 yields a constant sequence from that term on. - M. F. Hasler, Jan 15 2014

			28 + 8 = 36, 36 + 6 = 42.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Apart from initial term, same as A002081.

LinearRecurrence[{2,-2,2,-1},{1,2,4,8,16},60] (* Harvey P. Dale, Jul 02 2022 *)

print1(a=1);for(i=1,99,print1(","a+=a%10)) \\ M. F. Hasler, Jan 14 2014

Vec(x*(5*x^4+2*x^3+2*x^2+1)/((x-1)^2*(x^2+1)) + O(x^100)) \\ Colin Barker, Sep 20 2014

a(n) = if(n==1, 1, (-10-(1-I/2)*(-I)^n-(1+I/2)*I^n+5*n)) \\ Colin Barker, Oct 18 2015

a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>5. G.f.: x*(5*x^4+2*x^3+2*x^2+1) / ((x-1)^2*(x^2+1)). - Colin Barker, Sep 20 2014

a(n) = (-10 - (1-i/2)*(-i)^n - (1+i/2)*i^n + 5*n) for n>1, where i = sqrt(-1). - Colin Barker, Oct 18 2015

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

A235700 a(n+1) = a(n) + (a(n) mod 5), a(1)=1.

Original entry on oeis.org

1, 2, 4, 8, 11, 12, 14, 18, 21, 22, 24, 28, 31, 32, 34, 38, 41, 42, 44, 48, 51, 52, 54, 58, 61, 62, 64, 68, 71, 72, 74, 78, 81, 82, 84, 88, 91, 92, 94, 98, 101, 102, 104, 108, 111, 112, 114, 118, 121, 122, 124, 128, 131, 132, 134, 138, 141, 142, 144, 148, 151, 152, 154, 158, 161, 162, 164, 168, 171, 172, 174, 178, 181, 182, 184, 188, 191

Offset: 1

M. F. Hasler, Jan 14 2014

Although the present sequence has not been thought of via "writing a(n) in base b", this could be seen as "base 5" version of A102039 (base 10) and A001651 (base 3), A047235 (base 6), A047350 (base 7) and A007612 (base 9). For 4 or 8 one would get a sequence constant from that (3rd resp. 4th) term on.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A001651, A007612, A047235, A047350, A102039.

NestList[#+Mod[#,5]&,1,80] (* Harvey P. Dale, Oct 20 2024 *)

is_A235700(n) = bittest(278,n%10) \\ 278=2^1+2^2+2^4+2^8

PARI
```
A235700 = n -> 2^((n-1)%4)+(n-1)\4*10
```

print1(a=1);for(i=1,99,print1(","a+=a%5))

Vec(x*(2*x^3+2*x^2+1)/((x-1)^2*(x^2+1)) + O(x^100)) \\ Colin Barker, Jan 16 2014

a(n) = 2^(n-1 mod 4) + 10*floor((n-1)/4).
From Colin Barker, Jan 16 2014: (Start)
a(n) = (-10+(1+2*i)*(-i)^n+(1-2*i)*i^n+10*n)/4 where i=sqrt(-1).
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4).
G.f.: x*(2*x^3+2*x^2+1) / ((x-1)^2*(x^2+1)). (End)
E.g.f.: (4 + 5*exp(x)*(x - 1) + cos(x) + 2*sin(x))/2. - Stefano Spezia, Feb 22 2025

A302588 a(n) = a(n-3) + 7*(n-2), a(0)=1, a(1)=2, a(2)=4.

Original entry on oeis.org

1, 2, 4, 8, 16, 25, 36, 51, 67, 85, 107, 130, 155, 184, 214, 246, 282, 319, 358, 401, 445, 491, 541, 592, 645, 702, 760, 820, 884, 949, 1016, 1087, 1159, 1233, 1311, 1390, 1471, 1556, 1642, 1730, 1822, 1915, 2010

Offset: 0

Paul Curtz, Jul 17 2018

nonn

Third of a family after A000124 and A084684. Built from the second differences. The fourth sequence is 1, 2, 4, 8, 16, 32, 49, 91, ..., from periodic [1, 2, 4, 8].

0

Index entries for linear recurrences with constant coefficients, signature (2, -1, 1, -2, 1).

Cf. A069705, A047350 (first differences), A130518.

CoefficientList[ Series[-(4x^4 +x^3 +x^2 +1)/((x -1)^3 (x^2 +x +1)), {x, 0, 50}], x] (* or *)
LinearRecurrence[{2, -1, 1, -2, 1}, {1, 2, 4, 8, 16}, 50] (* Robert G. Wilson v, Jul 18 2018 *)

Repeat the second differences of 1, 2, 4, 8, 16, i.e., repeat [1, 2, 4].
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
a(n) = A069705(n) + 7*A130518(n).

A306969 Numbers congruent to 1, 2, or 4 mod 7 that are not represented by x^2+7y^2+7z^2.

Original entry on oeis.org

2, 22, 46, 58, 85, 93, 102, 205, 298, 310, 330, 358, 466, 478, 697, 862, 949, 1222, 1402, 1513, 1957, 1978, 2962, 3502, 7165, 10558

Offset: 1

N. J. A. Sloane, Mar 26 2019

Irving Kaplansky, The first nontrivial genus of positive definite ternary forms, Mathematics of Computation, Vol. 64 (1995): 341-345.

Cf. A047350 (numbers congruent to {1, 2, 4} mod 7).

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A047235 Numbers that are congruent to {2, 4} mod 6.

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A102039 a(n) = a(n-1) + last digit of a(n-1), starting at 1.

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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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A235700 a(n+1) = a(n) + (a(n) mod 5), a(1)=1.

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A302588 a(n) = a(n-3) + 7*(n-2), a(0)=1, a(1)=2, a(2)=4.

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A306969 Numbers congruent to 1, 2, or 4 mod 7 that are not represented by x^2+7*y^2+7*z^2.

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A306969 Numbers congruent to 1, 2, or 4 mod 7 that are not represented by x^2+7y^2+7z^2.