cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A359375 Numbers that are neither multiples of 4 nor of the form 6u+3.

Original entry on oeis.org

1, 2, 5, 6, 7, 10, 11, 13, 14, 17, 18, 19, 22, 23, 25, 26, 29, 30, 31, 34, 35, 37, 38, 41, 42, 43, 46, 47, 49, 50, 53, 54, 55, 58, 59, 61, 62, 65, 66, 67, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 89, 90, 91, 94, 95, 97, 98, 101, 102, 103, 106, 107, 109, 110, 113, 114, 115, 118, 119, 121, 122, 125
Offset: 1

Views

Author

Antti Karttunen, Dec 31 2022

Keywords

Comments

Numbers that are neither odd multiples of 3 nor multiples of 4.
Numbers k for which A252463(k) is odd.
Numbers that are congruent to {1, 2, 5, 6, 7, 10, 11} mod 12. - Amiram Eldar, Jan 24 2023

Links

Crossrefs

Cf. A000035, A064989, A252463, A359374 (characteristic function), A359376 (complement).
Cf. also A359380.

Programs

  • Mathematica
    Select[Range[125], Mod[#, 6] != 3 && Mod[#, 4] > 0 &] (* Amiram Eldar, Jan 24 2023 *)
  • PARI
    isA359375(n) = A359374(n);

A187074 a(n) = 0 if and only if n is of the form 3*k or 4*k + 2, otherwise a(n) = 1.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0
Offset: 1

Views

Author

Michael Somos, Mar 07 2011

Keywords

Comments

Characteristic function of A359380, numbers that are neither multiples of 3 nor of the form 4u+2. - Antti Karttunen, Dec 31 2022

Examples

			x + x^4 + x^5 + x^7 + x^8 + x^11 + x^13 + x^16 + x^17 + x^19 + x^20 + ...

Links

Crossrefs

Characteristic function of A359380.
Cf. A000661, A011655, A152822, A359374, A359422 (Dirichlet inverse).

Programs

  • Mathematica
    PadRight[{}, 120, {1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0}] (* or *)
Table[If[MemberQ[{0, 2, 3, 6, 9, 10}, Mod[n, 12]], 0, 1], {n, 120}] (* or *)
Table[Boole@ Or[CoprimeQ[n, 12], MemberQ[{4, 8}, Mod[n, 12]]], {n, 120}] (* or *)
Rest@ CoefficientList[Series[x (1 + x^4) (1 + x^3 + x^6)/(1 - x^12), {x, 0, 121}], x] (* Michael De Vlieger, Jul 16 2016 *)
Table[Which[Mod[n,3]==0,0,Mod[n,4]==2,0,True,1],{n,120}] (* Harvey P. Dale, Aug 02 2021 *)
  • PARI
    {a(n) = [0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1][n%12 + 1]};
  • PARI
    {a(n) = n = abs(n); sumdiv( 12, k, (n%k == 0) * [ 1, -1, -1, 1, 0, 1, 0, 0, 0, 0, 0, -1][k] )};
  • PARI
    A187074(n) = ((n%3)&&(2!=(n%4))); \\ Antti Karttunen, Dec 31 2022

Formula

Euler transform of length 12 sequence [0, 0, 1, 1, 0, 0, 0, -1, -1, 0, 0, 1].
Moebius transform is length 12 sequence [1, -1, -1, 1, 0, 1, 0, 0, 0, 0, 0, -1].
a(n) is multiplicative with a(2^e) = 1 except a(2) = 0, a(3^e) = 0^e, a(p^e) = 1 if p>3.
G.f.: x * (1 + x^4) * (1 + x^3 + x^6) / (1 - x^12). a(n + 12) = a(-n) = a(n). a(3*n) = a(4*n + 2) = 0.
Dirichlet g.f. zeta(s)*(1-3^(-s))*(1+4^(-s)-2^(-s)). - R. J. Mathar, Mar 31 2011
a(n+5) = A000661(n)(mod 2). - John M. Campbell, Jul 15 2016
a(n) = A011655(n) * A152822(n). - Antti Karttunen, Dec 31 2022

A374490 Greatest common divisor of sums of n consecutive cubes.

Original entry on oeis.org

1, 1, 9, 4, 5, 9, 7, 8, 27, 5, 11, 36, 13, 7, 45, 16, 17, 27, 19, 20, 63, 11, 23, 72, 25, 13, 81, 28, 29, 45, 31, 32, 99, 17, 35, 108, 37, 19, 117, 40, 41, 63, 43, 44, 135, 23, 47, 144, 49, 25, 153, 52, 53, 81, 55, 56, 171, 29, 59, 180, 61
Offset: 1

Views

Author

Kaleb Williams, Jul 09 2024

Keywords

Comments

A quasipolynomial of order 12 and degree 2. - Charles R Greathouse IV, Jul 11 2024

Examples

			For n=3, the sum of 3 consecutive cubes is S(x) = x^3 + (x+1)^3 + (x+2)^3 which has S(0) = 9 and thereafter remains a multiple of 9 since S(x) - S(x-1) = 9*(x^2 + x + 1), so that the GCD of all S(x) is a(3) = 9.

Links

Crossrefs

Cf. A026741 (for consecutive integers), A060789 (for consecutive squares).
Cf. A359380.

Programs

  • PARI
    f(n,x='x)=n*x^3 + (3/2*n^2 - 3/2*n)*x^2 + (n^3 - 3/2*n^2 + 1/2*n)*x + (1/4*n^4 - 1/2*n^3 + 1/4*n^2)
Polya(P)=my(x=variable(P),D=poldegree(P),f=D!,t=0); forstep(d=D,0,-1, my(c=polcoef(P,d,x)*d!); P-=c*binomial(x,d); t=gcd(t,c); f/=max(d,1)); t
a(n)=Polya(f(n)) \\ Charles R Greathouse IV, Jul 09 2024

Formula

From Stefano Spezia, Jul 10 2024: (Start)
G.f.: x*(1 + x + 9*x^2 + 4*x^3 + 5*x^4 + 9*x^5 + 7*x^6 + 8*x^7 + 27*x^8 + 5*x^9 + 11*x^10 + 36*x^11 + 11*x^12 + 5*x^13 + 27*x^14 + 8*x^15 + 7*x^16 + 9*x^17 + 5*x^18 + 4*x^19 + 9*x^20 + x^21 + x^22)/((1 - x)^2*(1 + x)^2*(1 + x^2)^2*(1 - x + x^2)^2*(1 + x + x^2)^2*(1 - x^2 + x^4)^2).
a(A359380(n)) = A359380(n). (End)
a(n) = n/2 if n is 2 or 10 mod 12; a(n) = 3n if n is 0, 3, or 9 mod 12; a(n) = 3n/2 if n = 6 mod 12; and a(n) = n otherwise (if n is 1, 4, 5, 7, 8, or 11 mod 12). In particular, n/2 <= a(n) <= 3n. - Charles R Greathouse IV, Jul 11 2024

Extensions

a(41)-a(61) from Charles R Greathouse IV, Jul 09 2024
Showing 1-3 of 3 results.

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