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.

User: Jianing Song

Jianing Song's wiki page.

Jianing Song has authored 1076 sequences. Here are the ten most recent ones:

A385552 Period of {binomial(N,n) mod 5: N in Z}.

Original entry on oeis.org

1, 5, 5, 5, 5, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125
Offset: 0

Views

Author

Jianing Song, Jul 03 2025

Keywords

Comments

a(n) is the smallest power of 5 > n. For the general result, see A349593.
Since the modulus (5) is a prime, the remainder of binomial(N,n) is given by Lucas's theorem.

Examples

			For N == 0, 1, ..., 24 (mod 5), binomial(N,5) == {0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4} (mod 5).

Links

Crossrefs

Column 5 of A349593. A062383, A064235 (if offset 0), A385553, and A385554 are respectively columns 2, 3, 6, and 10.

Programs

  • PARI
    a(n) = if(n, 5^(logint(n,5)+1), 1)
  • Python
    from sympy import integer_log
def A385552(n): return 5*5**(integer_log(n,5)[0]) if n else 1 # Chai Wah Wu, Jul 06 2025

A385553 Period of {binomial(N,n) mod 6: N in Z}.

Original entry on oeis.org

1, 6, 12, 36, 72, 72, 72, 72, 144, 432, 432, 432, 432, 432, 432, 432, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 2592, 2592, 2592, 2592, 2592, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184
Offset: 0

Views

Author

Jianing Song, Jul 03 2025

Keywords

Examples

			For N == 0, 1, ..., 35 (mod 36), binomial(N,3) == {0, 0, 0, 1, 4, 4, 2, 5, 2, 0, 0, 3, 4, 4, 4, 5, 2, 2, 0, 3, 0, 4, 4, 1, 2, 2, 2, 3, 0, 0, 4, 1, 4, 2, 2, 5} (mod 6).
For N == 0, 1, ..., 71 (mod 72), binomial(N,4) == {0, 0, 0, 0, 1, 5, 3, 5, 4, 0, 0, 0, 3, 1, 5, 3, 2, 4, 0, 0, 3, 3, 1, 5, 0, 2, 4, 0, 3, 3, 3, 1, 2, 0, 2, 4, 3, 3, 3, 3, 4, 2, 0, 2, 1, 3, 3, 3, 0, 4, 2, 0, 5, 1, 3, 3, 0, 0, 4, 2, 3, 5, 1, 3, 0, 0, 0, 4, 5, 3, 5, 1} (mod 6).

Links

Crossrefs

Column 6 of A349593. A062383, A064235 (if offset 0), A385552, and A385554 are respectively columns 2, 3, 5, and 10.

Programs

  • PARI
    a(n) = if(n, (2^(logint(n,2)+1)) * (3^(logint(n,3)+1)), 1)

Formula

a(n) = (the smallest power of 2 > n) * (the smallest power of 3 > n) = A062383(n) * A064235(n+1). For the general result, see A349593.

A385554 Period of {binomial(N,n) mod 10: N in Z}.

Original entry on oeis.org

1, 10, 20, 20, 40, 200, 200, 200, 400, 400, 400, 400, 400, 400, 400, 400, 800, 800, 800, 800, 800, 800, 800, 800, 800, 4000, 4000, 4000, 4000, 4000, 4000, 4000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000
Offset: 0

Views

Author

Jianing Song, Jul 03 2025

Keywords

Examples

			For N == 0, 1, ..., 19 (mod 20), binomial(N,3) == {0, 0, 0, 1, 4, 0, 0, 5, 6, 4, 0, 5, 0, 6, 4, 5, 0, 0, 6, 9} (mod 10).
For N == 0, 1, ..., 39 (mod 40), binomial(N,4) == {0, 0, 0, 0, 1, 5, 5, 5, 0, 6, 0, 0, 5, 5, 1, 5, 0, 0, 0, 6, 5, 5, 5, 5, 6, 0, 0, 0, 5, 1, 5, 5, 0, 0, 6, 0, 5, 5, 5, 1} (mod 10).

Links

Crossrefs

Column 10 of A349593. A062383, A064235 (if offset 0), A385552, and A385553 are respectively columns 2, 3, 5, and 6.

Programs

  • PARI
    a(n) = if(n, (2^(logint(n,2)+1)) * (5^(logint(n,5)+1)), 1)

Formula

a(n) = (the smallest power of 2 > n) * (the smallest power of 5 > n) = A062383(n) * A385552(n). For the general result, see A349593.

A385556 Period of {binomial(N,4) mod n: N in Z}.

Original entry on oeis.org

1, 8, 9, 16, 5, 72, 7, 32, 27, 40, 11, 144, 13, 56, 45, 64, 17, 216, 19, 80, 63, 88, 23, 288, 25, 104, 81, 112, 29, 360, 31, 128, 99, 136, 35, 432, 37, 152, 117, 160, 41, 504, 43, 176, 135, 184, 47, 576, 49, 200, 153, 208, 53, 648, 55, 224, 171, 232, 59, 720
Offset: 1

Views

Author

Jianing Song, Jul 03 2025

Keywords

Examples

			For N == 0, 1, ..., 31 (mod 32), binomial(N,4) == {0, 0, 0, 0, 1, 5, 7, 3, 6, 6, 2, 2, 7, 3, 1, 5, 4, 4, 4, 4, 5, 1, 3, 7, 2, 2, 6, 6, 3, 7, 5, 1} (mod 8).

Links

Crossrefs

Row n = 4 of A349593. A022998, A385555, A385557, A385558, A385559, and A385560 are respectively rows n = 2, 3, 5-6, 7, 8, and 9-10.
Cf. A000034, A089128.

Programs

  • Mathematica
    A385556[n_] := If[n == 1, 1, n*Product[p^Floor[Log[p, 4]], {p, FactorInteger[n][[All, 1]]}]];
Array[A385556, 100] (* Paolo Xausa, Jul 07 2025 *)
a[n_] := n * GCD[n, 6] * (2 - Mod[n, 2]); Array[a, 100] (* Amiram Eldar, Jul 07 2025 *)
  • PARI
    a(n, {choices=4}) = my(r=1, f=factor(n)); for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]); r *= p^(logint(choices, p)+e)); return(r)

Formula

Multiplicative with a(2^e) = 2^(e+2), a(3^e) = 3^(e+1), and a(p^e) = p^e for primes p >= 5.
From Amiram Eldar, Jul 07 2025: (Start)
a(n) = n * gcd(6, n) * (2 - (n mod 2)) = n * A089128(n) * A000034(n-1).
Dirichlet g.f.: zeta(s-1) * (1 + 3/2*(s-1)) * (1 + 2/3*(s-1)).
Sum_{k=1..n} a(k) ~ (25/12) * n^2. (End)

A385555 Period of {binomial(N,3) mod n: N in Z}.

Original entry on oeis.org

1, 4, 9, 8, 5, 36, 7, 16, 27, 20, 11, 72, 13, 28, 45, 32, 17, 108, 19, 40, 63, 44, 23, 144, 25, 52, 81, 56, 29, 180, 31, 64, 99, 68, 35, 216, 37, 76, 117, 80, 41, 252, 43, 88, 135, 92, 47, 288, 49, 100, 153, 104, 53, 324, 55, 112, 171, 116, 59, 360
Offset: 1

Views

Author

Jianing Song, Jul 03 2025

Keywords

Examples

			For N == 0, 1, ..., 26 (mod 27), binomial(N,3) == {0, 0, 0, 1, 4, 1, 2, 8, 2, 3, 3, 3, 4, 7, 4, 5, 2, 5, 6, 6, 6, 7, 1, 7, 8, 5, 8} (mod 4).

Links

Crossrefs

Row n = 3 of A349593. A022998, A385556, A385557, A385558, A385559, and A385560 are respectively rows n = 2, 4, 5-6, 7, 8, and 9-10.
Cf. A089128.

Programs

  • Mathematica
    A385555[n_] := If[n == 1, 1, n*Product[p^Floor[Log[p, 3]], {p, FactorInteger[n][[All, 1]]}]];
Array[A385555, 100] (* Paolo Xausa, Jul 07 2025 *)
a[n_] := n * GCD[n, 6]; Array[a, 100] (* Amiram Eldar, Jul 07 2025 *)
  • PARI
    a(n, {choices=3}) = my(r=1, f=factor(n)); for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]); r *= p^(logint(choices, p)+e)); return(r)

Formula

Multiplicative with a(2^e) = 2^(e+1), a(3^e) = 3^(e+1), and a(p^e) = p^e for primes p >= 5.
From Amiram Eldar, Jul 07 2025: (Start)
a(n) = n * gcd(6, n) = n * A089128(n).
Dirichlet g.f.: zeta(s-1) * (1 + 1/2*(s-1)) * (1 + 2/3*(s-1)).
Sum_{k=1..n} a(k) ~ (5/4) * n^2. (End)

A385557 Period of {binomial(N,5) mod n: N in Z}. Also, period of {binomial(N,6) mod n: N in Z}.

Original entry on oeis.org

1, 8, 9, 16, 25, 72, 7, 32, 27, 200, 11, 144, 13, 56, 225, 64, 17, 216, 19, 400, 63, 88, 23, 288, 125, 104, 81, 112, 29, 1800, 31, 128, 99, 136, 175, 432, 37, 152, 117, 800, 41, 504, 43, 176, 675, 184, 47, 576, 49, 1000, 153, 208, 53, 648, 275, 224, 171, 232, 59, 3600
Offset: 1

Views

Author

Jianing Song, Jul 03 2025

Keywords

Examples

			For N == 0, 1, ..., 71 (mod 72), binomial(N,5) == {0, 0, 0, 0, 0, 1, 0, 3, 2, 0, 0, 0, 0, 3, 4, 3, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 2, 0, 0, 3, 0, 3, 4, 0, 0, 2, 0, 3, 0, 3, 0, 4, 0, 0, 2, 3, 0, 3, 0, 0, 4, 0, 0, 5, 0, 3, 0, 0, 0, 4, 0, 3, 2, 3, 0, 0, 0, 0, 4, 3, 0, 5} (mod 6), and binomial(N,6) == {0, 0, 0, 0, 0, 0, 1, 1, 4, 0, 0, 0, 0, 0, 3, 1, 4, 4, 0, 0, 0, 0, 3, 3, 4, 4, 4, 0, 0, 0, 3, 3, 0, 4, 4, 4, 0, 0, 3, 3, 0, 0, 4, 4, 4, 0, 3, 3, 0, 0, 0, 4, 4, 4, 3, 3, 0, 0, 0, 0, 4, 4, 1, 3, 0, 0, 0, 0, 0, 4, 1, 1} (mod 6).

Links

Crossrefs

Rows n = 5 and 6 of A349593. A022998, A385555, A385556, A385558, A385559, and A385560 are respectively rows 2, 3, 4, 7, 8, and 9-10.

Programs

  • Mathematica
    A385557[n_] := If[n == 1, 1, n*Product[p^Floor[Log[p, 5]], {p, FactorInteger[n][[All, 1]]}]];
Array[A385557, 100] (* Paolo Xausa, Jul 07 2025 *)
a[n_] := n * GCD[n, 30] * (2 - Mod[n, 2]); Array[a, 100] (* Amiram Eldar, Jul 07 2025 *)
  • PARI
    a(n, {choices=5}) = my(r=1, f=factor(n)); for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]); r *= p^(logint(choices, p)+e)); return(r)

Formula

Multiplicative with a(2^e) = 2^(e+2), a(3^e) = 3^(e+1), a(5^e) = 5^(e+1), and a(p^e) = p^e for primes p >= 7.
From Amiram Eldar, Jul 07 2025: (Start)
a(n) = n * gcd(30, n) * (2 - (n mod 2)).
Dirichlet g.f.: zeta(s-1) * (1 + 3/2*(s-1)) * (1 + 2/3*(s-1)) * (1 + 4/5*(s-1)).
Sum_{k=1..n} a(k) ~ (15/4) * n^2. (End)

A385558 Period of {binomial(N,7) mod n: N in Z}.

Original entry on oeis.org

1, 8, 9, 16, 25, 72, 49, 32, 27, 200, 11, 144, 13, 392, 225, 64, 17, 216, 19, 400, 441, 88, 23, 288, 125, 104, 81, 784, 29, 1800, 31, 128, 99, 136, 1225, 432, 37, 152, 117, 800, 41, 3528, 43, 176, 675, 184, 47, 576, 343, 1000, 153, 208, 53, 648, 275, 1568, 171, 232, 59, 3600
Offset: 1

Views

Author

Jianing Song, Jul 03 2025

Keywords

Examples

			For N == 0, 1, ..., 48 (mod 49), binomial(N,7) == {0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6} (mod 7).

Links

Crossrefs

Row n = 7 of A349593. A022998, A385555, A385556, A385557, A385559, and A385560 are respectively rows 2, 3, 4, 5-6, 8, and 9-10.

Programs

  • Mathematica
    A385558[n_] := If[n == 1, 1, n*Product[p^Floor[Log[p, 7]], {p, FactorInteger[n][[All, 1]]}]];
Array[A385558, 100] (* Paolo Xausa, Jul 07 2025 *)
a[n_] := n * GCD[n, 210] * (2 - Mod[n, 2]); Array[a, 100] (* Amiram Eldar, Jul 07 2025 *)
  • PARI
    a(n, {choices=7}) = my(r=1, f=factor(n)); for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]); r *= p^(logint(choices, p)+e)); return(r)

Formula

Multiplicative with a(2^e) = 2^(e+2), a(3^e) = 3^(e+1), a(5^e) = 5^(e+1), a(7^e) = 7^(e+1), and a(p^e) = p^e for primes p >= 11.
From Amiram Eldar, Jul 07 2025: (Start)
a(n) = n * gcd(210, n) * (2 - (n mod 2)).
Dirichlet g.f.: zeta(s-1) * (1 + 3/2*(s-1)) * (1 + 2/3*(s-1)) * (1 + 4/5*(s-1)) * (1 + 6/7*(s-1)).
Sum_{k=1..n} a(k) ~ (195/28) * n^2. (End)

A385559 Period of {binomial(N,8) mod n: N in Z}.

Original entry on oeis.org

1, 16, 9, 32, 25, 144, 49, 64, 27, 400, 11, 288, 13, 784, 225, 128, 17, 432, 19, 800, 441, 176, 23, 576, 125, 208, 81, 1568, 29, 3600, 31, 256, 99, 272, 1225, 864, 37, 304, 117, 1600, 41, 7056, 43, 352, 675, 368, 47, 1152, 343, 2000, 153, 416, 53, 1296, 275, 3136, 171, 464, 59, 7200
Offset: 1

Views

Author

Jianing Song, Jul 03 2025

Keywords

Examples

			For N == 0, 1, ..., 63 (mod 64), binomial(N,8) == {0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 5, 7, 7, 3, 3, 6, 6, 6, 6, 2, 2, 2, 2, 7, 7, 3, 3, 1, 1, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 1, 1, 3, 3, 7, 7, 2, 2, 2, 2, 6, 6, 6, 6, 3, 3, 7, 7, 5, 5, 1, 1} (mod 8).

Links

Crossrefs

Row n = 8 of A349593. A022998, A385555, A385556, A385557, A385558, and A385560 are respectively rows 2, 3, 4, 5-6, 7, and 9-10.

Programs

  • Mathematica
    A385559[n_] := If[n == 1, 1, n*Product[p^Floor[Log[p, 8]], {p, FactorInteger[n][[All, 1]]}]];
Array[A385559, 100] (* Paolo Xausa, Jul 07 2025 *)
a[n_] := n * GCD[n, 210] * (4 - 3 * Mod[n, 2]); Array[a, 100] (* Amiram Eldar, Jul 07 2025 *)
  • PARI
    a(n, {choices=8}) = my(r=1, f=factor(n)); for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]); r *= p^(logint(choices, p)+e)); return(r)

Formula

Multiplicative with a(2^e) = 2^(e+3), a(3^e) = 3^(e+1), a(5^e) = 5^(e+1), a(7^e) = 7^(e+1), and a(p^e) = p^e for primes p >= 11.
From Amiram Eldar, Jul 07 2025: (Start)
a(n) = n * gcd(210, n) * (4 - 3 * (n mod 2)).
Dirichlet g.f.: zeta(s-1) * (1 + 7/2*(s-1)) * (1 + 2/3*(s-1)) * (1 + 4/5*(s-1)) * (1 + 6/7*(s-1)).
Sum_{k=1..n} a(k) ~ (351/28) * n^2. (End)

A385560 Period of {binomial(N,9) mod n: N in Z}. Also, period of {binomial(N,10) mod n: N in Z}.

Original entry on oeis.org

1, 16, 27, 32, 25, 432, 49, 64, 81, 400, 11, 864, 13, 784, 675, 128, 17, 1296, 19, 800, 1323, 176, 23, 1728, 125, 208, 243, 1568, 29, 10800, 31, 256, 297, 272, 1225, 2592, 37, 304, 351, 1600, 41, 21168, 43, 352, 2025, 368, 47, 3456, 343, 2000, 459, 416, 53, 3888, 275, 3136, 513, 464, 59, 21600
Offset: 1

Views

Author

Jianing Song, Jul 03 2025

Keywords

Examples

			For N == 0, 1, ..., 80 (mod 81), binomial(N,9) == {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 4, 4, 4, 1, 1, 1, 2, 2, 2, 8, 8, 8, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 7, 7, 7, 4, 4, 4, 5, 5, 5, 2, 2, 2, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 1, 1, 1, 7, 7, 7, 8, 8, 8, 5, 5, 5, 8, 8, 8} (mod 9), and binomial(N,10) == {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 7, 2, 6, 7, 8, 0, 2, 4, 6, 5, 4, 3, 5, 7, 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 4, 8, 3, 1, 8, 6, 1, 5, 0, 5, 1, 6, 8, 1, 3, 8, 4, 0, 6, 3, 0, 6, 3, 0, 6, 3, 0, 7, 5, 3, 4, 5, 6, 4, 2, 0, 8, 7, 6, 2, 7, 3, 2, 1} (mod 9).

Links

Crossrefs

Rows n = 9 and 10 of A349593. A022998, A385555, A385556, A385557, A385558, and A385559 are respectively rows 2, 3, 4, 5-6, 7, and 8.

Programs

  • Mathematica
    A385560[n_] := If[n == 1, 1, n*Product[p^Floor[Log[p, 9]], {p, FactorInteger[n][[All, 1]]}]];
Array[A385560, 100] (* Paolo Xausa, Jul 07 2025 *)
a[n_] := n * GCD[n, 6] * GCD[n, 210] * (2 - Mod[n, 2]); Array[a, 100] (* Amiram Eldar, Jul 07 2025 *)
  • PARI
    a(n, {choices=10}) = my(r=1, f=factor(n)); for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]); r *= p^(logint(choices, p)+e)); return(r)

Formula

Multiplicative with a(2^e) = 2^(e+3), a(3^e) = 3^(e+2), a(5^e) = 5^(e+1), a(7^e) = 7^(e+1), and a(p^e) = p^e for primes p >= 11.
From Amiram Eldar, Jul 07 2025: (Start)
a(n) = n * gcd(6, n) * gcd(210, n) * (2 - (n mod 2)).
Dirichlet g.f.: zeta(s-1) * (1 + 7/2*(s-1)) * (1 + 8/3*(s-1)) * (1 + 4/5*(s-1)) * (1 + 6/7*(s-1)).
Sum_{k=1..n} a(k) ~ (3861/140) * n^2. (End)

A380482 a(n) is the multiplicative order of -3 modulo prime(n); a(2) = 0 for completion.

Original entry on oeis.org

1, 0, 4, 3, 10, 6, 16, 9, 22, 28, 15, 9, 8, 21, 46, 52, 58, 5, 11, 70, 12, 39, 82, 88, 48, 100, 17, 106, 54, 112, 63, 130, 136, 69, 148, 25, 39, 81, 166, 172, 178, 90, 190, 16, 196, 99, 105, 111, 226, 114, 232, 238, 120, 250, 256, 262, 268, 15, 138, 280
Offset: 1

Views

Author

Jianing Song, Jun 27 2025

Keywords

Links

Crossrefs

Cf. A105875 (primes having primitive root -3).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), this sequence, A380531, A380532, A380533, A380540, A380541, A380542, A385222.

Programs

  • Mathematica
    A380482[n_] := If[n == 2, 0, MultiplicativeOrder[-3, Prime[n]]];
Array[A380482, 100] (* Paolo Xausa, Jun 29 2025 *)
  • PARI
    a(n,{k=-3}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

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