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 1-7 of 7 results.

A344973 a(n) = A344875(n) mod A011772(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 4, 3, 0, 0, 0, 0, 13, 0, 8, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 9, 0, 10, 0, 0, 16, 0, 0, 0, 16, 0, 6, 5, 20, 0, 30, 0, 0, 15, 6, 0, 24, 0, 42, 0, 0, 0, 11, 0, 28, 21, 0, 23, 5, 0, 0, 21, 12, 0, 57, 0, 0, 0, 14, 18, 0, 0, 60, 0, 0, 0, 36, 30, 40, 27, 22, 0, 26, 7, 16, 0, 44, 15, 0, 0, 0, 36
Offset: 1

Views

Author

Antti Karttunen, Jun 04 2021

Keywords

Links

Crossrefs

Cf. A011772, A344875, A344876, A344969, A344970, A344971, A344972.
Cf. A344974 (positions of zeros).

Programs

  • Mathematica
    b[n_] := If[n == 1, 1, Module[{p, e}, Product[{p, e} = pe;
     If[p == 2, 2^(1 + e) - 1, p^e - 1], {pe, FactorInteger[n]}]]];
c[n_] := Module[{m = 1}, While[Not[IntegerQ[m (m + 1)/(2 n)]], m++]; m];
a[n_] := Mod[b[n], c[n]];
Array[a, 100] (* Jean-François Alcover, Jun 12 2021 *)
  • PARI
    A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
A344973(n) = (A344875(n)%A011772(n));

Formula

a(n) = A344875(n) mod A011772(n) = A344876(n) mod A011772(n).

A344969 a(n) = gcd(A011772(n), A344875(n)).

Original entry on oeis.org

1, 3, 2, 7, 4, 3, 6, 15, 8, 4, 10, 2, 12, 1, 1, 31, 16, 8, 18, 1, 6, 1, 22, 15, 24, 12, 26, 7, 28, 3, 30, 63, 1, 16, 2, 8, 36, 1, 12, 15, 40, 4, 42, 2, 1, 1, 46, 2, 48, 24, 1, 3, 52, 3, 10, 6, 18, 28, 58, 1, 60, 1, 3, 127, 1, 1, 66, 16, 1, 4, 70, 3, 72, 36, 24, 14, 3, 12, 78, 4, 80, 40, 82, 12, 2, 1, 1, 2, 88, 1, 1, 1, 30
Offset: 1

Views

Author

Antti Karttunen, Jun 03 2021

Keywords

Links

Crossrefs

Cf. A011772, A344875, A344876, A344970, A344971, A344972, A344973.

Programs

  • Mathematica
    A011772[n_] := Module[{m = 1}, While[Not[IntegerQ[m(m+1)/(2n)]], m++]; m];
A344875[n_] := Product[{p, e} = pe; If[p == 2, 2^(1+e)-1, p^e-1], {pe, FactorInteger[n]}];
a[n_] := GCD[A011772[n], A344875[n]];
Array[a, 100] (* Jean-François Alcover, Jun 12 2021 *)
  • PARI
    A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
A344969(n) = gcd(A011772(n), A344875(n));

Formula

a(n) = gcd(A011772(n), A344875(n)).
a(n) = gcd(A011772(n), A344876(n)) = gcd(A344875(n), A344876(n)) = gcd(A011772(n), A344973(n)).

A344974 Numbers k such that A011772(k) divides A344875(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 39, 40, 41, 43, 47, 49, 50, 53, 55, 57, 58, 59, 61, 64, 67, 68, 71, 73, 74, 75, 78, 79, 81, 82, 83, 89, 93, 96, 97, 98, 100, 101, 103, 106, 107, 109, 111, 113, 120, 121, 122, 125, 127, 128, 129, 131, 136, 137
Offset: 1

Views

Author

Antti Karttunen, Jun 04 2021

Keywords

Crossrefs

Cf. A011772, A344875, A344884 (characteristic function).
Positions of ones in A344970, of zeros in A344973.
Union of A000961 and A344975. Complement of A344980.
Cf. also A344595 (subsequence).

Programs

A344971 a(n) = A344875(n) / gcd(A011772(n), A344875(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 7, 1, 18, 8, 1, 1, 3, 1, 28, 2, 30, 1, 2, 1, 3, 1, 6, 1, 8, 1, 1, 20, 3, 12, 7, 1, 54, 2, 4, 1, 9, 1, 35, 32, 66, 1, 31, 1, 3, 32, 28, 1, 26, 4, 15, 2, 3, 1, 56, 1, 90, 16, 1, 48, 60, 1, 7, 44, 18, 1, 40, 1, 3, 2, 9, 20, 6, 1, 31, 1, 3, 1, 7, 32, 126, 56, 75, 1, 96, 72, 154, 2, 138, 72
Offset: 1

Views

Author

Antti Karttunen, Jun 04 2021

Keywords

Comments

Numerator of the ratio A344875(n)/A011772(n): 1/1, 3/3, 2/2, 7/7, 4/4, 6/3, 6/6, 15/15, 8/8, 12/4, 10/10, 14/8, 12/12, 18/7, 8/5, 31/31, 16/16, 24/8, 18/18, 28/15, 12/6, 30/11, ... = 1/1, 1/1, 1/1, 1/1, 1/1, 2/1, 1/1, 1/1, 1/1, 3/1, 1/1, 7/4, 1/1, 18/7, 8/5, 1/1, 1/1, 3/1, 1/1, 28/15, 2/1, 30/11, etc.

Links

Crossrefs

Cf. A011772, A344875, A344969, A344970 (denominators), A344972 (ratio A344875/A011772 floored down), A344973 (and their remainder).

Programs

  • PARI
    A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
A344971(n) = { my(u=A344875(n)); (u/gcd(u, A011772(n))); };

Formula

a(n) = A344875(n) / A344969(n) = A344875(n) / gcd(A011772(n), A344875(n)).

A344972 a(n) = floor(A344875(n) / A011772(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 2, 1, 3, 1, 6, 1, 1, 1, 1, 1, 3, 1, 7, 1, 2, 2, 4, 1, 1, 1, 2, 3, 2, 1, 1, 1, 3, 1, 2, 1, 2, 4, 1, 2, 3, 1, 3, 1, 2, 1, 1, 1, 5, 1, 7, 1, 3, 1, 1, 1, 3, 2, 2, 2, 6, 1, 1, 1, 3, 1, 1, 1, 2, 1, 4, 1, 2, 5, 6, 2, 2, 3, 2, 1, 3, 1, 7, 1, 1, 1, 2, 3
Offset: 1

Views

Author

Antti Karttunen, Jun 04 2021

Keywords

Links

Crossrefs

Cf. A011772, A344875, A344876, A344969, A344970, A344971, A344973.

Programs

  • PARI
    A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
A344972(n) = floor(A344875(n)/A011772(n));

Formula

a(n) = (A344875(n)-A344973(n)) / A011772(n) = floor(A344875(n) / A011772(n)).
a(n) = floor(A344971(n) / A344970(n)). - Antti Karttunen, Jun 20 2021

A344978 Numbers k such that A011772(k) is equal to A344878(k), but k is not a power of prime.

Original entry on oeis.org

21, 26, 39, 50, 57, 74, 75, 78, 93, 98, 111, 122, 129, 146, 147, 150, 183, 194, 201, 205, 218, 219, 222, 237, 242, 291, 294, 301, 305, 309, 314, 327, 338, 362, 363, 366, 381, 386, 405, 417, 438, 452, 453, 458, 471, 482, 489, 505, 507, 543, 554, 578, 579, 582, 597, 605, 610, 626, 633, 654, 657, 669, 674, 676, 687
Offset: 1

Views

Author

Antti Karttunen, Jun 04 2021

Keywords

Comments

Numbers k for which A344976(k) = 0, and A001221(k) > 1.

Crossrefs

Cf. A000961, A001221, A011772, A344878, A344970, A344976.
Subsequence of A344975. Intersection of A024619 and A344979.

Programs

  • PARI
    A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A344878(n) = if(1==n,n, my(f=factor(n)~); lcm(vector(#f, i, (f[1, i]^(f[2, i]+(2==f[1, i]))-1))));
isA344978(n) = (omega(n)>1&&(A344878(n)==A011772(n)));

Formula

For all n, A344970(a(n)) = 1.

A344980 Numbers k such that A011772(k) does not divide A344875(k).

Original entry on oeis.org

12, 14, 15, 20, 22, 30, 33, 35, 38, 42, 44, 45, 46, 48, 51, 52, 54, 56, 60, 62, 63, 65, 66, 69, 70, 72, 76, 77, 80, 84, 85, 86, 87, 88, 90, 91, 92, 94, 95, 99, 102, 104, 105, 108, 110, 112, 114, 115, 116, 117, 118, 119, 123, 124, 126, 130, 132, 133, 134, 135, 138, 140, 141, 142, 143, 144, 145, 148, 152, 153, 154, 156
Offset: 1

Views

Author

Antti Karttunen, Jun 05 2021

Keywords

Comments

The first term not in A344883 is 60. First terms included in A344883, but not here are: 900, 1260, 1560, 3740, 6552, 6669, etc. (A344694). See also comments in A344975.

Crossrefs

Cf. A011772, A344694, A344875, A344975.
Complement of A344974. Positions of nonzero terms in A344973, and of terms > 1 in A344970.

Programs

  • PARI
    A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
isA344980(n) = (A344875(n)%A011772(n));
Showing 1-7 of 7 results.

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