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 1-9 of 9 results.

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

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

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 0, 0, 8, 0, 6, 0, 11, 3, 0, 0, 16, 0, 13, 6, 19, 0, 15, 0, 24, 0, 35, 0, 9, 0, 0, 9, 32, 10, 48, 0, 35, 12, 45, 0, 16, 0, 38, 23, 43, 0, 30, 0, 48, 15, 45, 0, 51, 30, 42, 18, 56, 0, 41, 0, 59, 21, 0, 23, 49, 0, 96, 21, 52, 0, 57, 0, 72, 24, 70, 39, 60, 0, 60, 0, 80, 0, 36, 30, 83, 27, 118, 0, 61, 59, 131
Offset: 1

Views

Author

Antti Karttunen, Jun 03 2021

Keywords

Comments

Apparently A000961 gives the positions of zeros.

Links

Crossrefs

Cf. A000961, A011772, A344875, A344969, A344973, A344976.
Cf. also A344763, A344765, A344874.

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_] := If[n == 1, 0, A344875[n] - A011772[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)); };
A344876(n) = (A344875(n)-A011772(n));

Formula

a(n) = A344875(n) - A011772(n).
a(n) >= A344976(n).

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

A344975 Numbers k such that A011772(k) divides A344875(k), but k is not a power of prime (in A000961).

Original entry on oeis.org

6, 10, 18, 21, 24, 26, 28, 34, 36, 39, 40, 50, 55, 57, 58, 68, 74, 75, 78, 82, 93, 96, 98, 100, 106, 111, 120, 122, 129, 136, 146, 147, 150, 155, 162, 164, 171, 178, 183, 194, 196, 201, 202, 203, 205, 218, 219, 222, 224, 226, 237, 242, 250, 253, 274, 288, 291, 292, 294, 298, 300, 301, 305, 309, 314, 324, 327, 333
Offset: 1

Views

Author

Antti Karttunen, Jun 04 2021

Keywords

Comments

This is not a subsequence of A344882. The first terms of this sequence that do not occur there are: 900, 1260, 1560, ... etc, see A344694. First terms of A344882 not present here are: 60, 66, 88, 92, 105, etc.

Crossrefs

Cf. A000961, A011772, A344875, A344882, A344973.
Intersection of A024619 and A344974.
Cf. also A344595, A344694, A344978 (subsequences).

Programs

A344595 Numbers k such that A011772(k) > A344878(k) and A011772(k) is a divisor of A344875(k).

Original entry on oeis.org

900, 1260, 1302, 1560, 2100, 3906, 4440, 6300, 6552, 6669, 9680, 11544, 12987, 15368, 18981, 19240, 19880, 24120, 26208, 35784, 36080, 42680, 46104, 57720, 59040, 59640, 62238, 62244, 71136, 74592, 76840, 79376, 81872, 84700, 101680, 103730, 108500, 124488, 128040, 145188, 160160, 168020, 171740, 178920, 185724, 201608
Offset: 1

Views

Author

Antti Karttunen, Jun 05 2021

Keywords

Comments

Numbers k for which A344973(k) = 0 and A344976(k) < 0.
It seems that in these cases, by necessity A011772(k) < A344875(k), i.e., A011772(k) is a proper divisor of A344875(k).
Has many terms common with A344694.

Crossrefs

Cf. A011772, A344694, A344875, A344878, A344973, A344976.
Intersection of A024619, A344974 and A344977.
Intersection of A344975 and A344977.

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)); };
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))));
isA344595(n) = { my(u=A011772(n)); (u>A344878(n)&&0==(A344875(n)%u)); };

A344694 Numbers k, not powers of primes, for which A011772(k) divides A344875(k), and for all proper divisors d of k, A011772(d) < A011772(k).

Original entry on oeis.org

900, 1260, 1560, 3740, 6552, 6669, 9680, 18981, 19880, 35784, 36080, 59040, 62238, 62244, 81872, 103730, 108500, 118910, 134420, 160160, 171740, 185724, 211072, 217833, 222224, 225929, 227528, 259325, 351072, 384944, 404294, 414778, 422604, 425178, 446600, 456228, 463008, 488205, 490105, 527100, 574308, 581184, 598400
Offset: 1

Views

Author

Antti Karttunen, Jun 05 2021

Keywords

Comments

Has many terms common with A344595.

Crossrefs

Cf. A011772, A344880, A344973.
Intersection of A024619, A344881 and A344974. Intersection of A344881 and A344975.
Cf. also A344595, A344980.

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)); };
A344880(n) = { my(x=A011772(n)); fordiv(n, d, if(A011772(d)==x, return(d==n))); };
A344973(n) = (A344875(n)%A011772(n));
isA344694(n) = ((n>1)&&!isprimepower(n)&&(0==A344973(n))&&A344880(n));

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 1-9 of 9 results.

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