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: Mihir Mathur

Mihir Mathur's wiki page.

Mihir Mathur has authored 10 sequences.

A235697 Harshad numbers which when divided by their digital sum, give a quotient which is a Multiple Harshad Numbers-2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100, 108, 120, 162, 180, 200, 210, 216, 240, 243, 270, 300, 324, 360, 378, 400, 405, 420, 432, 450, 480, 486, 500, 540, 600, 630, 648, 700, 720, 756, 800, 810, 840, 864, 900, 972, 1000, 1080, 1200, 1296, 1458
Offset: 1

Views

Author

Mihir Mathur, Jan 14 2014

Keywords

Comments

These numbers are also called MHN-3 or Multiple Harshad Numbers-3.
A Multiple Harshad number is a Harshad number that, when divided by the sum of its digits, produces another Harshad number.
Starts to differ from A235507 at position n=70. - R. J. Mathar, Jun 13 2024

Examples

			756 is a term as it gives quotient 42 on division by the digital sum (i.e. 18). 42 gives quotient 7 on division by its digital sum (i.e. 6). As 7 is also a Harshad Number, thus 756 is a MHN-3.

Links

Crossrefs

Cf. A005349, A235507.

A235591 Numbers which on division by their digital root give quotient which is divisible by its digital root.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 19, 20, 21, 24, 27, 28, 30, 36, 37, 38, 40, 42, 45, 46, 50, 54, 55, 56, 57, 60, 63, 64, 70, 72, 73, 74, 76, 80, 81, 82, 84, 90, 91, 92, 95, 100, 108, 109, 110, 111, 112, 114, 118, 120, 127, 128, 133, 136, 138, 140, 145, 146, 148, 152, 154, 162, 163, 164, 165, 168, 171, 172
Offset: 1

Views

Author

Mihir Mathur, Jan 12 2014

Keywords

Examples

			 108 is a term as on division by its digital root i.e. 9, gives 12, which is again divisible by its digital root i.e. 3.

Programs

  • Mathematica
    dr[n_] := 1 + Mod[n-1, 9]; Select[Range@172, IntegerQ[q = #/dr@#] && Mod[q, dr@q] == 0 &] (* Giovanni Resta, Jan 14 2014 *)

A235507 Harshad numbers which when divided by sum of their digits, give a quotient which is a Harshad number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100, 108, 120, 162, 180, 200, 210, 216, 240, 243, 270, 300, 324, 360, 378, 400, 405, 420, 432, 450, 480, 486, 500, 540, 600, 630, 648, 700, 720
Offset: 1

Views

Author

Mihir Mathur, Jan 14 2014

Keywords

Comments

These numbers are also called MHN-2 or Multiple Harshad Numbers-2.

Examples

			486 is a MHN as it is divisible by the sum of its digits i.e. 18. The quotient obtained, 27, is also divisible by the sum of its digits, i.e. 9.

Links

Crossrefs

Cf. A005349, A235697.

Programs

  • Mathematica
    mhnQ[n_]:=Module[{s=Total[IntegerDigits[n]]},Divisible[n,s]&&Divisible[ n/s,Total[IntegerDigits[n/s]]]]; Select[Range[800],mhnQ] (* Harvey P. Dale, Sep 02 2017 *)

A234814 Numbers that are divisible by their digital sum but not by their digital root.

Original entry on oeis.org

195, 209, 247, 266, 285, 375, 392, 407, 465, 476, 481, 518, 555, 592, 605, 629, 644, 645, 715, 735, 736, 782, 803, 825, 880, 915, 935, 1066, 1095, 1148, 1168, 1183, 1185, 1274, 1275, 1365, 1394, 1417, 1455, 1526, 1534, 1545, 1547, 1635, 1651, 1652, 1679, 1725, 1744, 1815, 1853, 1886, 1898, 1904, 1905
Offset: 1

Views

Author

Mihir Mathur, Dec 31 2013

Keywords

Comments

These are the Harshad numbers which are missing from A234474.

Examples

			195 is a term as it is divisible by its digital sum i.e. 15 but not by its digital root i.e. 6.

Links

Crossrefs

Cf. A005349, A234474.
Cf. A064807, A007953, A010888.

Programs

  • Haskell
    a234814 n = a234814_list !! (n-1)
a234814_list = filter (\x -> x `mod` a007953 x == 0 &&
                             x `mod` a010888 x /= 0) [1..]
-- Reinhard Zumkeller, Mar 04 2014
  • Mathematica
    Select[Range@1905, Mod[#, 1 + Mod[#-1, 9]] > 0 && Mod[#, Plus@@ IntegerDigits@ #] == 0 &] (* Giovanni Resta, Jan 03 2014 *)

A234474 Numbers that are divisible by their digital sum and digital root.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100, 102, 108, 110, 111, 112, 114, 117, 120, 126, 132, 133, 135, 140, 144, 150, 152, 153, 156, 162, 171, 180, 190, 192, 198, 200, 201, 204
Offset: 1

Views

Author

Mihir Mathur, Dec 26 2013

Keywords

Comments

Though a large number of initial terms match, it is different from A005349. First missing term is A005349(57) = 195.

Examples

			198 is a term of the sequence as it is divisible by its digital sum i.e., 18 and by its digital root i.e., 9.

Links

Crossrefs

Intersection of A064807 and A005349.
Cf. A010888, A007953.

Programs

  • Mathematica
    Select[Range[200],Divisible[#,Total[IntegerDigits[#]] &&  Divisible[#, #-9*Floor[(#-1)/9]]]&] (* Indranil Ghosh, Mar 04 2017 *)
  • PARI
    is(n)=my(d=sumdigits(n)); n%d==0 && n%((d-1)%9+1)==0 \\ Charles R Greathouse IV, Dec 26 2013

A226485 Integer part of length of median to hypotenuse of primitive Pythagorean triangles sorted on hypotenuse.

Original entry on oeis.org

2, 6, 8, 12, 14, 18, 20, 26, 30, 32, 32, 36, 42, 42, 44, 48, 50, 54, 56, 62, 68, 72, 72, 74, 78, 84, 86, 90, 92, 92, 96, 98, 102, 102, 110, 110, 114, 116, 120, 128, 132, 132, 134, 138, 140, 144, 146, 152, 152, 156, 158, 162, 162, 168, 174, 176, 182, 182
Offset: 1

Views

Author

Mihir Mathur, Jun 09 2013

Keywords

Comments

The median to hypotenuse is equal to the circumradius.
The length of the median is sqrt((a^2)/2 + (b^2)/2 - (c^2)/4) where a,b,c are sides of the triangle. In case of Pythagorean triangles, m=h/2 were h is the hypotenuse.

Examples

			a(1)=2 as it is the integer portion of the length of the median to hypotenuse of triangle having sides 3,4,5.
Similarly, a(5)=14 as it is the integer portion of the length of the median to hypotenuse of triangle having sides 20,21,29.

Links

Crossrefs

Cf. A020882.

Formula

a(n) = floor(A020882(n)/2).

A224317 a(n) = a(n-1) + 3 - a(n-1)!.

Original entry on oeis.org

1, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2
Offset: 1

Views

Author

Mihir Mathur, Apr 03 2013

Keywords

Comments

a(0) is taken as 1. The sequence remains essentially the same for a(0)=0,1,2,3.
Even though each term is dependent on the previous term, the sequence is repetitive.

Examples

			For n=1, a(1) = a(1-1) + 3 - a(1-1)! = 1 + 3 - 1 = 3.
For n=2, a(2) = a(2-1) + 3 - a(2-1)! = 3 + 3 - 6 = 0.
For n=3, a(3) = a(3-1) + 3 - a(3-1)! = 0 + 3 - 1 = 2.

Links

Programs

Formula

a(n) = a(n-1) + 3 - a(n-1)!.
For n > 1, a(n) = (5+4*cos(2*(n+1)*Pi/3) - 2*sqrt(3)*sin(2*(n+1)*Pi/3))/3. - Wesley Ivan Hurt, Sep 27 2017
a(n) = (floor((4*n - 1)/3) + signum(n - 1)) mod 4. - Lechoslaw Ratajczak, Jul 01 2023

A224302 Sorted perimeters of primitive Heronian triangles.

Original entry on oeis.org

12, 16, 18, 30, 32, 36, 36, 40, 42, 42, 44, 48, 50, 50, 54, 54, 54, 56, 60, 64, 64, 64, 66, 68, 70, 70, 72, 76, 78, 80, 80, 84, 84, 84, 84, 84, 90, 90, 90, 96, 98, 98, 98, 98, 98, 98, 100, 100, 100, 104, 104, 108, 108, 108, 108, 108, 110, 112, 112, 112, 112
Offset: 1

Views

Author

Mihir Mathur, Apr 04 2013

Keywords

Comments

Here a primitive Heronian triangle has integer sides a,b,c with gcd(a,b,c) = 1 and integral area.
The perimeters of primitive Heronian triangles are even [Wenzel Šimerka, 1869]. - Mo Li, Feb 02 2020

Examples

			a(1) = 12 as it is the perimeter of the Heronian triangle having sides 3,4,5 and is the smallest Heronian triangle perimeter.
a(2) = 16 as it is the perimeter of the Heronian triangle having sides 5,5,6 and is the 2nd smallest Heronian triangle perimeter.

References

  • L. E. Dickson, History of the Theory of Numbers, vol. II: Diophantine Analysis, Dover, 2005, p. 196. [21a]

Links

Crossrefs

Cf. A120131, A120132, A120133, A224301.

Programs

  • Mathematica
    hQ[a_, b_, c_] := IntegerQ@Sqrt@Block[{s = (a + b + c)/2}, s (s - a) (s - b) (s - c)];
Sort[Reap[Do[If[GCD[a, b, c] == 1 && hQ[a, b, c], Sow@(a + b + c)], {a, 100}, {b, a}, {c, a - b + 1, b}]][[2, 1]]] (* The last numbers given may not be exactly in the right place. *) (* Jinyuan Wang, Feb 02 2020 *)

Extensions

Corrected and extended by Giovanni Resta, Apr 04 2013

A223857 Ordered products of the perimeter and the sides of primitive Pythagorean triangles.

Original entry on oeis.org

720, 23400, 81600, 235200, 852600, 1305360, 1328400, 5314320, 8414280, 9434880, 16893240, 18498480, 33918720, 43995600, 45561600, 46652760, 57757440, 106226640, 108617760, 154736400, 155263680, 184041000, 235227600, 361712400, 417740400, 451760400, 471711240
Offset: 1

Views

Author

Mihir Mathur, Apr 02 2013

Keywords

Comments

Considering the set of primitive Pythagorean triangles with sides (A, B, C), the sequence gives the values (A+B+C)*(A*B*C), in increasing order.
It is a challenge to find a pair of primitive Pythagorean triangles such that product of perimeter and the sides is equal.

Examples

			a(1) = (3+4+5)*(3*4*5) = 720.
a(2) = (5+12+13)*(5*12*13) = 23400.

Links

Crossrefs

Cf. A063011, A024364.

Extensions

Corrected and extended by Giovanni Resta, Apr 03 2013

A224301 Sorted areas of primitive integer Heronian triangles.

Original entry on oeis.org

6, 12, 12, 24, 30, 36, 36, 42, 60, 60, 60, 60, 66, 72, 84, 84, 84, 84, 90, 90, 114, 120, 120, 120, 126, 126, 126, 132, 156, 156, 168, 168, 168, 180, 180, 198, 204, 210, 210, 210, 210, 210, 210, 216, 234, 240, 252, 252, 252, 264, 264, 270, 288, 300, 300, 306
Offset: 1

Views

Author

Mihir Mathur, Apr 03 2013

Keywords

Comments

The sequence gives the sorted areas of primitive triangles which have integer side lengths and integer areas.

Examples

			The smallest Heronian triangle is (3,4,5) as perimeter and area are integers. The first term of the sequence is thus the area of this triangle, which is 6.

Links

Crossrefs

Cf. A120131, A120132, A120133, A072294, A188158.

Programs

  • Mathematica
    AMax=400;
Do[
  c=p/b;
  a1=Sqrt[b^2+c^2+2Sqrt[b^2c^2-4A^2]];
  a2=Sqrt[b^2+c^2-2Sqrt[b^2c^2-4A^2]];
  If[IntegerQ[a2]&&GCD[a2,b,c]==1&&a1>a2>=b,A//Sow(*{A,a2,b,c}//Sow*)];
  If[IntegerQ[a1]&&GCD[a1,b,c]==1,A//Sow(*{A,a1,b,c}//Sow*)];
  ,{A,6,AMax,6}
  ,{p,4A^2//Divisors//Select[#,EvenQ[#]&&#>=2A&]&//#/2+2A^2/#&//Select[#,IntegerQ]&}
  ,{b,p//Divisors//Select[#,#^2>=p&]&}
]//Reap//Last//Last
{a1,a2,c}=.;
(* Albert Lau, May 20 2016 *)

Extensions

Definition corrected by Giovanni Resta, Apr 03 2013
More terms from Giovanni Resta, Apr 03 2013

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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