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.

Previous Showing 11-19 of 19 results.

A324934 Inverse permutation to A324931.

Original entry on oeis.org

1, 2, 4, 3, 10, 6, 9, 5, 12, 15, 35, 8, 24, 14, 26, 7, 41, 17, 23, 20, 25, 47, 52, 13, 58, 34, 28, 19, 79, 37, 184, 11, 87, 61, 53, 22, 56, 33, 60, 30, 145, 36, 92, 70, 65, 75, 164, 18, 51, 82, 98, 46, 54, 39, 178, 29, 59, 106, 293, 49, 122, 245, 63, 16, 125
Offset: 1

Views

Author

Gus Wiseman, Mar 21 2019

Keywords

Crossrefs

Cf. A003963, A120383, A196050, A317713, A324846, A324849, A324850, A324922, A324923, A324924, A324925, A324931.

A324856 Numbers divisible by exactly one of their prime indices.

Original entry on oeis.org

2, 10, 14, 15, 22, 26, 34, 38, 45, 46, 50, 55, 58, 62, 70, 74, 82, 86, 94, 98, 105, 106, 118, 119, 122, 130, 134, 135, 142, 146, 154, 158, 166, 170, 178, 182, 190, 194, 195, 202, 206, 207, 214, 218, 226, 230, 242, 250, 254, 255, 262, 266, 274, 275, 278, 285
Offset: 1

Views

Author

Gus Wiseman, Mar 21 2019

Keywords

Comments

Numbers n such that A324848(n) = 1.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
If k is in A324846, then k*prime(k) is in the sequence. - Robert Israel, Mar 22 2019

Examples

			The sequence of terms together with their prime indices begins:
   2: {1}
  10: {1,3}
  14: {1,4}
  15: {2,3}
  22: {1,5}
  26: {1,6}
  34: {1,7}
  38: {1,8}
  45: {2,2,3}
  46: {1,9}
  50: {1,3,3}
  55: {3,5}
  58: {1,10}
  62: {1,11}
  70: {1,3,4}
  74: {1,12}
  82: {1,13}
  86: {1,14}
  94: {1,15}
  98: {1,4,4}

Links

Crossrefs

Cf. A000720, A003963, A112798, A120383, A323440, A324694, A324704, A324846, A324847, A324848, A324849, A324850, A324926, A324929.

Programs

  • Maple
    filter:= proc(n) local F;
  F:= select(t -> n mod numtheory:-pi(t[1])=0, ifactors(n)[2]);
  nops(F)=1 and F[1][2]=1
end proc:
select(filter, [$2..1000]); # Robert Israel, Mar 22 2019
  • Mathematica
    Select[Range[100],Total[Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k/;Divisible[#,PrimePi[p]]]]==1&]

A324853 First number divisible by n of its own distinct prime indices.

Original entry on oeis.org

1, 2, 6, 30, 330, 4290, 60060, 1021020, 29609580, 917896980, 33962188260, 1290563153880, 52913089309080, 2275262840290440, 106937353493650680, 6309303856125390120, 422723358360401138040, 30013358443588480800840, 2190975166381959098461320
Offset: 0

Views

Author

Gus Wiseman, Mar 18 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
a(n) is the first position of n in A324852.

Examples

			a(6) = 60060 = 2^2 * 3 * 5 * 7 * 11 * 13 has prime indices {1,1,2,3,4,5,6}, and is less than any other number divisible by six of its own distinct prime indices.

Links

Crossrefs

Cf. A000720, A001222, A003963, A056239, A120383, A323440.
Cf. A324771, A324846, A324847, A324848, A324849, A324850, A324852, A324856.

Programs

  • C
    See Links section.
  • Mathematica
    nn=10000;
With[{mgs=Table[Count[If[n==1,{},FactorInteger[n]],{p_,_}/;Divisible[n,PrimePi[p]]],{n,nn}]},Table[Position[mgs,i][[1,1]],{i,0,5}]]
  • PARI
    isok(k,n) = {my(f=factor(k)[,1]); sum(j=1, #f, !(k % primepi(f[j]))) == n;}
a(n) = {my(k=1); while (!isok(k, n), k++); k;} \\ Michel Marcus, Mar 20 2019

Extensions

a(8)-a(9) from Rémy Sigrist, Mar 19 2019
a(10)-a(18) from Michel Lagneau, Aug 19 2019

A323440 Numbers divisible by exactly one of their distinct prime indices.

Original entry on oeis.org

2, 4, 8, 10, 14, 15, 16, 20, 22, 26, 32, 34, 38, 40, 44, 45, 46, 50, 52, 55, 58, 62, 64, 68, 70, 74, 75, 76, 80, 82, 86, 88, 92, 94, 98, 100, 104, 105, 106, 116, 118, 119, 122, 124, 128, 130, 134, 135, 136, 142, 146, 148, 154, 158, 160, 164, 166, 170, 172, 176
Offset: 1

Views

Author

Gus Wiseman, Mar 21 2019

Keywords

Comments

Numbers n such that A324852(n) = 1.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The sequence of terms together with their prime indices begins:
   2: {1}
   4: {1,1}
   8: {1,1,1}
  10: {1,3}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  20: {1,1,3}
  22: {1,5}
  26: {1,6}
  32: {1,1,1,1,1}
  34: {1,7}
  38: {1,8}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}
  46: {1,9}
  50: {1,3,3}
  52: {1,1,6}
  55: {3,5}

Links

Crossrefs

Cf. A000720, A003963, A112798, A120383, A324704, A324846, A324847, A324848, A324849, A324850, A324856, A324926, A324929.

Programs

  • Mathematica
    Select[Range[100],Count[If[#==1,{},FactorInteger[#]],{p_,_}/;Divisible[#,PrimePi[p]]]==1&]
  • PARI
    isok(n) = my(f=factor(n)[,1]); sum(k=1, #f, (n % primepi(f[k])) == 0) == 1; \\ Michel Marcus, Mar 22 2019

A324771 Numbers divisible by at least one of their prime indices > 1.

Original entry on oeis.org

6, 12, 15, 18, 24, 28, 30, 36, 42, 45, 48, 54, 55, 56, 60, 66, 72, 75, 78, 84, 90, 96, 102, 105, 108, 110, 112, 114, 119, 120, 126, 132, 135, 138, 140, 144, 150, 152, 156, 162, 165, 168, 174, 180, 186, 192, 195, 196, 198, 204, 207, 210, 216, 220, 222, 224, 225
Offset: 1

Views

Author

Gus Wiseman, Mar 18 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n.

Examples

			The sequence of terms together with their prime indices begins:
   6: {1,2}
  12: {1,1,2}
  15: {2,3}
  18: {1,2,2}
  24: {1,1,1,2}
  28: {1,1,4}
  30: {1,2,3}
  36: {1,1,2,2}
  42: {1,2,4}
  45: {2,2,3}
  48: {1,1,1,1,2}
  54: {1,2,2,2}
  55: {3,5}
  56: {1,1,1,4}
  60: {1,1,2,3}
  66: {1,2,5}
  72: {1,1,1,2,2}
  75: {2,3,3}
  78: {1,2,6}
  84: {1,1,2,4}

Links

Crossrefs

Complement of A324849.
Cf. A003963, A056239, A112798, A120383, A289509, A290822, A304360.
Cf. A324695, A324846, A324847, A324848, A324850, A324851, A324852, A324853.

Programs

  • Mathematica
    Select[Range[100],Or@@Cases[If[#==1,{},FactorInteger[#]],{p_?(#>2&),_}:>Divisible[#,PrimePi[p]]]&]

A324933 Denominator in the division of n by the product of prime indices of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 4, 3, 5, 1, 6, 2, 2, 1, 7, 2, 8, 3, 8, 5, 9, 1, 9, 3, 8, 1, 10, 1, 11, 1, 10, 7, 12, 1, 12, 4, 4, 3, 13, 4, 14, 5, 4, 9, 15, 1, 16, 9, 14, 3, 16, 4, 3, 1, 16, 5, 17, 1, 18, 11, 16, 1, 18, 5, 19, 7, 6, 6, 20, 1, 21, 6, 6, 2, 20, 2, 22, 3
Offset: 1

Views

Author

Gus Wiseman, Mar 21 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The sequence of quotients n/A003963(n) begins: 1, 2, 3/2, 4, 5/3, 3, 7/4, 8, 9/4, 10/3, 11/5, 6, 13/6, 7/2, 5/2, 16, ...

Links

Crossrefs

The numerators are A324932.
Cf. A003963, A109129, A120383, A196050, A317713, A324846, A324849, A324850, A324922, A324923, A324924, A324925, A324931, A324934.

Programs

  • Mathematica
    Table[n/Times@@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]^k],{n,100}]//Denominator

A324845 Matula-Goebel numbers of rooted trees where the branches of no non-leaf branch of any terminal subtree form a submultiset of the branches of the same subtree.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 14, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 31, 32, 33, 34, 35, 38, 40, 43, 44, 46, 49, 50, 51, 53, 57, 58, 59, 62, 63, 64, 67, 68, 69, 70, 71, 73, 76, 77, 79, 80, 81, 83, 85, 86, 87, 88, 92, 93, 95, 97, 98, 99, 100, 103, 106
Offset: 1

Views

Author

Gus Wiseman, Mar 18 2019

Keywords

Examples

			The sequence of terms together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  10: (o((o)))
  11: ((((o))))
  14: (o(oo))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  20: (oo((o)))
  21: ((o)(oo))
  22: (o(((o))))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))

Links

Crossrefs

Cf. A000081, A007097, A290822, A306844, A318186.
Cf. A324694, A324738, A324744, A324749, A324754, A324759, A324765, A324768, A324838, A324842, A324844, A324846, A324847, A324849.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
qaQ[n_]:=And[And@@Table[!Divisible[n,x],{x,DeleteCases[primeMS[n],1]}],And@@qaQ/@primeMS[n]];
Select[Range[100],qaQ]

A324932 Numerator in the division of n by the product of prime indices of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 10, 11, 6, 13, 7, 5, 16, 17, 9, 19, 20, 21, 22, 23, 12, 25, 13, 27, 7, 29, 5, 31, 32, 33, 34, 35, 9, 37, 19, 13, 40, 41, 21, 43, 44, 15, 46, 47, 24, 49, 50, 51, 26, 53, 27, 11, 14, 57, 29, 59, 10, 61, 62, 63, 64, 65, 33, 67, 68, 23
Offset: 1

Views

Author

Gus Wiseman, Mar 21 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The sequence of quotients n/A003963(n) begins: 1, 2, 3/2, 4, 5/3, 3, 7/4, 8, 9/4, 10/3, 11/5, 6, 13/6, 7/2, 5/2, 16, ...

Links

Crossrefs

The denominators are A324933.
Cf. A003963, A109129, A120383, A196050, A324846, A324849, A324850, A324922, A324923, A324924, A324925, A324931, A324934.

Programs

  • Mathematica
    Table[n/Times@@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]^k],{n,100}]//Numerator

A325031 Numbers divisible by all prime indices of their prime indices.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 16, 18, 19, 20, 21, 24, 26, 27, 28, 30, 32, 33, 36, 38, 40, 42, 46, 48, 49, 50, 52, 53, 54, 56, 57, 60, 63, 64, 66, 68, 70, 72, 74, 76, 78, 80, 81, 84, 87, 90, 92, 96, 98, 99, 100, 104, 106, 108, 112, 114, 120, 122, 126, 128
Offset: 1

Views

Author

Gus Wiseman, Mar 25 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. For example, the prime indices of 55 are {3,5} with prime indices {{2},{3}}. Since 55 is not divisible by 2 or 3, it does not belong to the sequence.

Examples

			The sequence of multisets of multisets whose MM-numbers (see A302242) belong to the sequence begins:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  10: {{},{2}}
  12: {{},{},{1}}
  14: {{},{1,1}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  20: {{},{},{2}}
  21: {{1},{1,1}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  30: {{},{1},{2}}
  32: {{},{},{},{},{}}
  33: {{1},{3}}
  36: {{},{},{1},{1}}

Crossrefs

Cf. A000720, A003963, A112798, A120383, A302242, A320325.
Cf. A323440, A324846, A324847, A324849, A324850, A324856, A324926, A325032.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@Table[Divisible[#,i],{i,Union@@primeMS/@primeMS[#]}]&]
Previous Showing 11-19 of 19 results.

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