A324848 -id:A324848 - cp's OEIS frontend

A324926 Numbers not divisible by any prime indices of their prime indices.

1, 2, 4, 5, 8, 11, 16, 17, 22, 23, 25, 31, 32, 34, 41, 44, 47, 55, 59, 62, 64, 67, 73, 82, 83, 85, 88, 97, 103, 109, 115, 118, 121, 124, 125, 127, 128, 134, 137, 149, 157, 164, 166, 167, 176, 179, 187, 191, 194, 197, 205, 211, 218, 227, 233, 235, 236, 241, 242

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

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 belongs to the sequence.

			The sequence of multisets of multisets whose MM-numbers (see A302242) belong to the sequence begins:
   1: {}
   2: {{}}
   4: {{},{}}
   5: {{2}}
   8: {{},{},{}}
  11: {{3}}
  16: {{},{},{},{}}
  17: {{4}}
  22: {{},{3}}
  23: {{2,2}}
  25: {{2},{2}}
  31: {{5}}
  32: {{},{},{},{},{}}
  34: {{},{4}}
  41: {{6}}
  44: {{},{},{3}}
  47: {{2,3}}
  55: {{2},{3}}
  59: {{7}}
  62: {{},{5}}
  64: {{},{},{},{},{},{}}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001222, A003963, A112798, A120383, A302242, A304360, A324846, A324847, A324848, A324849, A324850, A324927, A324928, A324930.

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[#]}]&]

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

Gus Wiseman, Mar 21 2019

nonn

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

			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}

Robert Israel, Table of n, a(n) for n = 1..10000

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

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

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

Gus Wiseman, Mar 18 2019

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.

			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.

Rémy Sigrist, C program for A324853

Cf. A000720, A001222, A003963, A056239, A120383, A323440.

Cf. A324771, A324846, A324847, A324848, A324849, A324850, A324852, A324856.

C
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See Links section.
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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}]]

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

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

Gus Wiseman, Mar 21 2019

nonn

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.

			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}

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Select[Range[100],Count[If[#==1,{},FactorInteger[#]],{p_,_}/;Divisible[#,PrimePi[p]]]==1&]

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

Gus Wiseman, Mar 18 2019

nonn

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

			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}

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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

A324842 Matula-Goebel numbers of rooted trees where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 28, 32, 36, 48, 54, 56, 64, 72, 78, 84, 96, 108, 112, 128, 144, 152, 156, 162, 168, 192, 196, 216, 224, 234, 252, 256, 288, 304, 312, 324, 336, 384, 392, 432, 444, 448, 456, 468, 486, 504, 512, 576, 588, 608, 624, 648, 672, 702

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

			The sequence of rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
  12: (oo(o))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  48: (oooo(o))
  54: (o(o)(o)(o))
  56: (ooo(oo))
  64: (oooooo)
  72: (ooo(o)(o))
  78: (o(o)(o(o)))
  84: (oo(o)(oo))
  96: (ooooo(o))

A subsequence of A120383.
Cf. A000081, A007097, A112798, A279861, A290689, A290760, A290822, A318186.
Cf. A324704, A324736, A324748, A324753, A324843, A324847, A324848, A324854.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
qaQ[n_]:=And[And@@Table[Divisible[n,x],{x,primeMS[n]}],And@@qaQ/@primeMS[n]];
Select[Range[1000],qaQ]

cp's OEIS Frontend

A324926 Numbers not divisible by any prime indices of their prime indices.

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A324856 Numbers divisible by exactly one of their prime indices.

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Maple

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

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C

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

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Mathematica

PARI

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

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Mathematica

A324842 Matula-Goebel numbers of rooted trees where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.

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Author

Keywords

Examples

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Mathematica