A323440 -id:A323440 - cp's OEIS frontend

A120383 A number n is included if it satisfies: m divides n for all m's where the m-th prime divides n.

1, 2, 4, 6, 8, 12, 16, 18, 24, 28, 30, 32, 36, 48, 54, 56, 60, 64, 72, 78, 84, 90, 96, 108, 112, 120, 128, 144, 150, 152, 156, 162, 168, 180, 192, 196, 216, 224, 234, 240, 252, 256, 270, 288, 300, 304, 312, 324, 330, 336, 360, 384, 390, 392, 414, 420, 432, 444, 448

Offset: 1

Leroy Quet, Jun 29 2006

nonn

From Rémy Sigrist, Apr 08 2017: (Start)
If n is in the sequence, then 2*n is also in the sequence.
a(2) = 2 is the only prime number in the sequence.
a(1) = 1 is the only odd number in the sequence.
(End)
Numbers divisible by all of their prime indices. A prime index of n is a number m such that prime(m) divides n. For example, the prime indices of 78 = prime(1) * prime(2) * prime(6) are {1,2,6}, all of which divide 78, so 78 is in the sequence. - Gus Wiseman, Mar 23 2019

			28 = 2^2 * 7. 2 is the first prime, 7 is the 4th prime. Since 1 and 4 both divide 28, then 28 is included in the sequence.
78 = 2 * 3 * 13. 2 is the first prime, 3 is the 2nd prime and 13 is the 6th prime. Since 1 and 2 and 6 each divide 78, then 78 is in the sequence. (Note that 1 * 2 * 6 does not divide 78.)
From _Gus Wiseman_, Mar 23 2019: (Start)
The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  12: {1,1,2}
  16: {1,1,1,1}
  18: {1,2,2}
  24: {1,1,1,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  48: {1,1,1,1,2}
  54: {1,2,2,2}
  56: {1,1,1,4}
  60: {1,1,2,3}
  64: {1,1,1,1,1,1}
(End)

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A000720, A003963, A056239, A112798, A323440, A324846, A324847, A324848, A324850, A324852, A324856.

A000040inv := proc(n) local i; i:=1 ; while true do if ithprime(i) = n then RETURN(i) ; fi ; i := i+1 ; end ; end: isA120383 := proc(n) local pl,p,i,j ; pl := ifactors(n) ; pl := pl[2] ; for i from 1 to nops(pl) do p := pl[i] ; j := A000040inv(p[1]) ; if n mod j <> 0 then RETURN(false) ; fi ; od ; RETURN(true) ; end: for n from 2 to 800 do if isA120383(n) then printf("%d,",n); fi ; od ; # R. J. Mathar, Sep 02 2006

{1}~Join~Select[Range[2, 450], Function[n, AllTrue[PrimePi /@ FactorInteger[n][[All, 1]], Mod[n, #] == 0 &]]] (* Michael De Vlieger, Mar 24 2019 *)

ok(n) = my (f=factor(n)); for (i=1, #f~, if (n % primepi(f[i,1]), return (0))); return (1) \\ Rémy Sigrist, Apr 08 2017

More terms from R. J. Mathar, Sep 02 2006

Initial 1 prepended by Rémy Sigrist, Apr 08 2017

A324848 Number of prime indices of n (counted with multiplicity) that divide n.

Original entry on oeis.org

0, 1, 0, 2, 0, 2, 0, 3, 0, 1, 0, 3, 0, 1, 1, 4, 0, 3, 0, 2, 0, 1, 0, 4, 0, 1, 0, 3, 0, 3, 0, 5, 0, 1, 0, 4, 0, 1, 0, 3, 0, 2, 0, 2, 1, 1, 0, 5, 0, 1, 0, 2, 0, 4, 1, 4, 0, 1, 0, 4, 0, 1, 0, 6, 0, 2, 0, 2, 0, 1, 0, 5, 0, 1, 2, 2, 0, 3, 0, 4, 0, 1, 0, 4, 0, 1, 0

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

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 prime indices of 6776 are {1,1,1,4,5,5}, four of which {1,1,1,4} divide 6776, so a(6776) = 4.

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

The version for distinct prime indices is A324852.
Positions of zeros are A324846.
Positions of ones are A324856.
Cf. A003963, A056239, A112798, A120383, A323440, A324704, A324847, A324849, A324850, A324853.

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k/;Divisible[n,PrimePi[p]]]],{n,100}]

A324852 Number of distinct prime indices of n that divide n.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 1, 2, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

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.

			60060 has 7 prime indices {1,1,2,3,4,5,6}, all of which divide 60060, and 6 of which are distinct, so a(60060) = 6.

Alois P. Heinz, Table of n, a(n) for n = 1..65536

The version for all prime indices (counted with multiplicity) is A324848.
Positions of zeros are A324846.
Positions of ones are A323440.
Cf. A000720, A001222, A003963, A056239, A120383, A124012.
Cf. A324704, A324771, A324847, A324849, A324850, A324853, A324856.

a:= n-> add(`if`(irem(n, numtheory[pi](i[1]))=0, 1, 0), i=ifactors(n)[2]):
seq(a(n), n=1..120);  # Alois P. Heinz, Mar 19 2019

Table[Count[If[n==1,{},FactorInteger[n]],{p_,_}/;Divisible[n,PrimePi[p]]],{n,100}]

a(n) = {my(f = factor(n)[,1]); sum(k=1, #f, !(n % primepi(f[k])));} \\ Michel Marcus, Mar 19 2019

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} 1/(k*prime(k)) = 0.848969... (A124012). - Amiram Eldar, Jan 11 2025

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
```
See Links section.
```

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

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

Gus Wiseman, Mar 25 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 does not belong to the sequence.

			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}}

Cf. A000720, A003963, A112798, A120383, A302242, A320325.

Cf. A323440, A324846, A324847, A324849, A324850, A324856, A324926, A325032.

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[#]}]&]

cp's OEIS Frontend

A120383 A number n is included if it satisfies: m divides n for all m's where the m-th prime divides n.

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A324848 Number of prime indices of n (counted with multiplicity) that divide n.

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A324852 Number of distinct prime indices of n that divide n.

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A324856 Numbers divisible by exactly one of their prime indices.

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

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