A324853 -id:A324853 - cp's OEIS frontend

A324850 Numbers divisible by the product of their prime indices.

1, 2, 4, 6, 8, 12, 16, 24, 28, 30, 32, 36, 48, 56, 60, 64, 72, 96, 112, 120, 128, 144, 152, 156, 168, 180, 192, 216, 224, 240, 256, 288, 304, 312, 330, 336, 360, 384, 432, 448, 476, 480, 512, 576, 608, 624, 660, 672, 720, 768, 784, 828, 840, 848, 864, 888, 896

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, with product A003963(n). For example, the prime indices of 30 are {1,2,3}, with product 6, which divides 30, so 30 is in the sequence.

			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}
  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}
  56: {1,1,1,4}
  60: {1,1,2,3}
  64: {1,1,1,1,1,1}
  72: {1,1,1,2,2}
  96: {1,1,1,1,1,2}

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

Cf. A003963, A036844, A120383, A324847, A324756, A324758, A324847, A324848, A324849, A324852, A324853, A324856, A324923, A324924, A324925, A324931.

Select[Range[100],Divisible[#,Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]]&]

isok(n) = my(f=factor(n)); !(n % prod(k=1, #f~, primepi(f[k,1])^f[k,2])); \\ Michel Marcus, Mar 22 2019

n/A003963(n) = A324933(n)/A324934(n).

A324846 Positive integers divisible by none of their prime indices.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 121, 123, 125, 127, 129, 131, 133, 137

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

A prime index of n is a number m such that prime(m) divides n. For example, the prime indices of 5673 are {2,11,18}, none of which divides 5673, so 5673 belongs to the sequence.

			The sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  33: {2,5}
  35: {3,4}
  37: {12}
  39: {2,6}

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Complement of A324847.
Cf. A000720, A001222, A003963, A056239, A112798, A120383, A289509, A290822, A304360, A306844.
Cf. A324695, A324741, A324743, A324756, A324758, A324765, A324848, A324849, A324850, A324852, A324853.

q:= n-> ormap(i-> irem(n, numtheory[pi](i[1]))=0, ifactors(n)[2]):
remove(q, [$1..200])[];  # Alois P. Heinz, Mar 19 2019

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

isok(n) = {my(f = factor(n)[,1]); for (k=1, #f, if (!(n % primepi(f[k])), return (0));); return (1);} \\ Michel Marcus, Mar 19 2019

A324849 Positive integers divisible by none of their prime indices > 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 76, 77, 79, 80, 81, 82, 83, 85, 86, 87

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:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}

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

Complement of A324771.
Cf. A003963, A056239, A112798, A120383, A289509, A304360, A306844.
Cf. A324695, A324741, A324743, A324756, A324758, A324765, A324846, A324847, A324848, A324850, A324852, A324853, A324856.

filter:= proc(n) andmap(t -> not ((n/numtheory:-pi(t))::integer), numtheory:-factorset(n) minus {2}) end proc:
select(filter, [$1..200]); # Robert Israel, Mar 20 2019

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

is(n) = my(f=factor(n)[, 1]~, idc=[]); for(k=1, #f, idc=concat(idc, [primepi(f[k])])); for(t=1, #idc, if(idc[t]==1, next); if(n%idc[t]==0, return(0))); 1 \\ Felix Fröhlich, Mar 21 2019

A324847 Numbers divisible by at least one of their prime indices.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

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

If n is in the sequence, then so are all multiples of n. - Robert Israel, Mar 19 2019

			The sequence of terms together with their prime indices begins:
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
  34: {1,7}
  36: {1,1,2,2}

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

Complement of A324846.
Cf. A003963, A056239, A112798, A120383, A289509, A290822, A304360, A306844.
Cf. A324695, A324741, A324743, A324847, A324756, A324758, A324765, A324848, A324849, A324850, A324852, A324853.

filter:= proc(n) local F;
  F:= map(numtheory:-pi, numtheory:-factorset(n));
  ormap(t -> n mod t = 0, F);
end proc:
select(filter, [$1..200]); # Robert Israel, Mar 19 2019

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

isok(n) = {my(f = factor(n)[,1]); for (k=1, #f, if (!(n % primepi(f[k])), return (1));); return (0);} \\ Michel Marcus, Mar 19 2019

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

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]]]&]

cp's OEIS Frontend

A324850 Numbers divisible by the product of their prime indices.

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A324846 Positive integers divisible by none of their prime indices.

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A324849 Positive integers divisible by none of their prime indices > 1.

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A324847 Numbers divisible by at least one of their prime indices.

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

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