A324848 -id:A324848 - 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).

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

Original entry on oeis.org

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

A324851 Numbers > 1 divisible by the sum of their prime indices.

Original entry on oeis.org

2, 4, 6, 12, 15, 16, 20, 30, 35, 36, 42, 48, 56, 88, 99, 112, 120, 126, 130, 135, 143, 144, 160, 162, 180, 192, 210, 216, 220, 221, 228, 231, 242, 250, 256, 270, 275, 280, 288, 297, 300, 308, 322, 330, 338, 360, 396, 400, 408, 429, 435, 440, 455, 468, 480, 493

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 sum of prime indices of n is A056239(n). For example, the prime indices of 99 are {2,2,5}, with sum 9, a divisor of 99, so 99 is in the sequence.

For any k>=2, let d be a divisor of k such that d > A056239(k). Then 2^(d-A056239(k))*k is in the sequence. Similarly if k is in the sequence with d = A056239(k), then 2^d*k is in the sequence. - Robert Israel, Mar 19 2019

			The sequence of terms together with their prime indices begins:
    2: {1}
    4: {1,1}
    6: {1,2}
   12: {1,1,2}
   15: {2,3}
   16: {1,1,1,1}
   20: {1,1,3}
   30: {1,2,3}
   35: {3,4}
   36: {1,1,2,2}
   42: {1,2,4}
   48: {1,1,1,1,2}
   56: {1,1,1,4}
   88: {1,1,1,5}
   99: {2,2,5}
  112: {1,1,1,1,4}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  130: {1,3,6}
  135: {2,2,2,3}

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

Cf. A003963, A036844, A056239, A112798, A120383, A290822, A306844.

Cf. A324695, A324741, A324743, A324847, A324756, A324758, A324765, A324847, A324848, A324849, A324850, A324852.

filter:= proc(n) local t; n mod add(numtheory:-pi(t[1])*t[2],t=ifactors(n)[2]) = 0 end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 19 2019

Select[Range[2,100],Divisible[#,Plus@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]*k]]&]

isok(n) = {my(f = factor(n)); (n!=1) && !(n % sum(k=1, #f~, primepi(f[k,1])*f[k,2]));} \\ Michel Marcus, Mar 19 2019

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

A324922 a(n) = unique m such that m/A003963(m) = n, where A003963 is product of prime indices.

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 28, 8, 36, 60, 330, 24, 156, 56, 180, 16, 476, 72, 152, 120, 168, 660, 828, 48, 900, 312, 216, 112, 1740, 360, 10230, 32, 1980, 952, 840, 144, 888, 304, 936, 240, 6396, 336, 2408, 1320, 1080, 1656, 8460, 96, 784, 1800, 2856, 624, 848, 432

Offset: 1

Gus Wiseman, Mar 20 2019

Every positive integer has a unique factorization into factors q(i) = prime(i)/i, i > 0 given by the rows of A324924. Then a(n) is the number obtained by encoding this factorization as a standard factorization into prime numbers (A112798).

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

Sorting the sequence gives A324850.
Cf. A000081, A003963, A061775, A109129, A112798, A120383, A196050, A317713.
Cf. A324848, A324849, A324923, A324924, A324925, A324931, A324934.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
difac[n_]:=If[n==1,{},With[{m=Product[Prime[i]/i,{i,primeMS[n]}]},Sort[Join[primeMS[n],difac[n/m]]]]];
Table[Times@@Prime/@difac[n],{n,30}]

a(n) = my (f=factor(n)); prod (i=1, #f~, (f[i,1] * a(primepi(f[i,1])))^f[i,2]) \\ Rémy Sigrist, Jul 18 2019

a(n) = Product_t mg(t) where the product is over all (not necessarily distinct) terminal subtrees of the rooted tree with Matula-Goebel number n, and mg(t) is the Matula-Goebel number of t.

Completely multiplicative with a(prime(n)) = prime(n) * a(n). - Rémy Sigrist, Jul 18 2019

Keyword mult added by Rémy Sigrist, Jul 18 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

A324844 Number of unlabeled rooted trees with n nodes 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, 1, 2, 3, 7, 13, 32, 71, 170, 406, 1002, 2469, 6204, 15644, 39871, 102116, 263325, 682079, 1775600, 4640220

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

			The a(1) = 1 through a(6) = 13 rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (((o)))  (o(oo))    (o(ooo))
                          (((oo)))   (((ooo)))
                          ((o)(o))   ((o)(oo))
                          (o((o)))   ((o(oo)))
                          ((((o))))  (o((oo)))
                                     (oo((o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

The Matula-Goebel numbers of these trees are given by A324845.
Cf. A000081, A290689, A306844, A318185.
Cf. A324694, A324738, A324744, A324749, A324754, A324759, A324765, A324768, A324838, A324843, A324846, A324847, A324848, A324849.

submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
rallt[n_]:=Select[Union[Sort/@Join@@(Tuples[rallt/@#]&/@IntegerPartitions[n-1])],And@@Table[!submultQ[b,#],{b,DeleteCases[#,{}]}]&];
Table[Length[rallt[n]],{n,10}]

A324843 Number of unlabeled rooted trees with n nodes 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, 1, 1, 2, 2, 4, 4, 8, 9, 15, 17, 31, 35, 57, 70, 111, 136, 213, 265, 405, 517, 763, 987, 1458, 1893, 2736, 3611, 5161, 6836, 9702

Offset: 1

Gus Wiseman, Mar 18 2019

A subset of totally transitive rooted trees (A318185).

			The a(1) = 1 through a(8) = 8 rooted trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)    (oooooo)    (ooooooo)
                (o(o))  (oo(o))  (oo(oo))   (ooo(oo))   (ooo(ooo))
                                 (ooo(o))   (oooo(o))   (oooo(oo))
                                 (o(o)(o))  (oo(o)(o))  (ooooo(o))
                                                        (oo(o)(oo))
                                                        (ooo(o)(o))
                                                        (o(o)(o)(o))
                                                        (o(o)(o(o)))

The Matula-Goebel numbers of these trees are given by A324842.
Cf. A000081, A279861, A290689, A290822, A318185.
Cf. A324704, A324736, A324748, A324753, A324847, A324848, A324854.

submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
rallt[n_]:=Select[Union[Sort/@Join@@(Tuples[rallt/@#]&/@IntegerPartitions[n-1])],And@@Table[submultQ[b,#],{b,#}]&];
Table[Length[rallt[n]],{n,10}]

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

cp's OEIS Frontend

A324850 Numbers divisible by the product of their prime indices.

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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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A324851 Numbers > 1 divisible by the sum of their prime indices.

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A324846 Positive integers divisible by none of their prime indices.

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A324922 a(n) = unique m such that m/A003963(m) = n, where A003963 is product of 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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