A324771 -id:A324771 - cp's OEIS frontend

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

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

A324838 Number of unlabeled rooted trees with n nodes where the branches of no branch of the root form a submultiset of the branches of the root.

Original entry on oeis.org

1, 0, 1, 2, 5, 10, 28, 64, 169, 422, 1108, 2872, 7627, 20202, 54216, 145867, 395288

Offset: 1

Gus Wiseman, Mar 18 2019

			The a(1) = 1 through a(6) = 10 rooted trees:
  o  ((o))  ((oo))   ((ooo))    ((oooo))
            (((o)))  (((oo)))   (((ooo)))
                     ((o)(o))   ((o)(oo))
                     ((o(o)))   ((o(oo)))
                     ((((o))))  ((oo(o)))
                                ((((oo))))
                                (((o)(o)))
                                (((o(o))))
                                ((o((o))))
                                (((((o)))))

Cf. A000081, A290689, A304360, A306844, A317787, A318185.

Cf. A324694, A324696, A324704, A324738, A324744, A324758, A324759, A324765, A324768, A324771, A324839, A324840, A324844, A324846.

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

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

A324839 Number of unlabeled rooted identity trees with n nodes where the branches of no branch of the root form a subset of the branches of the root.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 8, 16, 35, 74, 166, 367, 831, 1878, 4299, 9857, 22775, 52777, 122957, 287337

Offset: 1

Gus Wiseman, Mar 18 2019

An unlabeled rooted tree is an identity tree if there are no repeated branches directly under the same root.
Also the number of finitary sets with n brackets where no element is also a subset. For example, the a(7) = 8 sets are (o = {}):
  {{{{{{o}}}}}}
  {{{{o,{o}}}}}
  {{{o,{{o}}}}}
  {{o,{{{o}}}}}
  {{o,{o,{o}}}}
  {{{o},{{o}}}}
  {{o},{{{o}}}}
  {{o},{o,{o}}}

			The a(1) = 1 through a(8) = 16 rooted identity trees:
  o  ((o))  (((o)))  ((o(o)))   (((o(o))))   ((o)(o(o)))    (((o))(o(o)))
                     ((((o))))  ((o((o))))   ((o(o(o))))    (((o)(o(o))))
                                (((((o)))))  ((((o(o)))))   (((o(o(o)))))
                                             (((o)((o))))   ((o)((o(o))))
                                             (((o((o)))))   ((o)(o((o))))
                                             ((o)(((o))))   ((o((o(o)))))
                                             ((o(((o)))))   ((o(o)((o))))
                                             ((((((o))))))  ((o(o((o)))))
                                                            (((((o(o))))))
                                                            ((((o)((o)))))
                                                            ((((o((o))))))
                                                            (((o)(((o)))))
                                                            (((o(((o))))))
                                                            ((o)((((o)))))
                                                            ((o((((o))))))
                                                            (((((((o)))))))

Cf. A000081, A290760, A304360, A306844, A317787.

Cf. A324694, A324696, A324704, A324738, A324744, A324758, A324759, A324767, A324770, A324771, A324838, A324840, A324844, A324846.

idall[n_]:=If[n==1,{{}},Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&]];
Table[Length[Select[idall[n],And@@Table[!SubsetQ[#,b],{b,#}]&]],{n,10}]

cp's OEIS Frontend

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

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A324838 Number of unlabeled rooted trees with n nodes where the branches of no branch of the root form a submultiset of the branches of the root.

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

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

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A324839 Number of unlabeled rooted identity trees with n nodes where the branches of no branch of the root form a subset of the branches of the root.

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Mathematica