A067695 -id:A067695 - cp's OEIS frontend

A353026 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that is a multiple of the smallest prime factor with minimal exponent of a(n-1) (cf. A067695).

1, 2, 4, 6, 8, 10, 12, 3, 9, 15, 18, 14, 16, 20, 5, 25, 30, 22, 24, 21, 27, 33, 36, 26, 28, 7, 35, 40, 45, 50, 32, 34, 38, 42, 44, 11, 55, 60, 39, 48, 51, 54, 46, 52, 13, 65, 70, 56, 49, 63, 77, 84, 57, 66, 58, 62, 64, 68, 17, 85, 75, 69, 72, 78, 74, 76, 19, 95, 80, 90, 82, 86, 88, 99, 110

Offset: 1

Scott R. Shannon, Apr 18 2022

nonn

The sequences is conjectured to be a permutation of the positive integers. In the first 500000 terms any time a prime p appears, where p>=5, the following term is 5p. It is unknown if this is true for all primes. In the same range the fixed points are 1, 2, 9, 39, 49, 1079, 4897, although it is possible more exist.

			a(4) = 6 as a(3) = 4 = 2*2 which has A067695(4) = 2 as the smallest prime factor with minimal exponent, and 6 is the smallest unused number that is a multiple of 2.
a(8) = 3 as a(7) = 12 = 2*2*3 which has A067695(12) = 3 as the smallest prime factor with minimal exponent, and 3 is the smallest unused number that is a multiple of 3.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 100000 terms. The green line is y = n.

Cf. A067695, A352793, A064413.

nn = 75, c[] = 0; Array[Set[{a[#], c[#]}, {#, #}] &, 2]; u = 3; Do[p = MinimalBy[FactorInteger@ a[i - 1], Last][[1, 1]]; k = u; While[Nand[c[k] == 0, Divisible[k, p]], k++]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, Length[s] + 1, nn}]; Array[a, nn] (* _Michael De Vlieger, Apr 18 2022 *)

A364191 Low co-mode in the multiset of prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 2, 6, 1, 2, 1, 7, 1, 8, 3, 2, 1, 9, 2, 3, 1, 2, 4, 10, 1, 11, 1, 2, 1, 3, 1, 12, 1, 2, 3, 13, 1, 14, 5, 3, 1, 15, 2, 4, 1, 2, 6, 16, 1, 3, 4, 2, 1, 17, 2, 18, 1, 4, 1, 3, 1, 19, 7, 2, 1, 20, 2, 21, 1, 2, 8, 4, 1, 22, 3, 2, 1

Offset: 1

Gus Wiseman, Jul 16 2023

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.
We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes in {a,a,b,b,b,c,c} are {a,c}.
Extending the terminology of A124943, the "low co-mode" in a multiset is the least co-mode.

			The prime indices of 2100 are {1,1,2,3,3,4}, with co-modes {2,4}, so a(2100) = 2.

For prime factors instead of indices we have A067695, high A359612.
For mode instead of co-mode we have A363486, high A363487, triangle A363952.
For median instead of co-mode we have A363941, high A363942.
Positions of 1's are A364158, counted by A364159.
The high version is A364192 = positions of 1's in A364061.
A112798 lists prime indices, length A001222, sum A056239.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
Ranking and counting partitions:
- A356862 = unique mode, counted by A362608
- A359178 = unique co-mode, counted by A362610
- A362605 = multiple modes, counted by A362607
- A362606 = multiple co-modes, counted by A362609
Cf. A124943, A241131, A327473, A327476, A360005, A360015, A363488.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==1,0,Min[comodes[prix[n]]]],{n,30}]

a(n) = A000720(A067695(n)).

A067695(n) = A000040(a(n)).

A364192 High (i.e., greatest) co-mode in the multiset of prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 3, 1, 7, 1, 8, 3, 4, 5, 9, 2, 3, 6, 2, 4, 10, 3, 11, 1, 5, 7, 4, 2, 12, 8, 6, 3, 13, 4, 14, 5, 3, 9, 15, 2, 4, 1, 7, 6, 16, 1, 5, 4, 8, 10, 17, 3, 18, 11, 4, 1, 6, 5, 19, 7, 9, 4, 20, 2, 21, 12, 2, 8, 5, 6, 22, 3, 2

Offset: 1

Gus Wiseman, Jul 16 2023

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.
We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes in {a,a,b,b,b,c,c} are {a,c}.
Extending the terminology of A124943, the "high co-mode" in a multiset is the greatest co-mode.

			The prime indices of 2100 are {1,1,2,3,3,4}, with co-modes {2,4}, so a(2100) = 4.

For prime factors instead of indices we have A359612, low A067695.
For mode instead of co-mode we have A363487 (triangle A363953), low A363486 (triangle A363952).
The version for median instead of co-mode is A363942, low A363941.
Positions of 1's are A364061, counted by A364062.
The low version is A364191, 1's at A364158 (counted by A364159).
A112798 lists prime indices, length A001222, sum A056239.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
Ranking and counting partitions:
- A356862 = unique mode, counted by A362608
- A359178 = unique co-mode, counted by A362610
- A362605 = multiple modes, counted by A362607
- A362606 = multiple co-modes, counted by A362609
Cf. A241131, A327473, A327476, A360005, A360015, A362612, A363740.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==1,0,Max[comodes[prix[n]]]],{n,30}]

a(n) = A000720(A359612(n)).

A359612(n) = A000040(a(n)).

A359612 Largest prime factor with minimal exponent in canonical prime factorization of n.

Original entry on oeis.org

2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 2, 17, 2, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 3, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 7, 2, 17, 13, 53, 2, 11, 7, 19, 29, 59, 5, 61, 31, 7, 2, 13, 11, 67, 17, 23, 7, 71, 3, 73, 37, 3

Offset: 2

Jens Ahlström, Jan 06 2023

nonn

When inspecting the minimal exponent of the canonical representation of n, a(n) is the largest of those primes, while A067695(n) is the smallest.
On the other hand if the maximal exponent is regarded similarly, A356840(n) is the largest of those primes and A356838(n) is the smallest.
18 is the smallest n, where a(n) differs from A006530(n), since a(18) = 2, while A006530(18) = 3.

			a(162) = a(2^1 * 3^4) = 2.
a(225) = a(3^2 * 5^2) = 5.

Alois P. Heinz, Table of n, a(n) for n = 2..20000

Cf. A006530, A067695, A356838, A356840.

a:= n-> (l-> (m-> max(map(i-> i[1], select(y->y[2]=m,
         l))))(min(map(x-> x[2], l))))(ifactors(n)[2]):
seq(a(n), n=2..75);  # Alois P. Heinz, Jan 25 2023

a[n_] := Module[{f = FactorInteger[n], e, ind}, e = f[[;; , 2]]; ind = Position[e, Min[e]][[-1, 1]]; f[[ind, 1]]]; Array[a, 100, 2] (* Amiram Eldar, Jan 07 2023 *)

a(n) = my(f=factor(n), e=vecmin(f[,2])); f[vecmax(select(x->(x==e), f[,2], 1)), 1]; \\ Michel Marcus, Jan 26 2023

from sympy import factorint
def a(n):
    max_factor = 0
    min_exponent = float("inf")
    for p, exponent in factorint(n).items():
        if exponent < min_exponent:
            max_factor = p
            min_exponent = exponent
        elif exponent == min_exponent:
            max_factor = max(max_factor, p)
    return max_factor

from sympy import factorint
def A359612(n): return (f:=list(map(tuple,zip(*sorted(factorint(n).items(),reverse=True)))))[0][f[1].index(min(f[1]))] # Chai Wah Wu, Feb 07 2023

cp's OEIS Frontend

A353026 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that is a multiple of the smallest prime factor with minimal exponent of a(n-1) (cf. A067695).

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A364191 Low co-mode in the multiset of prime indices of n.

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A364192 High (i.e., greatest) co-mode in the multiset of prime indices of n.

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A359612 Largest prime factor with minimal exponent in canonical prime factorization of n.

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