A367860 -id:A367860 - cp's OEIS frontend

A367859 Multiset multiplicity cokernel (MMC) of n. Product of (greatest prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n.

1, 2, 3, 2, 5, 9, 7, 2, 3, 25, 11, 6, 13, 49, 25, 2, 17, 6, 19, 10, 49, 121, 23, 6, 5, 169, 3, 14, 29, 125, 31, 2, 121, 289, 49, 9, 37, 361, 169, 10, 41, 343, 43, 22, 15, 529, 47, 6, 7, 10, 289, 26, 53, 6, 121, 14, 361, 841, 59, 50, 61, 961, 21, 2, 169, 1331

Offset: 1

Gus Wiseman, Dec 03 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 the multiset multiplicity cokernel MMC(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then max(S) has multiplicity |S| in MMC(m). For example, MMC({1,1,2,2,3,4,5}) = {2,2,5,5,5}, and MMC({1,2,3,4,5,5,5,5}) = {4,4,4,4,5}. As an operation on multisets MMC is represented by A367858, and as an operation on their ranks it is represented by A367859.

			90 has prime factorization 2^1*3^2*5^1, so for k = 1 we have 5^2, and for k = 2 we have 3^1, so a(90) = 75.

Positions of 2's are A000079 without 1.
Positions of 3's are A000244 without 1.
Positions of primes (including 1) are A000961.
Depends only on rootless base A052410, see A007916.
Positions of prime powers are A072774.
Positions of squarefree numbers are A130091.
For kernel instead of cokernel we have A367580, ranks of A367579.
Rows of A367858 have this rank, sum A367860, max A061395, min A367587.
A007947 gives squarefree kernel.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A071625 counts distinct prime exponents.
A124010 gives multiset of multiplicities (prime signature), sorted A118914.
Cf. A005117, A020639, A052409, A175781, A181819, A353742, A367584, A367585.

mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Times@@mmc[Join@@ConstantArray@@@FactorInteger[n]], {n,30}]

a(n^k) = a(n) for all positive integers n and k.
If n is squarefree, a(n) = A006530(n)^A001222(n).
A055396(a(n)) = A367587(n).
A056239(a(n)) = A367860(n).
A061395(a(n)) = A061395(n).
A001222(a(n)) = A001221(n).
A001221(a(n)) = A071625(n).
A071625(a(n)) = A323022(n).

A367858 Irregular triangle read by rows where row n is the multiset multiplicity cokernel (MMC) of the multiset of prime indices of n.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 4, 1, 2, 3, 3, 5, 1, 2, 6, 4, 4, 3, 3, 1, 7, 1, 2, 8, 1, 3, 4, 4, 5, 5, 9, 1, 2, 3, 6, 6, 2, 1, 4, 10, 3, 3, 3, 11, 1, 5, 5, 7, 7, 4, 4, 2, 2, 12, 8, 8, 6, 6, 1, 3, 13, 4, 4, 4, 14, 1, 5, 2, 3, 9, 9, 15, 1, 2, 4, 1, 3, 7, 7, 1, 6, 16, 1, 2

Offset: 1

Gus Wiseman, Dec 03 2023

Row n = 1 is empty.
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 the multiset multiplicity cokernel MMC(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then max(S) has multiplicity |S| in MMC(m). For example, MMC({1,1,2,2,3,4,5}) = {2,2,5,5,5}, and MMC({1,2,3,4,5,5,5,5}) = {4,4,4,4,5}.

			The first 45 rows:
     1: {}       16: {1}        31: {11}
     2: {1}      17: {7}        32: {1}
     3: {2}      18: {1,2}      33: {5,5}
     4: {1}      19: {8}        34: {7,7}
     5: {3}      20: {1,3}      35: {4,4}
     6: {2,2}    21: {4,4}      36: {2,2}
     7: {4}      22: {5,5}      37: {12}
     8: {1}      23: {9}        38: {8,8}
     9: {2}      24: {1,2}      39: {6,6}
    10: {3,3}    25: {3}        40: {1,3}
    11: {5}      26: {6,6}      41: {13}
    12: {1,2}    27: {2}        42: {4,4,4}
    13: {6}      28: {1,4}      43: {14}
    14: {4,4}    29: {10}       44: {1,5}
    15: {3,3}    30: {3,3,3}    45: {2,3}

Indices of empty and singleton rows are A000961.
Row lengths are A001221.
Depends only on rootless base A052410, see A007916.
Row maxima are A061395.
Rows have A071625 distinct elements.
Indices of constant rows are A072774.
Indices of strict rows are A130091.
Row minima are A367587.
Rows have Heinz numbers A367859.
Row sums are A367860.
Sorted row indices of first appearances are A367861, for kernel A367585.
A007947 gives squarefree kernel.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124010 lists prime multiplicities (prime signature), sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, reversed A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
Cf. A000720, A027746, A051904, A052409, A061395, A175781, A367582, A367583.

mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[mmc[PrimePi /@ Join@@ConstantArray@@@If[n==1, {},FactorInteger[n]]], {n,100}]

For all positive integers n and k, row n^k is the same as row n.

A367861 Numbers k whose multiset multiplicity cokernel (in which each prime exponent becomes the greatest prime factor with that exponent) is different from that of all positive integers less than k.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 17, 19, 20, 22, 23, 26, 28, 29, 30, 31, 34, 37, 38, 41, 42, 43, 44, 45, 46, 47, 52, 53, 58, 59, 60, 61, 62, 63, 66, 67, 68, 71, 73, 74, 76, 78, 79, 82, 83, 84, 86, 89, 90, 92, 94, 97, 99, 101, 102, 103, 106, 107, 109, 113

Offset: 1

Gus Wiseman, Dec 03 2023

nonn

			The terms together with their prime indices begin:
     1: {}         23: {9}        47: {15}
     2: {1}        26: {1,6}      52: {1,1,6}
     3: {2}        28: {1,1,4}    53: {16}
     5: {3}        29: {10}       58: {1,10}
     6: {1,2}      30: {1,2,3}    59: {17}
     7: {4}        31: {11}       60: {1,1,2,3}
    10: {1,3}      34: {1,7}      61: {18}
    11: {5}        37: {12}       62: {1,11}
    12: {1,1,2}    38: {1,8}      63: {2,2,4}
    13: {6}        41: {13}       66: {1,2,5}
    14: {1,4}      42: {1,2,4}    67: {19}
    17: {7}        43: {14}       68: {1,1,7}
    19: {8}        44: {1,1,5}    71: {20}
    20: {1,1,3}    45: {2,2,3}    73: {21}
    22: {1,5}      46: {1,9}      74: {1,12}

Contains all primes A000040 but no other perfect powers A001597.
All terms are rootless A007916 (have no positive integer roots).
For kernel instead of cokernel we have A367585, sorted version of A367584.
The MMC triangle is A367858, sum A367860, min A367857, max A061395.
Sorted positions of first appearances in A367859.
A007947 gives squarefree kernel.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A071625 counts distinct prime exponents.
A124010 gives prime signature, sorted A118914.
Cf. A020639, A051904, A072774, A073485, A130091, A181819, A367582, A367768.

nn=100;
mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&], {i,mts}]]];
qq=Table[Times@@mmc[Join@@ConstantArray@@@FactorInteger[n]], {n,nn}];
Select[Range[nn], FreeQ[Take[qq,#-1],qq[[#]]]&]

cp's OEIS Frontend

A367859 Multiset multiplicity cokernel (MMC) of n. Product of (greatest prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A367858 Irregular triangle read by rows where row n is the multiset multiplicity cokernel (MMC) of the multiset of prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A367861 Numbers k whose multiset multiplicity cokernel (in which each prime exponent becomes the greatest prime factor with that exponent) is different from that of all positive integers less than k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica