cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A367587 Least element in row n of A367858 (multiset multiplicity cokernel).

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Dec 03 2023

Keywords

Comments

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.

Crossrefs

Indices of first appearances are A008578.
Depends only on rootless base A052410, see A007916.
For kernel instead of cokernel we have A055396.
For maximum instead of minimum element we have A061395.
The opposite version is A367583.
Row-minima of A367858.
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, sorted A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
A367579 lists MMK, rank A367580, sum A367581, max A367583, min A055396.
Cf. A000720, A027746, A051904, A071625, A072774, A130091, A367582, A367584, A367585, A367586.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&],{i,mts}]]];
Table[If[n==1,0,Min@@mmc[prix[n]]],{n,100}]

Formula

a(n) = A055396(A367859(n)).
a(n^k) = a(n) for all positive integers n and k.
If n is a power of a squarefree number, a(n) = A061395(n).

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.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Dec 03 2023

Keywords

Comments

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.

Examples

			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.

Crossrefs

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.

Programs

  • Mathematica
    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}]

Formula

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).

A367860 Sum of the multiset multiplicity cokernel (in which each multiplicity becomes the greatest element of that multiplicity) of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 4, 4, 1, 2, 6, 5, 3, 6, 8, 6, 1, 7, 3, 8, 4, 8, 10, 9, 3, 3, 12, 2, 5, 10, 9, 11, 1, 10, 14, 8, 4, 12, 16, 12, 4, 13, 12, 14, 6, 5, 18, 15, 3, 4, 4, 14, 7, 16, 3, 10, 5, 16, 20, 17, 7, 18, 22, 6, 1, 12, 15, 19, 8, 18, 12, 20, 3, 21, 24, 5, 9, 10
Offset: 1

Views

Author

Gus Wiseman, Dec 03 2023

Keywords

Comments

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.

Examples

			The multiset multiplicity cokernel of {1,2,2,3} is {2,3,3}, so a(90) = 8.

Crossrefs

Positions of 1's are A000079 without 1.
Depends only on rootless base A052410, see A007916, A052409.
For kernel instead of cokernel we have A367581, row-sums of A367579.
For minimum instead of sum we have A367587, opposite A367583.
The triangle A367858 has these as row sums, ranks A367859.
A007947 gives squarefree kernel.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, reverse A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
Cf. A000720, A051904, A055396, A061395, A071625, A072774, A130091, A367580, A175781, A367582, A367584.

Programs

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

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

Views

Author

Gus Wiseman, Dec 03 2023

Keywords

Comments

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.

Examples

			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}

Crossrefs

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.

Programs

  • Mathematica
    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[[#]]]&]
Showing 1-4 of 4 results.

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