A238747 -id:A238747 - cp's OEIS frontend

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

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

Gus Wiseman, Dec 03 2023

nonn

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.

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.

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}]

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).

A238744 Irregular table read by rows: T (n, k) gives the number of primes p such that p^k divides n; table omits all zero values.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2

Offset: 2

Matthew Vandermast, Apr 28 2014

If the prime signature of n (nonincreasing version) is viewed as a partition, row n gives the conjugate partition.

			24 = 2^3*3 is divisible by two prime numbers (2 and 3), one square of a prime (4 = 2^2), and one cube of a prime (8 = 2^3); therefore, row 24 of the table is {2,1,1}.
From _Gus Wiseman_, Mar 31 2022: (Start)
Rows begin:
     1: ()        16: (1,1,1,1)    31: (1)
     2: (1)       17: (1)          32: (1,1,1,1,1)
     3: (1)       18: (2,1)        33: (2)
     4: (1,1)     19: (1)          34: (2)
     5: (1)       20: (2,1)        35: (2)
     6: (2)       21: (2)          36: (2,2)
     7: (1)       22: (2)          37: (1)
     8: (1,1,1)   23: (1)          38: (2)
     9: (1,1)     24: (2,1,1)      39: (2)
    10: (2)       25: (1,1)        40: (2,1,1)
    11: (1)       26: (2)          41: (1)
    12: (2,1)     27: (1,1,1)      42: (3)
    13: (1)       28: (2,1)        43: (1)
    14: (2)       29: (1)          44: (2,1)
    15: (2)       30: (3)          45: (2,1)
(End)

Row lengths are A051903(n); row sums are A001222(n).
Cf. A217171.
These partitions are ranked by A238745.
For prime indices A296150 instead of exponents we get A321649, rev A321650.
A000700 counts self-conjugate partitions, ranked by A088902.
A003963 gives product of prime indices, conjugate A329382.
A008480 gives number of permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798.
A124010 gives prime signature, sorted A118914, length A001221.
A352486-A352490 are sets related to the fixed points of A122111.
Cf. A000701, A000720, A046682, A238747, A258116, A319005, A330644, A352491.

Table[Length/@Table[Select[Last/@FactorInteger[n],#>=k&],{k,Max@@Last/@FactorInteger[n]}],{n,2,100}] (* Gus Wiseman, Mar 31 2022 *)

Row n is identical to row A124859(n) of table A212171.

A082090 Length of iteration sequence if function A056239, a pseudo-logarithm is iterated and started at n. Fixed point equals zero for all initial values.

Original entry on oeis.org

2, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 5, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 7, 6, 6, 6, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 6, 7, 6, 6, 6, 6, 6, 7, 6, 7, 6, 7, 6, 7, 7, 6, 6, 6, 6, 7, 6, 6, 7, 7, 6, 6, 7, 6, 6, 7, 6, 6, 7, 7, 6, 7, 6, 7, 6, 6, 6, 7, 6, 7, 6, 6, 7

Offset: 1

Labos Elemer, Apr 09 2003

nonn

From Gus Wiseman, Dec 01 2023: (Start)
Conjecture:
- The position of first appearance of k is n = A007097(k-2).
- The position of last appearance of k is n = A014221(k-2) = 2^^(k-2).
- The number of times k appears is: 1, 1, 2, 8, 435, ...
(End)

			n=127:list={127,31,11,5,3,2,1,0},a[127]=8

Mohammad K. Azarian, On the Fixed Points of a Function and the Fixed Points of its Composite Functions, International Journal of Pure and Applied Mathematics, Vol. 46, No. 1, 2008, pp. 37-44. Mathematical Reviews, MR2433713 (2009c:65129), March 2009. Zentralblatt MATH, Zbl 1160.65015.
Mohammad K. Azarian, Fixed Points of a Quadratic Polynomial, Problem 841, College Mathematics Journal, Vol. 38, No. 1, January 2007, p. 60. Solution published in Vol. 39, No. 1, January 2008, pp. 66-67.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A008475, A001414, A082083-A082086, A082091.
A112798 lists prime indices, length A001222, sum A056239.
A304038 lists distinct prime indices, length A001221, sum A066328.
Cf. A000079, A000720, A052410, A181819, A225485, A238747, A288636, A296150.

f:= n-> add (numtheory[pi](i[1])*i[2], i=ifactors(n)[2]):
a:= n-> 1+ `if`(n=1, 1, a(f(n))):
seq (a(n), n=1..120);  # Alois P. Heinz, Aug 09 2012

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] bpi[x_] := Table[PrimePi[Part[ba[x], j]], {j, 1, lf[x]}] api[x_] := Apply[Plus, ep[x]*bpi[x]] Table[Length[FixedPointList[api, w]]-1, {w, 2, 128}]
Table[Length[FixedPointList[Total[PrimePi/@Join@@ ConstantArray@@@FactorInteger[#]]&,n]]-1, {n,100}] (* Gus Wiseman, Dec 01 2023 *)

A367683 Numbers whose sorted prime signature is the same as the multiset multiplicity kernel of their prime indices.

Original entry on oeis.org

1, 2, 6, 9, 10, 12, 14, 18, 22, 26, 30, 34, 38, 40, 42, 46, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 102, 106, 110, 112, 114, 118, 122, 125, 126, 130, 134, 138, 142, 146, 154, 158, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 225

Offset: 1

Gus Wiseman, Nov 30 2023

nonn

We define the multiset multiplicity kernel (MMK) of a positive integer n to be the product of (least prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n. For example, 90 has prime factorization 2^1 * 3^2 * 5^1, so for k = 1 we have 2^2, and for k = 2 we have 3^1, so MMK(90) = 12. As an operation on multisets MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   18: {1,2,2}
   22: {1,5}
   26: {1,6}
   30: {1,2,3}
   34: {1,7}
   38: {1,8}
   40: {1,1,1,3}
   42: {1,2,4}
   46: {1,9}
   58: {1,10}
   62: {1,11}
   66: {1,2,5}
   70: {1,3,4}

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, 20 X 20 color coded list of terms, color function: gray = 1, red = prime, gold = composite prime power, green = squarefree composite, blue-purple = numbers neither squarefree nor prime powers. Bright green = primorial, light green = even squarefree semiprime, light blue = highly composite, middle blue = in A055932, purple = squareful but not a prime power.
Michael De Vlieger, 485 X 485 = 235225-term raster with the same color code as above.

Squarefree terms are A039956.
The LHS is A118914, unsorted A124010.
Prime-power terms are A307539.
The RHS is A367579, ranks A367580, sum A367581, max A367583.
Partitions of this type are counted by A367682.
A007947 gives squarefree kernel.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, reversed A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
A367582 counts partitions by sum of multiset multiplicity kernel.
Cf. A000720, A000961, A005117, A051904, A052409, A071625, A072774, A130091, A367584, A367586, A367587.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Select[Range[100], #==1||Sort[Last/@FactorInteger[#]] == mmk[PrimePi/@Join@@ConstantArray@@@FactorInteger[#]]&]

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.

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

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.

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

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.

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}]

cp's OEIS Frontend

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

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A238744 Irregular table read by rows: T (n, k) gives the number of primes p such that p^k divides n; table omits all zero values.

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A082090 Length of iteration sequence if function A056239, a pseudo-logarithm is iterated and started at n. Fixed point equals zero for all initial values.

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A367683 Numbers whose sorted prime signature is the same as the multiset multiplicity kernel of their prime indices.

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

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

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