Displaying 1-10 of 17 results found.
Least number whose multiset multiplicity kernel (in which each prime exponent becomes the least prime factor with that exponent) is n. First position of n in A367580.
+20
13
1, 2, 3, 6, 5, 12, 7, 30, 15, 20, 11, 90, 13, 28, 45, 210, 17, 60, 19, 150, 63, 44, 23, 630, 35, 52, 105, 252, 29, 360, 31, 2310, 99, 68, 175, 2100, 37, 76, 117, 1050, 41, 504, 43, 396, 525, 92, 47, 6930, 77, 140, 153, 468, 53, 420, 275, 1470, 171, 116, 59
COMMENTS
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 the triangle A367579, and as an operation on their ranks it is represented by A367580.
FORMULA
a(p) = p for all primes p.
EXAMPLE
The least number with multiset multiplicity kernel 9 is 15, so a(9) = 15.
The terms together with their prime indices begin:
1 -> 1: {}
2 -> 2: {1}
3 -> 3: {2}
4 -> 6: {1,2}
5 -> 5: {3}
6 -> 12: {1,1,2}
7 -> 7: {4}
8 -> 30: {1,2,3}
9 -> 15: {2,3}
10 -> 20: {1,1,3}
11 -> 11: {5}
12 -> 90: {1,2,2,3}
13 -> 13: {6}
14 -> 28: {1,1,4}
15 -> 45: {2,2,3}
16 ->210: {1,2,3,4}
MATHEMATICA
nn=1000;
mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q, #]==i&], {i, mts}]]];
spnm[y_]:=Max@@NestWhile[Most, Sort[y], Union[#]!=Range[Max@@#]&];
qq=Table[Times@@mmk[Join@@ConstantArray@@@FactorInteger[n]], {n, nn}];
Table[Position[qq, i][[1, 1]], {i, spnm[qq]}]
CROSSREFS
Positions of squarefree numbers are A000961. Contains no nonprime prime powers A246547. Positions of first appearances in A367580. A071625 counts distinct prime exponents.
Numbers whose prime indices have a multiset multiplicity kernel (in which each prime exponent becomes the least prime factor with that exponent) that is all ones {1,1,...}. Positions of powers of 2 in A367580.
+20
9
1, 2, 4, 6, 8, 10, 14, 16, 22, 26, 30, 32, 34, 36, 38, 42, 46, 58, 62, 64, 66, 70, 74, 78, 82, 86, 94, 100, 102, 106, 110, 114, 118, 122, 128, 130, 134, 138, 142, 146, 154, 158, 166, 170, 174, 178, 182, 186, 190, 194, 196, 202, 206, 210, 214, 216, 218, 222
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 kernel MMK(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 min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.
FORMULA
Consists of 1 and all even terms of A072774 (powers of squarefree numbers).
EXAMPLE
We have MMK({1,1,2,2}) = {1,1} so 36 is in the sequence.
The terms together with their prime indices begin:
1: {}
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
10: {1,3}
14: {1,4}
16: {1,1,1,1}
22: {1,5}
26: {1,6}
30: {1,2,3}
32: {1,1,1,1,1}
34: {1,7}
36: {1,1,2,2}
38: {1,8}
42: {1,2,4}
MATHEMATICA
Select[Range[100], #==1||EvenQ[#]&&SameQ@@Last/@FactorInteger[#]&]
CROSSREFS
Partitions of this type (uniform containing 1) are counted by A097986. Positions of all one rows {1,1,...} in A367579. Positions of powers of 2 in A367580. A071625 counts distinct prime exponents.
Numbers k such that MMK(k) = MMK(i) for some i < k, where MMK is multiset multiplicity kernel A367580.
+20
3
4, 8, 9, 10, 14, 16, 18, 21, 22, 24, 25, 26, 27, 32, 33, 34, 36, 38, 39, 40, 42, 46, 48, 49, 50, 51, 54, 55, 56, 57, 58, 62, 64, 65, 66, 69, 70, 72, 74, 75, 78, 80, 81, 82, 84, 85, 86, 87, 88, 91, 93, 94, 95, 96, 98, 100, 102, 104, 106, 108, 110, 111, 112, 114
COMMENTS
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.
EXAMPLE
The terms together with their prime indices begin:
4: {1,1}
8: {1,1,1}
9: {2,2}
10: {1,3}
14: {1,4}
16: {1,1,1,1}
18: {1,2,2}
21: {2,4}
22: {1,5}
24: {1,1,1,2}
25: {3,3}
26: {1,6}
27: {2,2,2}
32: {1,1,1,1,1}
33: {2,5}
34: {1,7}
36: {1,1,2,2}
MATHEMATICA
nn=100;
mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q, #]==i&], {i, mts}]]];
qq=Table[Times@@mmk[Join @@ ConstantArray@@@FactorInteger[n]], {n, nn}];
Select[Range[nn], MemberQ[Take[qq, #-1], qq[[#]]]&]
CROSSREFS
Sorted positions of non-first appearances in A367580. A071625 counts distinct prime exponents.
Irregular triangle read by rows where row n is the multiset multiplicity kernel (MMK) of the multiset of prime indices of n.
+10
15
1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 1, 5, 1, 2, 6, 1, 1, 2, 2, 1, 7, 1, 2, 8, 1, 3, 2, 2, 1, 1, 9, 1, 2, 3, 1, 1, 2, 1, 4, 10, 1, 1, 1, 11, 1, 2, 2, 1, 1, 3, 3, 1, 1, 12, 1, 1, 2, 2, 1, 3, 13, 1, 1, 1, 14, 1, 5, 2, 3, 1, 1, 15, 1, 2, 4, 1, 3, 2, 2, 1, 6, 16, 1, 2
COMMENTS
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 kernel MMK(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 min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}.
Note: I chose the word 'kernel' because, as with A007947 and A304038, MMK(m) is constructed using the same underlying elements as m and has length equal to the number of distinct elements of m. However, it is not necessarily a submultiset of m.
FORMULA
For all positive integers n and k, row n^k is the same as row n.
EXAMPLE
The first 45 rows:
1: {} 16: {1} 31: {11}
2: {1} 17: {7} 32: {1}
3: {2} 18: {1,2} 33: {2,2}
4: {1} 19: {8} 34: {1,1}
5: {3} 20: {1,3} 35: {3,3}
6: {1,1} 21: {2,2} 36: {1,1}
7: {4} 22: {1,1} 37: {12}
8: {1} 23: {9} 38: {1,1}
9: {2} 24: {1,2} 39: {2,2}
10: {1,1} 25: {3} 40: {1,3}
11: {5} 26: {1,1} 41: {13}
12: {1,2} 27: {2} 42: {1,1,1}
13: {6} 28: {1,4} 43: {14}
14: {1,1} 29: {10} 44: {1,5}
15: {2,2} 30: {1,1,1} 45: {2,3}
MATHEMATICA
mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q, #]==i&], {i, mts}]]];
Table[mmk[PrimePi/@Join@@ConstantArray@@@If[n==1, {}, FactorInteger[n]]], {n, 100}]
CROSSREFS
Indices of empty and singleton rows are A000961. Rows have A071625 distinct elements. Indices of constant rows are A072774. Indices of strict rows are A130091. Index of first row with Heinz number n is A367584. Sorted row indices of first appearances are A367585. Indices of rows of the form {1,1,...} are A367586. A367582 counts partitions by sum of multiset multiplicity kernel.
Cf. A000720, A001597, A005117, A027746, A027748, A051904, A052409, A061395, A062770, A175781, A288636, A289023.
Sum of the multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity) of the prime indices of n.
+10
14
0, 1, 2, 1, 3, 2, 4, 1, 2, 2, 5, 3, 6, 2, 4, 1, 7, 3, 8, 4, 4, 2, 9, 3, 3, 2, 2, 5, 10, 3, 11, 1, 4, 2, 6, 2, 12, 2, 4, 4, 13, 3, 14, 6, 5, 2, 15, 3, 4, 4, 4, 7, 16, 3, 6, 5, 4, 2, 17, 5, 18, 2, 6, 1, 6, 3, 19, 8, 4, 3, 20, 3, 21, 2, 5, 9, 8, 3, 22, 4, 2, 2
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 kernel MMK(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 min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets, MMK is represented by A367579, and as an operation on their Heinz numbers, it is represented by A367580.
FORMULA
a(n^k) = a(n) for all positive integers n and k.
EXAMPLE
The multiset multiplicity kernel of {1,2,2,3} is {1,1,2}, so a(90) = 4.
MATHEMATICA
mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q, #]==i&], {i, mts}]]];
Table[Total[mmk[PrimePi/@Join@@ConstantArray@@@FactorInteger[n]]], {n, 100}]
CROSSREFS
Positions of 1's are A000079 without 1. Positions of first appearances are A008578. The triangle for this rank statistic is A367582. Cf. A000720, A005117, A051904, A055396, A061395, A071625, A072774, A130091, A175781, A367584, A367585.
Greatest element in row n of A367579 (multiset multiplicity kernel).
+10
12
0, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 2, 6, 1, 2, 1, 7, 2, 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, 3, 2, 6, 16, 2, 3, 4, 2, 1, 17, 2, 18, 1, 4, 1, 3, 1, 19, 7, 2, 1, 20, 2, 21, 1, 3, 8, 4, 1, 22, 3, 2, 1
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 kernel MMK(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 min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}.
FORMULA
a(n^k) = a(n) for all positive integers n and k.
If n is a power of a squarefree number, a(n) = A055396(n).
EXAMPLE
For 450 = 2^1 * 3^2 * 5^2, we have MMK({1,2,2,3,3}) = {1,2,2} so a(450) = 2.
MATHEMATICA
mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q, #]==i&], {i, mts}]]];
Table[If[n==1, 0, Max@@mmk[PrimePi/@Join@@ConstantArray@@@If[n==1, {}, FactorInteger[n]]]], {n, 1, 100}]
CROSSREFS
Positions of first appearances are A008578. For minimum instead of maximum element we have A055396. Positions of 1's are A367586 (powers of even squarefree numbers). A072774 lists powers of squarefree numbers.
A363486 gives least prime index of greatest exponent.
A363487 gives greatest prime index of greatest exponent.
A364191 gives least prime index of least exponent.
A364192 gives greatest prime index of least exponent.
Cf. A000720, A000961, A051904, A061395, A071625, A130091, A289023, A367581, A367584, A367585, A367683.
Triangle read by rows where T(n,k) is the number of integer partitions of n whose multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity), sums to k.
+10
11
1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 2, 2, 1, 0, 1, 3, 3, 2, 1, 1, 0, 1, 1, 4, 3, 3, 2, 1, 0, 1, 3, 5, 4, 4, 3, 1, 1, 0, 1, 2, 6, 4, 8, 3, 3, 2, 1, 0, 1, 3, 7, 9, 6, 7, 4, 3, 1, 1, 0, 1, 1, 8, 7, 11, 9, 9, 4, 3, 2, 1
COMMENTS
We define the multiset multiplicity kernel MMK(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 min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets, MMK is represented by A367579, and as an operation on their Heinz numbers, it is represented by A367580.
EXAMPLE
Triangle begins:
1
0 1
0 1 1
0 1 1 1
0 1 2 1 1
0 1 1 2 2 1
0 1 3 3 2 1 1
0 1 1 4 3 3 2 1
0 1 3 5 4 4 3 1 1
0 1 2 6 4 8 3 3 2 1
0 1 3 7 9 6 7 4 3 1 1
0 1 1 8 7 11 9 9 4 3 2 1
0 1 5 10 11 13 10 11 6 5 3 1 1
0 1 1 10 11 17 14 18 10 9 4 3 2 1
0 1 3 12 17 19 18 22 14 12 8 4 3 1 1
0 1 3 12 15 27 19 31 19 19 10 9 5 3 2 1
0 1 4 15 23 27 31 33 24 26 18 12 8 4 3 1 1
0 1 1 14 20 35 33 48 32 37 25 20 11 10 4 3 2 1
Row n = 7 counts the following partitions:
(1111111) (61) (421) (52) (4111) (511) (7)
(2221) (331) (322) (43)
(22111) (31111) (3211)
(211111)
MATHEMATICA
mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q, #]==i&], {i, mts}]]];
Table[Length[Select[IntegerPartitions[n], Total[mmk[#]]==k&]], {n, 0, 10}, {k, 0, n}]
CROSSREFS
A072233 counts partitions by number of parts.
A116608 counts partitions by number of distinct parts.
A116861 counts partitions by sum of distinct parts.
Numbers k whose multiset multiplicity kernel (in which each prime exponent becomes the least prime factor with that exponent) is different from that of all positive integers less than k.
+10
10
1, 2, 3, 5, 6, 7, 11, 12, 13, 15, 17, 19, 20, 23, 28, 29, 30, 31, 35, 37, 41, 43, 44, 45, 47, 52, 53, 59, 60, 61, 63, 67, 68, 71, 73, 76, 77, 79, 83, 89, 90, 92, 97, 99, 101, 103, 105, 107, 109, 113, 116, 117, 124, 127, 131, 137, 139, 140, 143, 148, 149, 150
COMMENTS
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.
EXAMPLE
The terms together with their prime indices begin:
1: {} 28: {1,1,4} 60: {1,1,2,3}
2: {1} 29: {10} 61: {18}
3: {2} 30: {1,2,3} 63: {2,2,4}
5: {3} 31: {11} 67: {19}
6: {1,2} 35: {3,4} 68: {1,1,7}
7: {4} 37: {12} 71: {20}
11: {5} 41: {13} 73: {21}
12: {1,1,2} 43: {14} 76: {1,1,8}
13: {6} 44: {1,1,5} 77: {4,5}
15: {2,3} 45: {2,2,3} 79: {22}
17: {7} 47: {15} 83: {23}
19: {8} 52: {1,1,6} 89: {24}
20: {1,1,3} 53: {16} 90: {1,2,2,3}
23: {9} 59: {17} 92: {1,1,9}
MATHEMATICA
nn=100;
mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q, #]==i&], {i, mts}]]];
qq=Table[Times@@mmk[Join@@ConstantArray@@@FactorInteger[n]], {n, nn}];
Select[Range[nn], FreeQ[Take[qq, #-1], qq[[#]]]&]
CROSSREFS
All terms are rootless A007916 (have no positive integer roots). Positions of squarefree terms appear to be A073485. Contains no nonprime prime powers A246547. Sorted positions of first appearances in A367580. A071625 counts distinct prime exponents.
Least element in row n of A367858 (multiset multiplicity cokernel).
+10
8
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
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.
FORMULA
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).
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}]
CROSSREFS
Indices of first appearances are A008578. For kernel instead of cokernel we have A055396. For maximum instead of minimum element we have A061395.
Numbers divisible by their multiset multiplicity kernel.
+10
6
1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 100, 101, 103, 104, 107
COMMENTS
First differs from A344586 in lacking 120. 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.
EXAMPLE
The terms together with their prime indices begin:
1: {}
2: {1}
3: {2}
4: {1,1}
5: {3}
7: {4}
8: {1,1,1}
9: {2,2}
11: {5}
12: {1,1,2}
13: {6}
16: {1,1,1,1}
17: {7}
18: {1,2,2}
19: {8}
20: {1,1,3}
23: {9}
24: {1,1,1,2}
MATHEMATICA
mmk[n_Integer]:= Product[Min[#]^Length[#]&[First/@Select[FactorInteger[n], Last[#]==k&]], {k, Union[Last/@FactorInteger[n]]}];
Select[Range[100], Divisible[#, mmk[#]]&]
CROSSREFS
The only squarefree terms are the primes A008578. Partitions of this type are counted by A367684. A071625 counts distinct prime exponents.
A124010 gives multiset of multiplicities (prime signature), sorted A118914.
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