A322529
Number of integer partitions of n whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.
Original entry on oeis.org
1, 1, 2, 2, 1, 3, 2, 3, 2, 2, 4, 2, 3, 3, 4, 4, 4, 3, 5, 4, 5, 6, 6, 6, 6, 6, 8, 6, 7, 9, 8, 11, 8, 11, 11, 11, 12, 13, 13, 15, 13, 17, 17, 18, 18, 17, 20, 22, 21, 24, 24, 24, 26, 29, 28, 33, 30, 35, 34, 38, 38, 45, 42, 43, 45, 48, 52, 54, 55, 59, 59, 65, 65, 72, 73
Offset: 0
The a(30) = 8 integer partitions:
(30),
(17,13),(19,11),(23,7),
(17,11,2),(23,5,2),
(13,7,5,3,2),
(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1).
Cf.
A002865,
A003963,
A005117,
A038041,
A064573,
A073576,
A302505,
A319056,
A320322,
A321717,
A321718,
A322526,
A322527,
A322528,
A322530,
A322531.
-
Table[Length[Select[IntegerPartitions[n],And[SameQ@@PrimeOmega/@#,SquareFreeQ[Times@@#]]&]],{n,30}]
A319877
Numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).
Original entry on oeis.org
1, 7, 9, 14, 18, 23, 25, 28, 36, 46, 50, 56, 72, 92, 97, 100, 112, 121, 144, 151, 161, 169, 175, 183, 184, 185, 194, 195, 200, 207, 224, 225, 227, 242, 288, 289, 302, 322, 338, 350, 366, 368, 370, 388, 390, 400, 414, 448, 450, 454, 484, 541, 576, 578, 604, 644
Offset: 1
The sequence of multiset multisystems whose MM-numbers belong to the sequence begins:
1: {}
7: {{1,1}}
9: {{1},{1}}
14: {{},{1,1}}
18: {{},{1},{1}}
23: {{2,2}}
25: {{2},{2}}
28: {{},{},{1,1}}
36: {{},{},{1},{1}}
46: {{},{2,2}}
50: {{},{2},{2}}
56: {{},{},{},{1,1}}
72: {{},{},{},{1},{1}}
92: {{},{},{2,2}}
97: {{3,3}}
100: {{},{},{2},{2}}
112: {{},{},{},{},{1,1}}
121: {{3},{3}}
144: {{},{},{},{},{1},{1}}
151: {{1,1,2,2}}
161: {{1,1},{2,2}}
169: {{1,2},{1,2}}
175: {{2},{2},{1,1}}
183: {{1},{1,2,2}}
184: {{},{},{},{2,2}}
185: {{2},{1,1,2}}
194: {{},{3,3}}
195: {{1},{2},{1,2}}
200: {{},{},{},{2},{2}}
Cf.
A003963,
A005117,
A005176,
A062503,
A064573,
A072774,
A295193,
A302505,
A319878,
A319899,
A320325,
A322526,
A322527,
A322530.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Or[#==1,SameQ[##,2]&@@Last/@FactorInteger[Times@@primeMS[#]]]&]
A320323
Numbers whose product of prime indices (A003963) is a perfect power and where each prime index has the same number of prime factors, counted with multiplicity.
Original entry on oeis.org
7, 9, 19, 23, 25, 27, 49, 53, 81, 97, 103, 121, 125, 131, 151, 161, 169, 225, 227, 243, 289, 311, 343, 361, 419, 529, 541, 625, 661, 679, 691, 719, 729, 827, 841, 961, 1009, 1089, 1127, 1159, 1183, 1193, 1321, 1331, 1369, 1427, 1543, 1589, 1619, 1681, 1849
Offset: 1
The terms together with their corresponding multiset multisystems (A302242):
7: {{1,1}}
9: {{1},{1}}
19: {{1,1,1}}
23: {{2,2}}
25: {{2},{2}}
27: {{1},{1},{1}}
49: {{1,1},{1,1}}
53: {{1,1,1,1}}
81: {{1},{1},{1},{1}}
97: {{3,3}}
103: {{2,2,2}}
121: {{3},{3}}
125: {{2},{2},{2}}
131: {{1,1,1,1,1}}
151: {{1,1,2,2}}
161: {{1,1},{2,2}}
169: {{1,2},{1,2}}
225: {{1},{1},{2},{2}}
Cf.
A000720,
A001222,
A003963,
A056239,
A064573,
A112798,
A302242,
A305551,
A306017,
A319056,
A319066,
A319071,
A320324,
A320325.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And[GCD@@FactorInteger[Times@@primeMS[#]][[All,2]]>1,SameQ@@PrimeOmega/@primeMS[#]]&]
-
is(n) = my (f=factor(n), pi=apply(primepi, f[,1]~)); #Set(apply(bigomega, pi))==1 && ispower(prod(i=1, #pi, pi[i]^f[i,2])) \\ Rémy Sigrist, Oct 11 2018
A320699
Numbers whose product of prime indices is a nonprime prime power (A246547).
Original entry on oeis.org
7, 9, 14, 18, 19, 21, 23, 25, 27, 28, 36, 38, 42, 46, 49, 50, 53, 54, 56, 57, 63, 72, 76, 81, 84, 92, 97, 98, 100, 103, 106, 108, 112, 114, 115, 121, 125, 126, 131, 133, 144, 147, 152, 159, 162, 168, 171, 184, 189, 194, 196, 200, 206, 212, 216, 224, 227, 228
Offset: 1
The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (4), (2,2), (4,1), (2,2,1), (8), (4,2), (9), (3,3), (2,2,2), (4,1,1), (2,2,1,1), (8,1), (4,2,1), (9,1), (4,4), (3,3,1), (16), (2,2,2,1), (4,1,1,1), (8,2), (4,2,2), (2,2,1,1,1), (8,1,1), (2,2,2,2), (4,2,1,1), (9,1,1), (25), (4,4,1), (3,3,1,1).
Cf.
A000720,
A000961,
A001597,
A003963,
A056239,
A064573,
A112798,
A246547,
A246655,
A279787,
A320322,
A320325,
A320698,
A320700.
-
Select[Range[100],With[{x=Times@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]^k]},!PrimeQ[x]&&PrimePowerQ[x]]&]
A320700
Odd numbers whose product of prime indices is a nonprime prime power (A246547).
Original entry on oeis.org
7, 9, 19, 21, 23, 25, 27, 49, 53, 57, 63, 81, 97, 103, 115, 121, 125, 131, 133, 147, 159, 171, 189, 227, 243, 289, 311, 343, 361, 371, 393, 399, 419, 441, 477, 513, 515, 529, 567, 575, 625, 661, 691, 719, 729, 917, 931, 933, 961, 1007, 1009, 1029, 1067, 1083
Offset: 1
The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (4), (2,2), (8), (4,2), (9), (3,3), (2,2,2), (4,4), (16), (8,2), (4,2,2), (2,2,2,2), (25), (27), (9,3), (5,5), (3,3,3), (32), (8,4), (4,4,2), (16,2), (8,2,2), (4,2,2,2), (49), (2,2,2,2,2)
Cf.
A000720,
A000961,
A001597,
A003963,
A056239,
A064573,
A112798,
A246547,
A279787,
A320322,
A320325,
A320698,
A320699.
-
Select[Range[1000],With[{x=Times@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]^k]},OddQ[#]&&!PrimeQ[x]&&PrimePowerQ[x]]&]
A322546
Numbers k such that every integer partition of k contains a 1 or a prime power.
Original entry on oeis.org
1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23
Offset: 1
24 does not belong to the sequence because there are integer partitions of 24 containing no 1's or prime powers, namely: (24), (18,6), (14,10), (12,12), (12,6,6), (6,6,6,6).
Cf.
A000607,
A002095,
A023893,
A023894,
A064573,
A078135,
A101417,
A246655,
A320322,
A322452,
A322454,
A322547.
-
nn=100;
ser=Product[If[n==1||PrimePowerQ[n],1,1/(1-x^n)],{n,nn}];
Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1
A322547
Numbers k such that every integer partition of k contains a 1, a squarefree number, or a prime power.
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 59, 61, 67, 71, 79
Offset: 1
48 does not belong to the sequence because there are integer partitions of 48 containing no 1's, squarefree numbers, or prime powers, namely: (48), (36,12), (28,20), (24,24), (24,12,12), (18,18,12), (12,12,12,12).
-
nn=100;
ser=Product[If[PrimePowerQ[n]||SquareFreeQ[n],1,1/(1-x^n)],{n,nn}];
Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1
A319878
Odd numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).
Original entry on oeis.org
1, 7, 9, 23, 25, 97, 121, 151, 161, 169, 175, 183, 185, 195, 207, 225, 227, 289, 541, 661, 679, 687, 781, 841, 847, 873, 957, 961, 1009, 1089, 1193, 1427, 1563, 1589, 1681, 1819, 1849, 1879, 1895, 2023, 2043, 2167, 2193, 2209, 2231, 2425, 2437, 2585, 2601
Offset: 1
The sequence of multiset partitions whose MM-numbers belong to the sequence begins:
1: {}
7: {{1,1}}
9: {{1},{1}}
23: {{2,2}}
25: {{2},{2}}
97: {{3,3}}
121: {{3},{3}}
151: {{1,1,2,2}}
161: {{1,1},{2,2}}
169: {{1,2},{1,2}}
175: {{2},{2},{1,1}}
183: {{1},{1,2,2}}
185: {{2},{1,1,2}}
195: {{1},{2},{1,2}}
207: {{1},{1},{2,2}}
225: {{1},{1},{2},{2}}
227: {{4,4}}
289: {{4},{4}}
541: {{1,1,3,3}}
661: {{5,5}}
679: {{1,1},{3,3}}
687: {{1},{1,3,3}}
781: {{3},{1,1,3}}
841: {{1,3},{1,3}}
847: {{1,1},{3},{3}}
873: {{1},{1},{3,3}}
957: {{1},{3},{1,3}}
961: {{5},{5}}
Cf.
A003963,
A005117,
A005176,
A062503,
A064573,
A072774,
A295193,
A302505,
A319877,
A319899,
A320325,
A322526,
A322527,
A322530.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,100,2],Or[#==1,SameQ[##,2]&@@Last/@FactorInteger[Times@@primeMS[#]]]&]
A330106
Number of integer partitions of n whose product is a powerful number.
Original entry on oeis.org
0, 0, 0, 0, 2, 2, 5, 5, 9, 11, 18, 19, 30, 36, 51, 62, 87, 104, 141, 171, 225, 271, 349, 419, 534, 643, 804, 965, 1197, 1431, 1766, 2106, 2571, 3063, 3719, 4410, 5325, 6305, 7567, 8939, 10678, 12572, 14961, 17567, 20804, 24389, 28775, 33626, 39551, 46106
Offset: 0
The a(4) = 2 through a(10) = 18 partitions:
(4) (41) (33) (331) (8) (9) (55)
(22) (221) (42) (421) (44) (81) (82)
(222) (2221) (422) (333) (91)
(411) (4111) (2222) (441) (433)
(2211) (22111) (3311) (4221) (442)
(4211) (22221) (811)
(22211) (33111) (3322)
(41111) (42111) (3331)
(221111) (222111) (4222)
(411111) (4411)
(2211111) (22222)
(42211)
(222211)
(331111)
(421111)
(2221111)
(4111111)
(22111111)
Partitions whose product is a perfect power are
A320322.
-
powQ[n_]:=Min@@Last/@FactorInteger[n]>1;
Table[Length[Select[IntegerPartitions[n],powQ[Times@@#]&]],{n,0,30}]
A330216
Number of strict integer partitions of n whose product is a powerful number.
Original entry on oeis.org
0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 3, 4, 5, 7, 8, 8, 10, 12, 12, 15, 18, 19, 20, 24, 25, 28, 38, 41, 43, 50, 55, 63, 79, 85, 88, 104, 116, 124, 143, 157, 173, 197, 214, 235, 274, 294, 319, 363, 393, 430, 487, 529, 577, 647, 692, 752, 856, 925, 992, 1099
Offset: 0
The a(n) partitions for n = 4, 9, 12, 13, 16, 17, 18:
(4) (9) (8,4) (9,4) (16) (9,8) (12,6)
(8,1) (9,3) (6,4,3) (9,4,3) (16,1) (16,2)
(6,3,2,1) (8,4,1) (12,3,1) (8,6,3) (9,8,1)
(9,3,1) (9,4,2,1) (9,6,2) (8,6,3,1)
(6,4,3,2,1) (10,5,2) (9,4,3,2)
(12,3,2) (9,6,2,1)
(9,4,3,1) (10,5,2,1)
(12,3,2,1)
Partitions whose product is a perfect power are
A320322.
-
powQ[n_]:=Min@@Last/@FactorInteger[n]>1;
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&powQ[Times@@#]&]],{n,0,30}]
Comments