A358332
Numbers whose prime indices have arithmetic and geometric mean differing by one.
Original entry on oeis.org
57, 228, 1064, 1150, 1159, 2405, 3249, 7991, 29785, 29999, 30153, 35378, 51984, 82211, 133931, 185193, 187039, 232471, 242592, 374599, 404225, 431457, 685207, 715129, 927288, 1132096, 1165519, 1322500, 1343281, 1555073, 1872413, 2016546, 2873687, 3468319, 4266421, 4327344
Offset: 1
The terms together with their prime indices begin:
57: {2,8}
228: {1,1,2,8}
1064: {1,1,1,4,8}
1150: {1,3,3,9}
1159: {8,18}
2405: {3,6,12}
3249: {2,2,8,8}
7991: {18,32}
29785: {3,4,9,12}
29999: {32,50}
30153: {2,8,9,9}
35378: {1,4,4,8,8}
51984: {1,1,1,1,2,2,8,8}
82211: {50,72}
133931: {4,8,8,16}
185193: {2,2,2,8,8,8}
187039: {72,98}
232471: {12,18,27}
The partitions with these Heinz numbers are counted by
A358331.
A003963 multiplies together prime indices.
A067539 counts partitions with integer geometric mean, ranked by
A326623.
A078175 lists numbers whose prime factors have integer average.
A320322 counts partitions whose product is a perfect power.
Cf.
A000720,
A051293,
A111233,
A215366,
A289508,
A289509,
A326027,
A326624,
A326028,
A326645,
A357710.
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primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],Mean[primeMS[#]]==1+GeometricMean[primeMS[#]]&]
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isok(k) = if (k>1, my(f=factor(k), vf=List()); for (i=1, #f~, for (j=1, f[i,2], listput(vf, primepi(f[i,1])))); my(v = Vec(vf)); vecsum(v)/#v == 1 + sqrtn(vecprod(v), #v);); \\ Michel Marcus, Nov 11 2022
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.
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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.
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powQ[n_]:=Min@@Last/@FactorInteger[n]>1;
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&powQ[Times@@#]&]],{n,0,30}]
A339555
Number of subsets of {2..n} such that the product of the elements is a perfect power.
Original entry on oeis.org
1, 1, 1, 1, 3, 3, 5, 5, 11, 25, 41, 41, 80, 80, 144, 284, 568, 568, 1147, 1147, 2339, 4667, 8763, 8763, 17548, 35196, 67964, 135918, 273806, 273806, 548956, 548956, 1097974, 2194294, 4291446, 8608698, 17216783, 17216783, 33993999, 67979983, 135956742
Offset: 0
a(8) = 11 subsets: {}, {4}, {8}, {2, 4}, {2, 8}, {4, 8}, {2, 3, 6}, {2, 4, 8}, {3, 6, 8}, {2, 3, 4, 6} and {3, 4, 6, 8}.
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