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.
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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[#]]&]
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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
A371732
Numbers n such that each binary index k (from row n of A048793) has the same sum of binary indices A029931(k).
Original entry on oeis.org
1, 2, 4, 8, 12, 16, 32, 64, 128, 144, 256, 288, 512, 576, 1024, 2048, 3072, 4096, 8192, 16384, 32768, 32800, 33024, 33056, 65536, 65600, 66048, 66112, 131072, 132096, 133120, 134144, 262144, 266240, 524288, 528384, 786432, 790528, 1048576, 1056768, 2097152
Offset: 1
The terms together with their binary expansions and binary indices begin:
1: 1 ~ {1}
2: 10 ~ {2}
4: 100 ~ {3}
8: 1000 ~ {4}
12: 1100 ~ {3,4}
16: 10000 ~ {5}
32: 100000 ~ {6}
64: 1000000 ~ {7}
128: 10000000 ~ {8}
144: 10010000 ~ {5,8}
256: 100000000 ~ {9}
288: 100100000 ~ {6,9}
512: 1000000000 ~ {10}
576: 1001000000 ~ {7,10}
1024: 10000000000 ~ {11}
2048: 100000000000 ~ {12}
3072: 110000000000 ~ {11,12}
4096: 1000000000000 ~ {13}
8192: 10000000000000 ~ {14}
16384: 100000000000000 ~ {15}
32768: 1000000000000000 ~ {16}
32800: 1000000000100000 ~ {6,16}
For prime instead of binary indices we have
A326534.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
Cf.
A035470,
A038041,
A237258,
A320324,
A321453,
A321455,
A326518,
A336137,
A371783,
A371791,
A371796.
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bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],SameQ@@Total/@bix/@bix[#]&]
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