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A372438
Least binary index equals greatest prime index.
8
6, 18, 20, 54, 56, 60, 100, 162, 168, 176, 180, 280, 300, 392, 416, 486, 500, 504, 528, 540, 840, 880, 900, 1088, 1176, 1232, 1248, 1400, 1458, 1500, 1512, 1584, 1620, 1936, 1960, 2080, 2432, 2500, 2520, 2640, 2700, 2744, 2912, 3264, 3528, 3696, 3744, 4200
OFFSET
1,1
COMMENTS
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
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.
Are there any squarefree terms > 6?
FORMULA
A001511(a(n)) = A061395(a(n)).
EXAMPLE
The binary indices of 60 are {3,4,5,6}, the prime indices are {1,1,2,3}, and 3 = 3, so 60 is in the sequence.
The terms together with their prime indices begin:
6: {1,2}
18: {1,2,2}
20: {1,1,3}
54: {1,2,2,2}
56: {1,1,1,4}
60: {1,1,2,3}
100: {1,1,3,3}
162: {1,2,2,2,2}
168: {1,1,1,2,4}
176: {1,1,1,1,5}
180: {1,1,2,2,3}
280: {1,1,1,3,4}
300: {1,1,2,3,3}
The terms together with their binary expansions and binary indices begin:
6: 110 ~ {2,3}
18: 10010 ~ {2,5}
20: 10100 ~ {3,5}
54: 110110 ~ {2,3,5,6}
56: 111000 ~ {4,5,6}
60: 111100 ~ {3,4,5,6}
100: 1100100 ~ {3,6,7}
162: 10100010 ~ {2,6,8}
168: 10101000 ~ {4,6,8}
176: 10110000 ~ {5,6,8}
180: 10110100 ~ {3,5,6,8}
280: 100011000 ~ {4,5,9}
300: 100101100 ~ {3,4,6,9}
MATHEMATICA
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], Min[bix[#]]==Max[prix[#]]&]
CROSSREFS
Same length: A071814, zeros of A372441.
Same sum: A372427, zeros of A372428.
Same maxima: A372436, zeros of A372442.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Sequence in context: A088724 A124276 A107405 * A077663 A025163 A186889
KEYWORD
nonn,base
AUTHOR
Gus Wiseman, May 04 2024
STATUS
approved