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A356232
Numbers whose prime indices are all odd and cover an initial interval of odd positive integers.
16
1, 2, 4, 8, 10, 16, 20, 32, 40, 50, 64, 80, 100, 110, 128, 160, 200, 220, 250, 256, 320, 400, 440, 500, 512, 550, 640, 800, 880, 1000, 1024, 1100, 1210, 1250, 1280, 1600, 1760, 1870, 2000, 2048, 2200, 2420, 2500, 2560, 2750, 3200, 3520, 3740, 4000, 4096, 4400
OFFSET
1,2
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.
Also positions of first appearances of rows in A356226.
EXAMPLE
The terms together with their prime indices begin:
1: {}
2: {1}
4: {1,1}
8: {1,1,1}
10: {1,3}
16: {1,1,1,1}
20: {1,1,3}
32: {1,1,1,1,1}
40: {1,1,1,3}
50: {1,3,3}
64: {1,1,1,1,1,1}
80: {1,1,1,1,3}
100: {1,1,3,3}
110: {1,3,5}
128: {1,1,1,1,1,1,1}
160: {1,1,1,1,1,3}
200: {1,1,1,3,3}
220: {1,1,3,5}
250: {1,3,3,3}
256: {1,1,1,1,1,1,1,1}
320: {1,1,1,1,1,1,3}
400: {1,1,1,1,3,3}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
Select[Range[1000], normQ[(primeMS[#]+1)/2]&]
CROSSREFS
The partitions with these Heinz numbers are counted by A053251.
This is the odd restriction of A055932.
A subset of A066208 (numbers with all odd prime indices).
This is the sorted version of A356603.
These are the positions of first appearances of rows in A356226. Other statistics are:
- length: A287170, firsts A066205
- minimum: A356227
- maximum: A356228
- bisected length: A356229
- standard composition: A356230
- Heinz number: A356231
- positions of first appearances: A356232 (this sequence)
A001221 counts distinct prime factors, with sum A001414.
A001223 lists the prime gaps, reduced A028334.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices, row sums of A112798.
A073491 lists numbers with gapless prime indices, complement A073492.
Sequence in context: A287178 A093547 A068382 * A025612 A335239 A102248
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 20 2022
STATUS
approved