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A329132 Numbers whose augmented differences of prime indices are a periodic sequence. 7
4, 8, 15, 16, 32, 55, 64, 90, 105, 119, 128, 225, 253, 256, 403, 512, 540, 550, 697, 893, 935, 1024, 1155, 1350, 1357, 1666, 1943, 2048, 2263, 3025, 3071, 3150, 3240, 3375, 3451, 3927, 3977, 4096, 4429, 5123, 5500, 5566, 6731, 7735, 8083, 8100, 8192, 9089 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
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.
A sequence is periodic if its cyclic rotations are not all different.
LINKS
EXAMPLE
The sequence of terms together with their augmented differences of prime indices begins:
4: (1,1)
8: (1,1,1)
15: (2,2)
16: (1,1,1,1)
32: (1,1,1,1,1)
55: (3,3)
64: (1,1,1,1,1,1)
90: (2,1,2,1)
105: (2,2,2)
119: (4,4)
128: (1,1,1,1,1,1,1)
225: (1,2,1,2)
253: (5,5)
256: (1,1,1,1,1,1,1,1)
403: (6,6)
512: (1,1,1,1,1,1,1,1,1)
540: (2,1,1,2,1,1)
550: (3,1,3,1)
697: (7,7)
893: (8,8)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
aperQ[q_]:=Array[RotateRight[q, #1]&, Length[q], 1, UnsameQ];
aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Select[Range[100], !aperQ[aug[primeMS[#]//Reverse]]&]
CROSSREFS
Complement of A329133.
These are the Heinz numbers of the partitions counted by A329143.
Periodic binary words are A152061.
Periodic compositions are A178472.
Numbers whose binary expansion is periodic are A121016.
Numbers whose prime signature is periodic are A329140.
Numbers whose differences of prime indices are periodic are A329134.
Sequence in context: A104101 A130826 A312708 * A312709 A312710 A312711
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 06 2019
STATUS
approved

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Last modified May 18 15:24 EDT 2024. Contains 372664 sequences. (Running on oeis4.)