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A329132
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Numbers whose augmented differences of prime indices are a periodic sequence.
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7
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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)
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MATHEMATICA
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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]]&]
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CROSSREFS
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These are the Heinz numbers of the partitions counted by A329143.
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.
Cf. A000961, A027375, A056239, A112798, A325356, A325389, A325394, A328594, A329135, A329136, A329139.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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