

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
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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

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.


KEYWORD

nonn


AUTHOR



STATUS

approved



