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A362003
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Squarefree composite numbers m such that k - m^2 < m, where k is the smallest number greater than m^2 such that rad(k) | m.
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0
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42, 66, 78, 362, 1086, 1122, 1254, 1794, 1810, 1846, 1974, 2534, 2730, 3318, 3982, 4890, 5538, 5590, 6006, 6214, 9230, 12922, 12990, 13515, 15510, 16205, 17430, 18642, 20306, 22170, 23170, 25098, 26962, 27030, 29274, 31070, 32142, 32410
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OFFSET
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1,1
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COMMENTS
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Most small squarefree m have k - m^2 > m. For prime m = p, k = p^3, hence (p^3 - p^2) > p.
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LINKS
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FORMULA
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EXAMPLE
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a(1) = 42 since 42 is the smallest squarefree number such that the smallest k > m^2 such that rad(k) | m also has difference k - m^2 < m.
Table showing a(n) = A120944(i) = m, A362045(i) = k, and the difference k-m^2.
i m k (k-m^2)
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14 42 1792 28
22 66 4374 18
27 78 6144 60
147 362 131072 28
478 1086 1179648 252
495 1122 1259712 828
558 1254 1572864 348
813 1794 3219264 828
822 1810 3276800 700
840 1846 3407872 156
900 1974 3898368 1692
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MATHEMATICA
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s = Select[Range[6, 400], And[CompositeQ[#], SquareFreeQ[#]] &]; Reap[Do[(m = #^2 + 1; While[! Divisible[#, Times @@ FactorInteger[m][[All, 1]]], m++]; If[m - #^2 < #, Sow[#]]) &[s[[i]]], {i, Length[s]}] ][[-1, -1]]
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PROG
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(PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
isok(m) = if (!isprime(m) && issquarefree(m), for (k=1+m^2, m+m^2, if (!(m % rad(k)), return(1)))); \\ Michel Marcus, Apr 21 2023
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CROSSREFS
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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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