A362003 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.

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

Offset: 1

Author

Michael De Vlieger, Apr 05 2023

Keywords

• nonn

Comments

Most small squarefree m have k - m^2 > m. For prime m = p, k = p^3, hence (p^3 - p^2) > p.

Examples

			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)
  -----------------------------
   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
		

Crossrefs

Cf. A007947, A120944, A362045.

Programs

Mathematica

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]]

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

Formula

This sequence is { m : A362045(n) - m^2 < m and m in A120944 }.