A362045 -id:A362045 - cp's OEIS frontend

A362044 a(n) = largest k such that k < m^2 and rad(k) | m, where rad(k) = A007947(k) and m = A120944(n).

32, 80, 128, 135, 343, 352, 512, 864, 891, 1088, 875, 1216, 1053, 1728, 2048, 2187, 1375, 2187, 2048, 2048, 3125, 4224, 2187, 4802, 4736, 3773, 5832, 5248, 4913, 5504, 7047, 4459, 7533, 8192, 6859, 10368, 10935, 8192, 11264, 8991, 12312, 12167, 8192, 5831, 8192, 9963, 10449, 16640, 16807, 17152, 18432

Offset: 1

Michael De Vlieger, Apr 05 2023

nonn

The largest k such that k < p^2 such that p is prime and rad(k) | p is p itself.

			a(1) = 32 since m = 6 and the largest k < m^2 such that rad(k) | 6 is 32. This is to say, the number that precedes 6^2 in A003586 is 32.
a(2) = 80 since m = 10 and the largest k < m^2 such that rad(k) | 10 is 80. This is to say, the number that precedes 10^2 in A003592 is 80.
Table of n = 1..12, m = A120944(n), a(n), and m^2.
   n    m    a(n)   m^2
  ---------------------
   1    6     32     36
   2   10     80    100
   3   14    128    196
   4   15    135    225
   5   21    343    441
   6   22    352    484
   7   26    512    676
   8   30    864    900
   9   33    891   1089
  10   34   1088   1156
  11   35    875   1225
  12   38   1216   1444

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Scatterplot of a(n), m^2, and b(n), n = 1..2^14, where b(n) = A362045(n) is shown in red, m^2 in black, and a(n) in blue.

Cf. A007947, A120944, A289280, A362045.

Table[m = k^2 - 1; While[! Divisible[k, Times @@ FactorInteger[m][[All, 1]]], m--]; m, {k, Select[Range[6, 133], And[CompositeQ[#], SquareFreeQ[#]] &]}]

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.

Original entry on oeis.org

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

Michael De Vlieger, Apr 05 2023

nonn

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

			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

Cf. A007947, A120944, A362045.

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]]

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

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

cp's OEIS Frontend

A362044 a(n) = largest k such that k < m^2 and rad(k) | m, where rad(k) = A007947(k) and m = A120944(n).

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