A188794 -id:A188794 - cp's OEIS frontend

A188836 Numbers n for which A188794(n)^2 = n.

4, 9, 25, 49, 121, 169, 289, 361, 625, 841, 961, 1369, 1681, 1849, 3721, 4489, 5041, 5329, 7921, 9409, 10201, 10609, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 32761, 36481, 37249, 38809, 39601, 44521, 52441, 57121, 58081, 63001, 73441

Offset: 1

Vladimir Shevelev, Apr 12 2011

nonn

The sequence contains many squares of primes.
Question 1: What is the sequence of primes whose squares are not in this sequence?  It begins: 23, 47, 53, 59, 79, 83, 107, ... A188833
Question 2: What is the sequence of composite numbers whose squares are in this sequence?  It begins: 25, 289, 361, 529, ...

Cf. A188550, A188579, A188794, A188833.

with(numtheory):
b:= proc(n) local h, i, k, m;
       m, i:= 0, 0;
       for k from 2 to floor(sqrt(n)) do
          h:= nops(select(x-> irem(x, k)=0,
                  [seq (n-d, d=divisors(n-k) minus{1})]));
          if h>m then m, i:= h, k fi
       od; i
    end:
a:= proc(n) option remember; local k;
      for k from 1+ `if` (n=1, 3, a(n-1))
      while not b(k)^2=k do od; k
    end:
seq(a(n), n=1..15);  # Alois P. Heinz, Apr 13 2011

b[n_] := Module[{h, i = 0, k, m = 0}, For[k = 2, k <= Floor[Sqrt[n]], k++, h = Length[Select[Table[n - d, {d, Rest[Divisors[n - k]]}], Mod[#, k] == 0 &]]; If[h > m, {m, i} = {h, k}]]; i];
Reap[For[n = 1, n <= 80000, n++, If[b[n]^2==n, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Nov 11 2020, after Alois P. Heinz *)

A226182 a(n) is the smallest integer k >= 2 such that the number of divisors d>1 of n + k with k|n + d equals A225867(n).

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 3, 2, 2, 4, 3, 2, 2, 3, 5, 2, 3, 2, 3, 2, 2, 4, 3, 2, 5, 4, 3, 2, 3, 2, 3, 2, 2, 4, 3, 2, 2, 2, 7, 2, 3, 2, 2, 2, 5, 4, 3, 2, 4, 4, 2, 2, 3, 2, 5, 6, 3, 4, 3, 2, 4, 8, 2, 2, 5, 4, 7, 2, 2, 4, 3, 2, 2, 4, 5, 2, 3, 2, 2, 6, 7, 4

Offset: 6

Vladimir Shevelev, May 30 2013

nonn

			Let n = 33. We begin with k = 2. Divisors>1 of 33 + 2 = 35 are d = 5,7,35. For all d, 33 + d is divisible by k = 2. But the number of such d is 3, while A225867(33)= 6. Therefore, a(33) > 2. Consider now k = 3. Divisors>1 of 33 + 3 = 36 are 2,3,4,6,9,12,18,36, but only for d = 3,6,9,12,18,36, 33 + d is divisible by k = 3. Since we have exactly A225867(33) = 6 such divisors, then k = 3 is required and a(33) = 3.

Peter J. C. Moses, Table of n, a(n) for n = 6..3005

Cf. A225867, A225868, A188550, A188794.

A226182 := proc(n)
    local ak,k,nd,kpiv ;
    ak := 0 ;
    kpiv := 2 ;
    for k from 2 to n/2-1 do
        nd := 0 ;
        for d in numtheory[divisors](n+k) minus {1} do
            if modp(n+d,k) = 0 then
                nd := nd+1;
            end if;
        end do:
        if nd > ak then
            ak := max(ak,nd) ;
            kpiv := k ;
        end if;
    end do:
    kpiv ;
end proc: # R. J. Mathar, Jul 04 2013

Table[NestWhile[#+1&, 2, Max[Map[Count[(n+Rest[Divisors[n+#]])/#, Integer]&, Range[2, Floor[(n-2)/2]]]]-Count[(n+Rest[Divisors[n+#]])/#, _Integer] =!= 0&], {n,6,55}] (* _Peter J. C. Moses, Jun 03 2013 *)

A188795 a(n) counts all integers k in [2,floor(sqrt(n))] such that the number of divisors d>1 of n-k with k|(n-d) equals A188550(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 3, 1, 2, 2, 3, 2, 1, 1, 1, 1, 1, 1, 4, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1

Offset: 4

Vladimir Shevelev, Apr 10 2011

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..10000

Cf. A188550, A188579, A188794.

with(numtheory):
a:= proc(n) option remember; local c, h, k, m;
       m, c:= 0, 0;
       for k from 2 to floor(sqrt(n)) do
          h:= nops(select(x-> irem(x, k)=0,
              [seq (n-d, d=divisors(n-k) minus{1})]));
          if h=m then c:=c+1 elif h>m then m, c:= h, 1 fi
       od; c
    end:
seq(a(n), n=4..120);  # Alois P. Heinz, Apr 10 2011

b[n_] := Max @ Table[Length @ Select[Table[n-d, {d, Divisors[n-k] // Rest} ], Mod[#, k] == 0&], {k, 2, Floor[Sqrt[n]]}];
a[n_] := a[n] = Count[Range[2, Floor[Sqrt[n]]], k_ /; Count[Rest @ Divisors[n-k], d_ /; Divisible[n-d, k]] == b[n]];
Table[a[n], {n, 4, 120}] (* Jean-François Alcover, Mar 27 2017, after Alois P. Heinz *)

A188833 Primes p such that p^2 is not in A188836.

Original entry on oeis.org

23, 47, 53, 59, 79, 83, 107, 163, 167, 173, 179, 223, 227, 233, 257, 263, 269, 277, 283, 293, 317, 347, 353, 359, 367, 373, 383, 389, 401, 431, 439, 443, 457, 467, 479, 499, 503, 509, 557, 563, 569, 587, 593, 607, 643, 647, 653, 677, 683, 691, 719, 727, 733

Offset: 1

Vladimir Shevelev, Apr 12 2011

nonn

Cf. A188550, A188579, A188794, A188836.

A188550[n_] := Max @ Table[Length @ Select[Table[n-d, {d, Divisors[n-k] // Rest}], Mod[#, k] == 0&], {k, 2, Floor[Sqrt[n]]}]; A188794[n_] := Module[{k=2, a1=A188550[n]}, While[DivisorSum[n-k,1&, #>1&&Divisible[n-#,k]&] != a1, k++];k]; s={}; Do[p=Prime[n]; p2=p^2; If[aa[p2]^2 != p2, AppendTo[s,p]], {n, 1, 130}]; s (* Amiram Eldar, Feb 06 2019 after Jean-François Alcover at A188550 *)

More terms from Amiram Eldar, Feb 06 2019

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A188836 Numbers n for which A188794(n)^2 = n.

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A226182 a(n) is the smallest integer k >= 2 such that the number of divisors d>1 of n + k with k|n + d equals A225867(n).

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A188795 a(n) counts all integers k in [2,floor(sqrt(n))] such that the number of divisors d>1 of n-k with k|(n-d) equals A188550(n).

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A188833 Primes p such that p^2 is not in A188836.

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