A188550 -id:A188550 - cp's OEIS frontend

A188794 a(n) is the smallest integer k >= 2 such that the number of divisors d>1 of n-k with k|(n-d) equals A188550(n).

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

Offset: 4

Vladimir Shevelev, Apr 10 2011

a(n) <= floor(sqrt(n)) follows from the definition of A188550.

Alois P. Heinz, Table of n, a(n) for n = 4..20004

Cf. A188550, A188579.

with(numtheory):
a:= 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:
seq(a(n), n=4..120);  # Alois P. Heinz, Apr 10 2011

a[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];
a /@ Range[4, 120] (* Jean-François Alcover, Oct 28 2020, after Alois P. Heinz *)

A188591 Records of A188550.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 32, 36, 40, 48, 60, 64, 72, 80, 84, 90, 96, 100, 108, 120, 128, 144, 160, 168, 180, 192, 200, 216

Offset: 1

Vladimir Shevelev, Apr 04 2011

Is this the same sequence as A002183, number of divisors of n-th highly composite number?

Cf. A188550, A188579.

max = 10^6; (* b = A188550 *) b[n_] := Max @ Table[Length @ Select[ Table[ n-d, {d, Divisors[n-k] // Rest}], Mod[#, k] == 0&], {k, 2, Floor[ Sqrt[n] ]}]; A188591 = Reap[For[record = 0; k = 1; n = 1, n <= max, n++, bn = b[n]; If[bn > record, record = bn; Print["a(", k++, ") = b(", n, ") = ", bn]; Sow[bn]]]][[2, 1]] (* Jean-François Alcover, Feb 07 2016 *)

More terms from Jean-François Alcover, Feb 07 2016

A188586 a(n) is the smallest m for which A188550(m)=n, or a(n)=0 if no such m exists.

Original entry on oeis.org

4, 6, 10, 14, 34, 26, 107, 50, 74, 98, 317, 122, 5443, 386, 290, 242, 1577, 362, 2837, 482, 1154, 6146, 3467, 722, 2594, 11027, 1802, 1922, 14177, 1442, 10397, 1682, 18434, 358406, 10370, 2522, 445541, 288293, 31187, 3362, 127577, 5762, 145603, 30722, 7202

Offset: 1

Vladimir Shevelev, Apr 04 2011

nonn

Question: For which primes p exist terms in the sequence of the form 2*p^2? The sequence of these primes starts with 5, 7, 11, 19, 31, 29, 41, ...

Cf. A188550, A188579.

More terms from Alois P. Heinz, Apr 04 2011

A188592 Places of records of A188550.

Original entry on oeis.org

4, 6, 10, 14, 26, 50, 74, 98, 122, 242, 362, 482, 722, 1442, 1682, 2522, 3362, 5042, 10082, 15122, 20162, 30242, 40322, 50402, 55442, 90722, 100802, 110882, 166322, 221762, 332642, 443522

Offset: 1

Vladimir Shevelev, Apr 04 2011

Questions:
1) Are there any terms after a(9) which are not of the form 10*k+2?
2) For which primes p do there exist terms in the sequence of the form 2*p^2? The sequence of these primes starts with 5, 7, 11, 19, 29, 41, 71, ...

Cf. A188550, A188579, A188591.

More terms from Alois P. Heinz, Apr 04 2011

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 *)

A188579 Numbers m for which max_{2<=k<(m-2)/2} Sum_{d>1: d|m-k, k|m-d} 1 = 3.

Original entry on oeis.org

10, 15, 17, 20, 23, 25, 29, 31, 37, 40, 41, 43, 53, 67, 71, 73, 79, 89, 97, 109, 127, 151, 157, 181, 193, 239, 241, 271, 313, 331, 337, 349, 373, 397, 421, 433, 449, 601, 613, 661, 673, 701, 757, 811, 1009, 1021, 1051, 1117, 1249, 1471, 1531, 1741

Offset: 1

Vladimir Shevelev, Apr 04 2011

nonn

All terms a(n) >= 41 are primes. - Vladimir Shevelev, May 12 2013

If prime p is in the sequence, then either (p-2,p) is a twin prime pair, or p-2 = q*r, where q and r are distinct primes, or p-2 is the cube of a prime. - Vladimir Shevelev, May 15 2013

			Let n=10. Then k takes the values 2 and 3. If k=3, then d=7 and k divides n-d; if k=2, then d = 2,4,8, n-d = 8,6,2 and k divides all these values. Since max(1,3) = 3, 10 is in the sequence. - _Vladimir Shevelev_, May 12 2013

Peter J. C. Moses, Table of n, a(n) for n = 1..215
Vladimir Shevelev, Proof that all terms >= 41 of A188579 are primes, SeqFan Mailing List, May 15, 2013.

Cf. A188550.

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

Original entry on oeis.org

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 *)

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

cp's OEIS Frontend

A188794 a(n) is the smallest integer k >= 2 such that the number of divisors d>1 of n-k with k|(n-d) equals A188550(n).

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A188591 Records of A188550.

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A188586 a(n) is the smallest m for which A188550(m)=n, or a(n)=0 if no such m exists.

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A188592 Places of records of A188550.

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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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A188579 Numbers m for which max_{2<=k<(m-2)/2} Sum_{d>1: d|m-k, k|m-d} 1 = 3.

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

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