A188579 -id:A188579 - cp's OEIS frontend

A188550 Maximal number of divisors d>1 of n-k such that n-d is a multiple of k, when k runs through values 2, 3, ..., floor(sqrt(n)).

1, 1, 2, 1, 2, 2, 3, 2, 2, 1, 4, 3, 2, 3, 4, 2, 3, 4, 4, 3, 2, 3, 6, 4, 4, 3, 4, 3, 4, 4, 5, 4, 4, 3, 6, 6, 3, 3, 6, 3, 4, 4, 4, 5, 4, 4, 8, 5, 6, 3, 4, 4, 4, 6, 6, 4, 4, 1, 8, 6, 4, 6, 6, 3, 5, 4, 4, 3, 4, 3, 9, 8, 6, 5, 6, 3, 4, 4, 8, 5, 6, 5, 8, 6, 4, 3, 6, 6, 6, 8, 6, 6, 4, 3, 10, 6, 8, 5, 6, 4, 6, 6, 6, 7, 4, 3, 8, 9, 4, 4, 8, 5, 6, 6, 4, 5, 4

Offset: 4

Vladimir Shevelev, Apr 04 2011

nonn

Conjecture: if the definition is changed so that k runs through values 2, 3, ..., floor((n-2)/2) then, beginning with n=6, the sequence remains without changes. - Vladimir Shevelev, Apr 10 2011
From Vladimir Shevelev, Jan 21 2013: (Start)
Other conjectures:
1) Primes 5, 7, 13 are only primes p for which a(p) = 1;
2) Primes 11 and 19 are only primes p for which a(p) = 2;
3) Let n = m^2 and m be the least value of k for which the number of divisors d > 1 of n-k, such that k|(n-d), equals a(n). Then m is prime or even power of a prime. (End)

Alois P. Heinz, Table of n, a(n) for n = 4..10004

Cf. A000005, A188579, A188586, A188591, A188592.

with(numtheory):
a:= n-> max(seq(nops(select(x-> irem(x, k)=0,
    [seq(n-d, d=divisors(n-k) minus{1})])), k=2..floor(sqrt(n)))):
seq(a(n), n=4..120);  # Alois P. Heinz, Apr 04 2011

a[n_] := Max @ Table[ Length @ Select[Table[n-d, {d, Divisors[n-k] // Rest} ], Mod[#, k] == 0&], {k, 2, Floor[Sqrt[n]]}]; Table[a[n], {n, 4, 120}] (* Jean-François Alcover, Feb 06 2016, after Alois P. Heinz *)

lim sup_{n -> infinity} a(n) = infinity. Indeed, it is easy to show that a(2^(2^n+1)) >= 2^n. Moreover, for n>5, we have a(2^(2^n+1)) > 2^n. - Vladimir Shevelev, Apr 09 2011

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).

Original entry on oeis.org

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 *)

A225868 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

6, 9, 12, 13, 16, 19, 24, 31, 32, 48, 53, 83, 89, 107, 113, 131, 139, 149, 167, 179, 191, 199, 227, 233, 251, 263, 409, 431, 449, 467, 479, 503, 587, 599, 631, 659, 683, 719, 769, 827, 983, 1019, 1091, 1259, 1367, 1409, 1439, 1487, 1499, 1511, 1583, 1619, 1979

Offset: 1

Vladimir Shevelev, May 18 2013

nonn

Terms >= 53 are primes p such that p+2 is either prime or semiprime or, relatively rarely, the cube of a prime. However, according to calculations by Peter J. C. Moses, up to 4.2*10^13 there are no numbers p in the sequence for which p+2 is cube of a prime. One can prove that if such a prime p exists, then it is necessary (but not sufficient) for all numbers of the quadruple {r, 2*r - 1, 4*r^2 - 6*r + 3, (2*r - 1)^3 - 2} to be primes, where r == 19 (mod 30) is defined by the equality (2r-1)^3 - 2 = p. The first 3 suitable values of r are 229, 3109, and 17449. But the corresponding p's are not in the sequence. We conjecture that all primes of the sequence are Chen primes, that is, all of them are in A109611.

Peter J. C. Moses, Table of n, a(n) for n = 1..300

Cf. A225867, A188579, A109611.

f[n_] := (m = 0; Do[s = Sum[ Boole[ Divisible[n+d, k]], {d, Divisors[n+k] // Rest}]; If[s > m, m = s], {k, 2, (n-2)/2}]; m); Reap[ For[n = 1, n <= 2000, n = If[n < 53, n+1, NextPrime[n]], If[f[n] == 3, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 09 2013, after Vladimir Shevelev *)

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

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 *)

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 *)

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

A188550 Maximal number of divisors d>1 of n-k such that n-d is a multiple of k, when k runs through values 2, 3, ..., floor(sqrt(n)).

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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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A225868 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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A188591 Records of A188550.

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

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

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