A225867 -id:A225867 - cp's OEIS frontend

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).

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 *)

A226977 Places n where A225867(n) <= 2.

Original entry on oeis.org

7, 8, 11, 17, 23, 29, 41, 47, 59, 71, 239, 359, 419, 839

Offset: 1

Vladimir Shevelev, Jun 27 2013

Except for a(2)=8, all terms are prime.
Is this sequence finite?
There are no other terms up to 86000. - Peter J. C. Moses, Jun 28 2013
There are no more terms up to 10^9. - Charles R Greathouse IV, Nov 25 2014

Cf. A225867, A049002.

is(n)=for(k=2,n\2-1,if(sumdiv(n+k,d,(n+d)%k==0 && d>1)>2, return(0))); n>6 \\ Charles R Greathouse IV, Nov 25 2014

A227010 Places n where A225867(n) = 4.

Original entry on oeis.org

10, 14, 15, 18, 20, 21, 25, 26, 27, 28, 35, 36, 37, 39, 40, 42, 49, 50, 52, 56, 63, 64, 65, 67, 72, 74, 79, 80, 84, 91, 96, 100, 101, 104, 109, 111, 112, 125, 127, 128, 144, 168, 181, 185, 192, 200, 211, 224, 256, 269, 281, 288, 293, 307, 336, 353, 379

Offset: 1

Vladimir Shevelev, Jun 27 2013

nonn

Note that the proportion of prime terms has a tendency to increase. For example,

1 from the first 20 terms; 6 from the first 40 terms; 16 from the first 60 terms; 34 from the first 80 terms; 53 from the first 100 terms; 90 from the first 140 terms. We conjecture that this proportion asymptotically tends to 1.

Cf. A225867, A225868, A226977.

A226727 Places n where A225867(n)=5.

Original entry on oeis.org

30, 43, 45, 60, 61, 73, 75, 90, 97, 120, 137, 151, 160, 169, 173, 180, 197, 240, 241, 257, 259, 311, 320, 331, 347, 360, 367, 389, 400, 421, 480, 491, 499, 569, 571, 601, 617, 640, 641, 647, 709, 720, 773, 800, 809, 857, 881, 919, 929, 947, 960, 971

Offset: 1

Vladimir Shevelev, Jun 29 2013

nonn

The proportion of prime terms has a tendency to increase. For example, 4 from the first 10 terms; 10 from the first 20 terms; 16 from the first 30 terms; 24 from the first 40 terms; 40 from the first 60 terms, 88 from the first 110 terms. We conjecture that this proportion asymptotically tends to 1.

Cf. A225867, A225868, A226977, A227010.

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 *)

A226653 Numbers n for which A226182(n^2) = n.

Original entry on oeis.org

3, 5, 7, 11, 17, 19, 23, 29, 31, 43, 47, 53, 59, 67, 71, 79, 83, 87, 89, 97, 101, 107, 109, 125, 127, 131, 137, 139, 149, 151, 167, 173, 179, 181, 191, 197, 199, 207, 223, 227, 229, 233, 239, 251, 263, 269, 271, 279, 281, 287, 293, 307, 311

Offset: 1

Vladimir Shevelev, Jun 14 2013

nonn

The sequence of primes which are not in this sequence begins: 2, 13, 37, 41, 61, 73, 103, 113, 157, 163, 193, 211, 241, ...

The sequence of composite numbers in this sequence begins: 87, 125, 207, 279, 287, 395, 489, 524, 543, 559, ... (Cf. comment in A188836)

Cf. A225867, A225868, A226182.

More terms from Peter J. C. Moses, Jul 01 2013.

cp's OEIS Frontend

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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A226977 Places n where A225867(n) <= 2.

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A227010 Places n where A225867(n) = 4.

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A226727 Places n where A225867(n)=5.

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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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A226653 Numbers n for which A226182(n^2) = n.

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