A225649 -id:A225649 - cp's OEIS frontend

A225648 Positions of ones in A225650, numbers n such that n and A000793(n) are coprime.

0, 1, 5, 7, 8, 9, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313

Offset: 1

Antti Karttunen, May 12 2013

nonn

Contains all primes from 5 onward. Are 8, 9 and 27 only composite numbers present?

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Complement: A225649. Cf. also A225650.

b[n_, i_] := b[n, i] = Module[{p}, p = If[i < 1, 1, Prime[i]]; If[n == 0 || i < 1, 1, Max[b[n, i - 1], Table[p^j*b[n - p^j, i - 1], {j, 1, Log[p, n] // Floor}]]]]; g[n_] := b[n, If[n < 8, 3, PrimePi[Ceiling[1.328*Sqrt[n* Log[n] // Floor]]]]]; Join[{0}, Position[Table[GCD[n, g[n]], {n, 1, 500} ], 1] // Flatten] (* Jean-François Alcover, Mar 03 2016, after Alois P. Heinz *)

A225651 Numbers k that divide A000793(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 12, 14, 15, 20, 21, 24, 30, 35, 36, 39, 40, 42, 44, 52, 55, 56, 60, 65, 66, 70, 72, 76, 77, 78, 84, 85, 90, 91, 95, 99, 102, 105, 110, 114, 115, 117, 119, 120, 126, 130, 132, 133, 136, 138, 140, 143, 152, 153, 154, 155, 156, 161, 165, 170

Offset: 1

Antti Karttunen, May 16 2013

nonn

After 1, a subset of A225649.

Also, for all n, A225650(a(n)) = a(n) and A225655(a(n)) = A000793(a(n)).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A225648, A225649, A225650, A225653, A225655-A225657.

b:= proc(n, i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0 or i<1, 1, max(b(n, i-1),
           seq(p^j*b(n-p^j, i-1), j=1..ilog[p](n))))
    end:
g:=n->b(n, `if`(n<8, 3, numtheory[pi](ceil(1.328*isqrt(n*ilog(n)))))):
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1))
      while not irem(g(k), k)=0 do od; k
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, May 22 2013

Reap[For[n=1, n <= 40, n++, If[Divisible[Max[LCM @@@ IntegerPartitions[n] ], n], Sow[n]]]][[2, 1]]
(* or, for a large number of terms: *)
b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n==0 || i<1, 1, Max[b[n, i-1], Table[p^j*b[n - p^j, i-1], {j, 1, Log[p, n] // Floor}]]]]; g[n_] := b[n, If[n<8, 3, PrimePi[Ceiling[1.328*Sqrt[n*Log[n] // Floor]]]]]; Reap[For[k=1, k <= 1000, k++, If[Divisible[g[k], k], Sow[ k]]]][[2, 1]] (* Jean-François Alcover, Feb 28 2016, after Alois P. Heinz *)

A225653 Numbers n such that A225634(n) = A225644(n).

Original entry on oeis.org

0, 1, 21, 30, 33, 35, 36, 40, 42, 44, 48, 51, 52, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 88, 91, 92, 96

Offset: 0

Antti Karttunen, May 16 2013

nonn

Positions of zeros in A225654.

Cf. A225648, A225649, A225650, A225651, A225654.

A255429 Numbers with a prime number of nontrivial divisors.

Original entry on oeis.org

6, 8, 10, 14, 15, 16, 21, 22, 26, 27, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 64, 65, 69, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 144, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 196

Offset: 1

Rory Glover, Feb 22 2015

nonn

Empirically, numbers in this sequence seem to have few divisors.

This sequence appears to be the union of A130763 and the squares of A225649. - Kellen Myers, Apr 21 2015

Cf. A063806, A070824, A130763, A225649.

[n: n in [1..200] | IsPrime(NumberOfDivisors(n)-2)]; // Vincenzo Librandi, Apr 21 2015

seq[n_] := Select[Range[n], PrimeQ[DivisorSigma[0, #] - 2] &] (* Kellen Myers, Apr 21 2015 *)

isok(m) = isprime(numdiv(m)-1); \\ Michel Marcus, Jan 13 2023

{n: A070824(n) in A000040}.

Terms fixed by Kellen Myers, Apr 21 2015

Name corrected by Michel Marcus, Jan 13 2023

cp's OEIS Frontend

A225648 Positions of ones in A225650, numbers n such that n and A000793(n) are coprime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A225651 Numbers k that divide A000793(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A225653 Numbers n such that A225634(n) = A225644(n).

Views

Author

Keywords

Comments

Crossrefs

A255429 Numbers with a prime number of nontrivial divisors.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions