cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A378664 Greatest divisor d of n such that sigma(d) <= 2*d < A003961(d), or 1 if no such divisor exists, where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 1, 8, 9, 10, 1, 6, 1, 14, 15, 16, 1, 9, 1, 10, 21, 1, 1, 8, 1, 1, 27, 28, 1, 15, 1, 32, 1, 1, 35, 9, 1, 1, 39, 10, 1, 21, 1, 44, 45, 1, 1, 16, 49, 50, 1, 52, 1, 27, 1, 28, 57, 1, 1, 15, 1, 1, 63, 64, 1, 6, 1, 68, 69, 35, 1, 9, 1, 1, 75, 76, 1, 39, 1, 16, 81, 1, 1, 28, 1, 1, 1, 44, 1, 45, 91, 92
Offset: 1

Views

Author

Antti Karttunen, Dec 06 2024

Keywords

Comments

Largest term of {1} U A341614 that divides n.

Links

Crossrefs

Cf. A000203, A003961, A337372, A341612, A341614, A378662, A378665, A378736 [= a(A005101(n))], A378737, A378738, A378739, A378740, A378741, A378742.
Positions of fixed points (where a(n)=n) is given by {1} U A341614.
Cf. A246281 (positions of 1's), A246282 (of terms > 0), A005101 (of terms that are neither 1 nor fixed points).

Programs

  • Mathematica
    Table[If[Length[#] == 0, 1, Max[#]] &@ Select[Divisors[n], DivisorSigma[1, #] <= 2 # < (Times @@ Map[Power @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi[p] + 1], e}] - Boole[# == 1]) &], {n, 92}] (* Michael De Vlieger, Dec 06 2024 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A341612(n) = ((sigma(n)<=(2*n))&&((2*n)<A003961(n)));
A378664(n) = { fordiv(n,d,if(A341612(n/d), return(n/d))); (1); };

Formula

a(n) <= A378665(n).

A378736 Greatest divisor d of the n-th abundant number such that sigma(d) <= 2*d < A003961(d).

Original entry on oeis.org

6, 9, 10, 8, 15, 9, 10, 21, 16, 27, 28, 15, 6, 35, 9, 39, 16, 28, 44, 45, 32, 50, 6, 52, 27, 28, 57, 15, 63, 44, 69, 35, 16, 75, 52, 32, 81, 28, 6, 44, 45, 6, 64, 98, 99, 50, 68, 52, 105, 27, 110, 6, 32, 76, 117, 16, 6, 63, 6, 130, 44, 135, 136, 92, 35, 6, 32, 147, 75, 152, 153, 154, 52, 6, 64, 81, 165, 28, 170, 171
Offset: 1

Views

Author

Antti Karttunen, Dec 06 2024

Keywords

Comments

There are no 1's in this sequence. See A378662, A378664 and A337372 for a proof.

Links

Crossrefs

Cf. A003961, A005101, A294935, A341612, A378664, A378735, A378737.
Cf. also A337372.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A294935(n) = (sigma(n)<=(2*n));
A341612(n) = ((sigma(n)<=(2*n))&&((2*n)<A003961(n)));
A378664(n) = { fordiv(n,d,if(A341612(n/d), return(n/d))); (1); };
k=0; n=0; while(k<20000, n++; if(!A294935(n), k++; print1(A378664(n),", ")));

Formula

a(n) = A378664(A005101(n)).
a(n) <= A378735(n).

Extensions

Unnecessary escape-clause removed from the definition by Antti Karttunen, Dec 12 2024

A378738 Primitively abundant numbers k for which A378665(k) > A378664(k).

Original entry on oeis.org

66, 102, 174, 186, 222, 246, 258, 282, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 748, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362, 1374, 1398, 1434, 1446, 1506, 1542, 1578, 1614, 1626, 1662, 1686, 1698, 1758, 1842, 1866
Offset: 1

Views

Author

Antti Karttunen, Dec 07 2024

Keywords

Comments

Subsequence of A378737: 1496 is its first term that does not occur here.
Equal to primitively abundant numbers k such that A032742(k) > A378664(k), because for primitively abundant numbers the greatest non-abundant divisor is the largest proper divisor, A378665(k) = A032742(k).
Question: What is the asymptotic density of these numbers among A091191? Does it tend to 1?
Conjecture: A001222(a(n)) = 3 <=> 3|a(n).

Examples

			Examples given in A378737 for 66, 748, 1866, and 1870 all work also here, because those four numbers are all in A091191.

Links

Crossrefs

Intersection of A091191 and A378737.
Cf. A001222, A032742, A294930, A337372, A341612, A341614, A378664, A378665, A378735, A378736, A378739 [= A378664(a(n))], A378741 (subsequence), A378742 (subsequence after its initial term).

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A341612(n) = ((sigma(n)<=(2*n))&&((2*n)<A003961(n)));
A378664(n) = { fordiv(n,d,if(A341612(n/d), return(n/d))); (1); };
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
is_A091191(n) = if(sigma(n)<=2*n, 0, fordiv(n,d,if(d2*d, return(0))); (1));
is_A378738(n) = (is_A091191(n) && (A378664(n)!=A032742(n)));

Formula

{k such that A294930(k) = 1 and A032742(k) > A378664(k)}.

A378735 Greatest non-abundant divisor of the n-th abundant number.

Original entry on oeis.org

6, 9, 10, 8, 15, 9, 10, 21, 16, 27, 28, 15, 33, 35, 9, 39, 16, 28, 44, 45, 32, 50, 51, 52, 27, 28, 57, 15, 63, 44, 69, 35, 16, 75, 52, 32, 81, 28, 87, 44, 45, 93, 64, 98, 99, 50, 68, 52, 105, 27, 110, 111, 32, 76, 117, 16, 123, 63, 129, 130, 44, 135, 136, 92, 35, 141, 32, 147, 75, 152, 153, 154, 52, 159, 64, 81, 165
Offset: 1

Views

Author

Antti Karttunen, Dec 06 2024

Keywords

Links

Crossrefs

Cf. A005101, A294935, A378665.
Cf. also A378736 and the points of differences A378737.

Programs

  • PARI
    A294935(n) = (sigma(n)<=(2*n));
A378665(n) = { fordiv(n,d,if(A294935(n/d), return(n/d))); (1); };
k=0; n=0; while(k<105, n++; if(!A294935(n), k++; print1(A378665(n),", ")));

Formula

a(n) = A378665(A005101(n)).
a(n) >= A378736(n).
Showing 1-4 of 4 results.

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