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.

A060654 a(n) = gcd(n, A060766(n)).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 5, 8, 1, 9, 1, 10, 7, 11, 1, 12, 5, 13, 9, 14, 1, 30, 1, 16, 11, 17, 7, 18, 1, 19, 13, 20, 1, 21, 1, 22, 15, 23, 1, 24, 7, 25, 17, 26, 1, 27, 11, 28, 19, 29, 1, 60, 1, 31, 21, 32, 13, 33, 1, 34, 23, 70, 1, 36, 1, 37, 25, 38, 11, 39, 1, 40, 27, 41
Offset: 2

Views

Author

Labos Elemer, Apr 25 2001

Keywords

Examples

			If n is prime p, then A060766(p) = p-1 and lcm(p, p-1) = 1. If n=2k then a(2k)=k or as an "anomaly", a(2k)=2k.
At n=30, D={1, 2, 3, 5, 6, 10, 15, 30}, dD={1, 1, 2, 1, 4, 5, 15}={1, 2, 4, 5, 15}, lcm(dD)=60, gcd(n, lcm(dD(n))) = gcd(30, 60) = 30 = n.
At n=36, D={1, 2, 3, 4, 6, 9, 12, 18, 36}, dD={1, 1, 1, 2, 3, 3, 6, 18}={1, 2, 3, 6, 18}, lcm(dD)=18, gcd(n, lcm(dD(n))) = gcd(36, 18) = 18 = n/2.

Links

Crossrefs

Cf. A000005, A060680-A060685, A060741, A060742, A060763-A060766.

Programs

  • Maple
    A060766:= proc(n) local F; F:= sort(convert(numtheory:-divisors(n),list));
   ilcm(op(F[2..-1] - F[1..-2])) end proc:
seq(igcd(n,A060766(n)),n=2..100); # Robert Israel, Dec 20 2015
  • Mathematica
    Table[GCD[n, LCM @@ Differences@ Divisors@ n], {n, 2, 82}] (* Michael De Vlieger, Dec 20 2015 *)

Formula

a(n) = gcd(n, lcm(dd(n))), where dd(n) is the first difference of divisors (ordered by size).

A060695 a(n) = gcd(2n, A060766(2n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 30, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 60, 31, 32, 33, 34, 70, 36, 37, 38, 39, 40, 41, 42, 43, 44, 90, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 126, 64, 65, 66, 67, 68, 69, 140, 71
Offset: 1

Views

Author

Labos Elemer, Apr 25 2001

Keywords

Examples

			n = 30: D = {1, 2, 3, 5, 6, 10, 15, 30}, dD = {1, 1, 2, 1, 4, 5, 15}={1, 2, 4, 5, 15}, lcm(dD) = 60, gcd(n, lcm(dD(n))) = gcd(30, 60) = 30 = n
n = 36: D = {1, 2, 3, 4, 6, 9, 12, 18, 36}, dD = {1, 1, 1, 2, 3, 3, 6, 18} = {1, 2, 3, 6, 18}, lcm(dD) = 18, gcd(n, lcm(dD(n))) = gcd(36, 18) = 18 = n/2.

Crossrefs

Cf. A000005, A060680-A060685, A060741, A060742, A060763-A060766.

Programs

  • Mathematica
    Table[GCD[2 n, LCM @@ Differences@ Divisors[2 n]], {n, 71}] (* Michael De Vlieger, Dec 20 2015 *)
  • PARI
    a(n) = my(d=divisors(2*n), dd = vector(#d-1, k, d[k+1] - d[k])); gcd(2*n, lcm(dd)); \\ Michel Marcus, Mar 22 2020

Formula

a(n) = a(2k) is either n = 2k or n/2 = k. a(n) = n/2 seems regular, a(n) = n seems "anomalous".

A298045 Integers equal to the least common multiple of the set of numbers generated by all the differences between their consecutive divisors, taken in increasing order.

Original entry on oeis.org

1, 60, 300, 504, 1500, 1512, 3528, 3660, 4536, 7500, 12240, 13608, 24696, 36720, 37500, 40824, 122472, 172872, 187500, 208080, 223260, 367416, 937500, 1102248, 1210104, 3306744, 3537360, 4687500, 8470728, 9920232, 12450312, 13618860, 23437500, 29760696
Offset: 1

Views

Author

Paolo P. Lava, Jan 11 2018

Keywords

Comments

Subset of A060765.
Fixed points of A060766.
Many terms m > 1 have omega(m) = 3 or 4, 60 and 3660 being the smallest of both, respectively. Is there a term with omega(m) = 5? - Michael De Vlieger, Jan 13 2018
The first two terms with 5 prime divisors are 149829840 and 1348395120. The sequence is infinite since it contains all the numbers of the form 72*7^k, for k>0. - Giovanni Resta, Jan 15 2018

Examples

			Divisors of 504 are 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 18, 21, 24, 28, 36, 42, 56, 63, 72, 84, 126, 168, 252 and 504.
Differences are: 2 - 1 = 1, 3 - 2 = 1, 4 - 3 = 1, 6 - 4 = 2, 7 - 6 = 1, 8 - 7 = 1, 9 - 8 = 1, 12 - 9 = 3, 14 - 12 = 2, 18 - 14 = 4, 21 - 18 = 3, 24 - 21 = 3, 28 - 24 = 4, 36 - 28 = 8, 42 - 36 = 6, 56 - 42 = 14, 63 - 56 = 7, 72 - 63 = 9, 84 - 72 = 12, 126 - 84 = 42, 168 - 126 = 42, 252 - 168 = 84, 504 - 252 = 252.
lcm(1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 42, 84, 252) is 504 again.

Links

Crossrefs

Cf. A060765, A060766.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,k,n; for n from 1 to q do a:=sort([op(divisors(n))]);
if n=lcm(op([seq(a[k+1]-a[k],k=1..nops(a)-1)])) then print(n); fi; od; end: P(10^6);
  • Mathematica
    {1}~Join~Select[Range[2, 10^6], LCM @@ Differences@ Divisors@ # == # &] (* Michael De Vlieger, Jan 13 2018 *)

Extensions

More terms from Michael De Vlieger, Jan 13 2018
a(31)-a(34) from Giovanni Resta, Jan 15 2018

A060700 "Anomalous" numbers k such that for even numbers 2k, gcd(2k, lcm(dd(2k)))=2k and not k, where dd(2k) is the first difference set of divisors of 2k.

Original entry on oeis.org

15, 30, 35, 45, 63, 70, 75, 77, 91, 99, 105, 117, 126, 135, 140, 143, 150, 153, 154, 165, 175, 182, 187, 189, 195, 198, 209, 221, 225, 231, 234, 245, 247, 252, 255, 273, 280, 285, 286, 297, 299, 306, 308, 315, 323, 325, 330, 345, 350, 351, 357, 364, 374, 375
Offset: 1

Views

Author

Labos Elemer, Apr 25 2001

Keywords

Examples

			63 is here because for 126 = 2*63, lcm(dd(126)) = lcm(1, 1, 3, 1, 2, 5, 4, 3, 21, 21, 63) = 1260, so gcd(126, lcm(dd(126))) = gcd(126, 1260) = 126.

Crossrefs

Cf. A060680, A060681, A060682, A060683, A060684, A060685, A060741, A060742, A060763, A060764, A060765, A060766, A000005.

Programs

  • PARI
    f(n) = {my(d = divisors(n), dd = vector(#d-1, k, d[k+1] - d[k])); gcd(n, lcm(dd));}
isok(n) = (f(2*n) == 2*n); \\ Michel Marcus, Mar 29 2018
Showing 1-4 of 4 results.

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