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 1-5 of 5 results.

A377952 Numbers k that divide A084190(k) = lcm{d-1 : d > 1 and d|k}.

Original entry on oeis.org

1, 30, 60, 90, 105, 132, 180, 210, 252, 264, 360, 380, 420, 495, 504, 520, 528, 546, 630, 660, 756, 840, 858, 870, 924, 990, 1040, 1056, 1092, 1140, 1224, 1260, 1320, 1365, 1485, 1512, 1530, 1560, 1638, 1656, 1716, 1722, 1740, 1785, 1820, 1848, 1900, 1980, 2040
Offset: 1

Views

Author

Amiram Eldar, Nov 12 2024

Keywords

Comments

After the first term a(1) = 1, the next odd term is a(5) = 105, the next term that is coprime to 6 is a(133) = 6545, and the next term that is coprime to 30 is a(322) = 19019.

Links

Crossrefs

Cf. A084190.
A377953 is a subsequence.
Similar sequences: A056954, A355331, A377950.

Programs

  • Mathematica
    Select[Range[2000], # == 1 || Divisible[LCM @@ (Rest @ Divisors[#] - 1), #] &]
  • PARI
    is(k) = !(lcm(apply(x -> if(x > 1, x-1, x), divisors(k))) % k);

A377953 Numbers k such that k | A084190(k) and (k+1) | A084190(k+1).

Original entry on oeis.org

310155, 2566025, 2853135, 5746455, 6515145, 7329608, 8459360, 11291091, 15446079, 16181535, 26782224, 26942475, 32364464, 34318844, 36951200, 38579442, 38596239, 38763900, 40564524, 41273154, 47308976, 47648600, 49309715, 50163735, 51177224, 52573520, 58524465, 63668079
Offset: 1

Views

Author

Amiram Eldar, Nov 12 2024

Keywords

Comments

Numbers k such that k and k+1 are both terms in A377952.

Links

Crossrefs

Cf. A084190.
Subsequence of A377952.
Similar sequences: A355332, A377949, A377951.

Programs

  • Mathematica
    q[n_] := q[n] = n == 1 || Divisible[LCM @@ (Rest @ Divisors[n] - 1), n] ; Select[Range[3*10^6], q[#] && q[# + 1] &]
  • PARI
    is1(k) = !(lcm(apply(x -> if(x > 1, x-1, x), divisors(k))) % k);
lista(kmax) = {my(q1 = is1(1), q2); for(k = 2, kmax, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A378056 a(n) = gcd(A057643(n), A084190(n)) = gcd(lcm{d+1 : d|n}, lcm{d-1 : d > 1 and d|n}).

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 2, 3, 4, 6, 2, 30, 2, 6, 4, 15, 2, 20, 2, 6, 4, 6, 2, 210, 6, 6, 4, 6, 2, 84, 2, 15, 4, 6, 12, 420, 2, 6, 4, 126, 2, 60, 2, 30, 8, 6, 2, 210, 8, 6, 4, 30, 2, 20, 12, 90, 4, 6, 2, 4620, 2, 6, 40, 45, 6, 84, 2, 6, 4, 36, 2, 420, 2, 6, 24, 30, 12
Offset: 1

Views

Author

Amiram Eldar, Nov 15 2024

Keywords

Links

Crossrefs

Cf. A000079, A057643, A084190, A378053, A378057.

Programs

  • Mathematica
    a[n_] := Module[{d = Divisors[n]}, GCD[LCM @@ (d + 1), LCM @@ (Rest @ d - 1)]]; a[1] = 1; Array[a, 100]
  • PARI
    a(n) = {my(d = divisors(n)); gcd(lcm(apply(x->x+1, d)), lcm(apply(x -> if(x > 1, x-1, x), d)));}

Formula

a(n) == 1 (mod 2) if and only if n is a power of 2 (A000079).
a(p) = 2 for an odd prime p. Composite numbers k such that a(k) = 2 are listed in A378057.

A084191 a(n) = A084190(A084190(n)).

Original entry on oeis.org

1, 1, 1, 2, 3, 36, 10, 60, 21, 157080, 36, 2394591020640, 330, 438900, 702, 185640, 105, 392201962392240, 680, 47347883887966200, 450574740, 3183089069160, 210
Offset: 1

Views

Author

Reinhard Zumkeller, May 18 2003

Keywords

Comments

Next term: a(24) = A084190(53130) > 10^91.
B-file goes up to n=59 because a(60) has 1503 digits. - Robert Israel, Sep 04 2019

Links

Crossrefs

Cf. A084190.

Programs

A258409 Greatest common divisor of all (d-1)'s, where the d's are the positive divisors of n.

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 2, 1, 22, 1, 4, 1, 2, 1, 28, 1, 30, 1, 2, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 2, 1, 46, 1, 6, 1, 2, 1, 52, 1, 2, 1, 2, 1, 58, 1, 60, 1, 2, 1, 4, 1, 66, 1, 2, 1, 70, 1, 72, 1, 2, 1, 2, 1, 78, 1
Offset: 2

Views

Author

Ivan Neretin, May 29 2015

Keywords

Comments

a(n) = 1 for even n; a(p) = p-1 for prime p.
a(n) is even for odd n (since all divisors of n are odd).
It appears that a(n) = A052409(A005179(n)), i.e., it is the largest integer power of the smallest number with exactly n divisors. - Michel Marcus, Nov 10 2015
Conjecture: GCD of all (p-1) for prime p|n. - Thomas Ordowski, Sep 14 2016
Conjecture is true, because the set of numbers == 1 (mod g) is closed under multiplication. - Robert Israel, Sep 14 2016
Conjecture: a(n) = A289508(A328023(n)) = GCD of the differences between consecutive divisors of n. See A328163 and A328164. - Gus Wiseman, Oct 16 2019

Examples

			65 has divisors 1, 5, 13, and 65, hence a(65) = gcd(1-1,5-1,13-1,65-1) = gcd(0,4,12,64) = 4.

Links

Crossrefs

Cf. A049559, A057237, A060680, A063994, A187730, A027750.
Cf. A084190 (similar but with LCM).
Looking at prime indices instead of divisors gives A328167.
Partitions whose parts minus 1 are relatively prime are A328170.
Cf. A000005, A060681, A060683, A193829, A289508.

Programs

  • Haskell
    a258409 n = foldl1 gcd $ map (subtract 1) $ tail $ a027750_row' n
-- Reinhard Zumkeller, Jun 25 2015
  • Maple
    f:= n -> igcd(op(map(`-`,numtheory:-factorset(n),-1))):
map(f, [$2..100]); # Robert Israel, Sep 14 2016
  • Mathematica
    Table[GCD @@ (Divisors[n] - 1), {n, 2, 100}]
  • PARI
    a(n) = my(g=0); fordiv(n, d, g = gcd(g, d-1)); g; \\ Michel Marcus, May 29 2015
  • PARI
    a(n) = gcd(apply(x->x-1, divisors(n))); \\ Michel Marcus, Nov 10 2015
  • PARI
    a(n)=if(n%2==0, return(1)); if(n%3==0, return(2)); if(n%5==0 && n%4 != 1, return(2)); gcd(apply(p->p-1, factor(n)[,1])) \\ Charles R Greathouse IV, Sep 19 2016
Showing 1-5 of 5 results.

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