A378053 -id:A378053 - cp's OEIS frontend

A378054 Numbers k that divide A378053(k) = gcd(Product_{d|k} (d + 1), Product_{d|k, d>1} (d - 1)).

1, 60, 90, 120, 144, 168, 180, 210, 240, 252, 280, 336, 360, 420, 504, 540, 560, 630, 660, 720, 840, 900, 924, 990, 1008, 1056, 1080, 1092, 1200, 1260, 1320, 1404, 1440, 1512, 1560, 1680, 1800, 1848, 1872, 1890, 1980, 2016, 2100, 2112, 2160, 2184, 2310, 2376, 2400

Offset: 1

Amiram Eldar, Nov 15 2024

nonn

After the first term a(1) = 1, the next odd term is a(71) = 3465, the next term that is coprime to 6 is a(1058) = 95095, and the next term that is coprime to 30 is a(12174) = 2263261.

			60 is a term since A378053(60) = 166320 = 60 * 2772 is divisible by 60.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A056954 and A355331.
A378055 is a subsequence.
Cf. A020696, A377484, A378053, A378058.

Select[Range[2500], And @@ Divisible[{Times @@ ((d = Divisors[#]) + 1), Times @@ (Rest @ d - 1)}, #] &]

is(k) = if(k == 1, 1, my(d = divisors(k)); !(gcd(prod(k=1, #d, d[k]+1), prod(k=2, #d, d[k]-1)) % k));

A378055 Numbers k such that k | A378053(k) and (k+1) | A378053(k+1).

Original entry on oeis.org

638000, 13466816, 14753024, 16092999, 19494189, 38137749, 63668079, 80061344, 86119704, 107232255, 112375899, 121550624, 127205000, 154466675, 294147854, 391738599, 553140224, 561712095, 682199595, 728999999, 871651143, 879207615, 911062844, 920985624, 1017609999

Offset: 1

Amiram Eldar, Nov 15 2024

nonn

Intersection of A355332 and A377949.
Subsequence of A378054.
Cf. A020696, A377484, A378053, A378059.

q[n_] := q[n] = And @@ Divisible[{Times @@ ((d = Divisors[n]) + 1), Times @@ (Rest@d - 1)}, n]; Select[Range[2*10^7], q[#] && q[# + 1] &]

is1(k) = if(k == 1, 1, my(d = divisors(k)); !(gcd(prod(k=1, #d, d[k]+1), prod(k=2, #d, d[k]-1)) % 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

Amiram Eldar, Nov 15 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

a[n_] := Module[{d = Divisors[n]}, GCD[LCM @@ (d + 1), LCM @@ (Rest @ d - 1)]]; a[1] = 1; Array[a, 100]

a(n) = {my(d = divisors(n)); gcd(lcm(apply(x->x+1, d)), lcm(apply(x -> if(x > 1, x-1, x), d)));}

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.

cp's OEIS Frontend

A378054 Numbers k that divide A378053(k) = gcd(Product_{d|k} (d + 1), Product_{d|k, d>1} (d - 1)).

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A378055 Numbers k such that k | A378053(k) and (k+1) | A378053(k+1).

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

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