A378056 -id:A378056 - cp's OEIS frontend

A378058 Numbers k that divide A378056(k) = gcd(lcm{d+1 : d|k}, lcm{d-1 : d > 1 and d|k}).

1, 60, 210, 360, 420, 504, 630, 660, 840, 924, 1092, 1260, 1320, 1560, 1848, 1980, 2184, 2310, 2520, 2640, 2772, 3080, 3120, 3276, 3465, 3960, 4080, 4284, 4620, 4680, 5320, 5460, 5544, 6006, 6552, 6732, 6840, 6864, 6930, 7140, 7800, 7854, 7920, 8190, 8280, 8568, 8580, 9240, 9360, 9828

Offset: 1

Amiram Eldar, Nov 15 2024

nonn

After the first term a(1) = 1, the next odd term is a(25) = 3465, the next term that is coprime to 6 is a(308) = 95095, and the next term that is coprime to 30 is a(13544) = 10023013.

			60 is a term since A378056(60) = 4620 = 60 * 77 is divisible by 60.

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

Intersection of A377950 and A377952.
A378059 is a subsequence.
Cf. A378056, A378054.

s[n_] := Module[{d = Divisors[n]}, GCD[LCM @@ (d + 1), LCM @@ (Rest @ d - 1)]]; s[1] = 1; Select[Range[10000], Divisible[s[#], #] &]

is(k) = {my(d = divisors(k)); !(lcm(apply(x->x+1, d)) % k) && !(lcm(apply(x -> if(x > 1, x-1, x), d)) % k);}

A378057 Composite numbers k such that A378056(k) = gcd(lcm{d+1 : d|k}, lcm{d-1 : d > 1 and d|k}) = 2.

Original entry on oeis.org

6, 481, 793, 949, 1417, 2041, 2257, 2509, 2701, 2977, 3133, 3589, 3601, 4033, 4069, 4453, 4849, 5161, 5317, 5809, 5917, 5941, 6697, 7033, 7081, 7141, 7501, 7957, 7969, 8593, 8917, 9217, 9529, 9577, 10249, 10573, 10777, 11041, 11401, 11461, 11581, 11773, 12469, 12913, 12961

Offset: 1

Amiram Eldar, Nov 15 2024

nonn

A378056(p) = 2 for all odd primes p.
6 is the only even term.
The least term that is not a semiprime is a(114) = 29341 = 13 * 37 * 61, and the least term that has more than 3 distinct prime factors is a(4087545) = 1038565321 = 37 * 61 * 421 * 1093.

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

Cf. A378056.

s[n_] := Module[{d = Divisors[n]}, GCD[LCM @@ (d + 1), LCM @@ (Rest @ d - 1)]]; s[1] = 1; Select[Range[13000], CompositeQ[#] && s[#] == 2 &]

is(k) = if(isprime(k), 0, my(d = divisors(k)); gcd(lcm(apply(x->x+1, d)), lcm(apply(x -> if(x > 1, x-1, x), d))) == 2);

A378059 Numbers k such k | A378056(k) and (k+1) | A378056(k+1).

Original entry on oeis.org

112375899, 871651143, 1525038515, 3524721824, 6058144224, 7616307699, 7929320399, 9778346864, 10799650575, 11536526000, 13663711775, 20596306224, 22326106256, 24442111385, 26385908912, 27394105760, 28476579725, 31552570400, 34148839725, 36045427040, 40916036304, 44037977984, 44430326199

Offset: 1

Amiram Eldar, Nov 15 2024

nonn

Intersection of A377951 and A377953.
Subsequence of A378058.
Cf. A378055, A378056.

is1(k) = {my(d = divisors(k)); !(lcm(apply(x->x+1, d)) % k) && !(lcm(apply(x -> if(x > 1, x-1, x), d)) % k);}
lista(kmax) = {my(q1 = is1(1), q2); for(k = 2, kmax, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A378053 a(n) = gcd(Product_{d|n} (d + 1), Product_{d|n, d>1} (d - 1)) = gcd(A020696(n), A377484(n)).

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 2, 3, 16, 36, 2, 30, 4, 6, 16, 45, 4, 80, 2, 108, 16, 6, 2, 210, 24, 12, 32, 18, 4, 1008, 2, 45, 64, 12, 48, 8400, 4, 18, 16, 2268, 4, 240, 2, 90, 512, 18, 2, 3150, 32, 216, 64, 540, 4, 160, 144, 2430, 32, 12, 2, 166320, 4, 6, 1280, 405, 48, 1344

Offset: 1

Amiram Eldar, Nov 15 2024

nonn

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

Cf. A000079, A002144, A002145, A020696, A377484, A378056.

a[n_] := GCD[Times @@ ((d = Divisors[n]) + 1), Times @@ (Rest@ d - 1)]; Array[a, 70]

a(n) = if(n == 1, 1, my(d = divisors(n)); gcd(prod(k=1, #d, d[k]+1), prod(k=2, #d, d[k]-1)));

a(n) = 2 if and only if n = 6 or n is a prime of the form 4*k+3 (A002145).
a(n) = 4 if and only if n is a prime of the form 4*k+1 (A002144).
a(n) == 1 (mod 2) if and only if n is a power of 2 (A000079).

cp's OEIS Frontend

A378058 Numbers k that divide A378056(k) = gcd(lcm{d+1 : d|k}, lcm{d-1 : d > 1 and d|k}).

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A378057 Composite numbers k such that A378056(k) = gcd(lcm{d+1 : d|k}, lcm{d-1 : d > 1 and d|k}) = 2.

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A378059 Numbers k such k | A378056(k) and (k+1) | A378056(k+1).

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A378053 a(n) = gcd(Product_{d|n} (d + 1), Product_{d|n, d>1} (d - 1)) = gcd(A020696(n), A377484(n)).

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