A377950 -id:A377950 - cp's OEIS frontend

A056954 Numbers k such that k^2 divides A056819(k).

1, 30, 60, 90, 105, 120, 132, 144, 168, 180, 210, 240, 252, 264, 280, 336, 360, 380, 396, 420, 495, 504, 520, 528, 540, 546, 552, 560, 612, 616, 630, 660, 720, 728, 756, 760, 792, 840, 858, 870, 900, 924, 990, 1008, 1040, 1050, 1056, 1080, 1092, 1104

Offset: 1

Leroy Quet, Sep 06 2000

From Amiram Eldar, Nov 12 2024: (Start)
Equivalently, numbers k that divide A377484(k) = Product_{d|k, d>1} (d - 1).
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(228) = 6545, and the next term that is coprime to 30 is a(574) = 19019. (End)

			30 is a term because 30^2 divides A056819(30) = 5320224000.

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

Cf. A056819, A377484.
A377949 is a subsequence.
Similar sequences: A355331, A377950, A377952.

Select[Range[1000], Divisible[Times @@ (Rest@ Divisors[#] - 1), #] &] (* Amiram Eldar, Nov 12 2024 *)

is(k) = if(k == 1, 1, my(d = divisors(k)); !(prod(i = 2, #d, d[i]-1) % k)); \\ Amiram Eldar, Nov 12 2024

A377951 Numbers k such that k | A057643(k) and (k+1) | A057643(k+1).

Original entry on oeis.org

1, 799799, 1204280, 2460975, 3382379, 6116175, 7050120, 8070699, 13339424, 20966049, 28460600, 41265680, 41463135, 52404624, 66108399, 68919080, 72946224, 81102944, 84479680, 102971924, 106663304, 110791736, 112375899, 115225439, 118333215, 131115984, 132073424

Offset: 1

Amiram Eldar, Nov 12 2024

nonn

Numbers k such that k and k+1 are both terms in A377950.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A057643.
Subsequence of A377950.
Similar sequences: A355332, A377949, A377953.

q[n_] := q[n] = Divisible[LCM @@ (Divisors[n] + 1), n]; Select[Range[4*10^6], q[#] && q[# + 1] &]

is1(k) = !(lcm(apply(x->x+1, 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);}

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

Amiram Eldar, Nov 12 2024

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.

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

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

Select[Range[2000], # == 1 || Divisible[LCM @@ (Rest @ Divisors[#] - 1), #] &]

is(k) = !(lcm(apply(x -> if(x > 1, x-1, x), divisors(k))) % k);

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

Original entry on oeis.org

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);}

cp's OEIS Frontend

A056954 Numbers k such that k^2 divides A056819(k).

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A377951 Numbers k such that k | A057643(k) and (k+1) | A057643(k+1).

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

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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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