A355331 -id:A355331 - cp's OEIS frontend

A355332 Numbers k such that k | A020696(k) and (k+1) | A020696(k+1).

1, 201824, 227799, 313599, 395199, 544824, 638000, 654975, 799799, 862784, 1056159, 1204280, 1269729, 1447550, 1890944, 2276351, 2460975, 2481115, 2781999, 2821272, 3348224, 3382379, 3403700, 3832191, 3864575, 3956120, 5142500, 5961950, 6116175, 6401024, 7050120

Offset: 1

Amiram Eldar, Jun 29 2022

nonn

Numbers k such that k and k+1 are both in A355331.
Are there 3 consecutive integers in A355331?
There are no such 3 consecutive integers below 10^10. - Amiram Eldar, Oct 12 2023

			1 is a term since A020696(1) = 2 is divisible by 1 and A020696(2) = 6 is divisible 2.

Amiram Eldar, Table of n, a(n) for n = 1..2000

Cf. A020696, A355331.

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

f(n) = my(d = divisors(n)); prod(i=1, #d, d[i]+1); \\ A020696
isok(k) = !(f(k) % k) && !(f(k+1) % (k+1)); \\ Michel Marcus, Jun 30 2022

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

Original entry on oeis.org

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

A377950 Numbers k that divide A057643(k) = lcm{d+1 : d|k}.

Original entry on oeis.org

1, 2, 6, 12, 42, 60, 84, 120, 140, 156, 168, 210, 220, 240, 280, 312, 360, 420, 440, 462, 468, 504, 600, 630, 660, 720, 770, 780, 840, 924, 936, 1008, 1064, 1092, 1170, 1200, 1260, 1320, 1404, 1428, 1540, 1560, 1680, 1683, 1800, 1806, 1848, 1860, 1980, 2016, 2160

Offset: 1

Amiram Eldar, Nov 12 2024

After the first term a(1) = 1, the next odd term is a(44) = 1683, the next term that is coprime to 6 is a(159) = 10465, and the next term that is coprime to 30 is a(1359) = 151487.

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

Cf. A057643.
A377951 is a subsequence.
Similar sequences: A056954, A355331, A377952.

Select[Range[2500], Divisible[LCM @@ (Divisors[#] + 1), #] &]

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

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);

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

Original entry on oeis.org

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));

cp's OEIS Frontend

A355332 Numbers k such that k | A020696(k) and (k+1) | A020696(k+1).

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

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A377950 Numbers k that divide A057643(k) = lcm{d+1 : d|k}.

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

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

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