A377949 -id:A377949 - 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);}

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

Amiram Eldar, Nov 12 2024

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..1500

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

q[n_] := q[n] = n == 1 || Divisible[LCM @@ (Rest @ Divisors[n] - 1), n] ; Select[Range[3*10^6], q[#] && q[# + 1] &]

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);}

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);}

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

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

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