cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A355331 Numbers k that divide A020696(k).

Original entry on oeis.org

1, 2, 6, 12, 20, 24, 42, 60, 72, 84, 90, 120, 126, 140, 144, 156, 168, 180, 210, 216, 220, 240, 252, 280, 312, 336, 342, 360, 420, 432, 440, 462, 468, 480, 504, 540, 560, 600, 624, 630, 660, 672, 684, 700, 720, 770, 780, 816, 840, 864, 880, 900, 924, 936, 945, 960, 990
Offset: 1

Views

Author

Amiram Eldar, Jun 29 2022

Keywords

Comments

If k and m are coprime terms then k*m is also a term.
The least odd term above 1 is a(55) = 945, the least term above 1 that is coprime to 6 is a(378) = 10465, least term above 1 that is coprime to 30 is a(3122) = 151487, and the least term above 1 that is coprime to 210 is a(6858) = 414713.

Examples

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

Links

Crossrefs

Cf. A020696.
A355332 is a subsequence.

Programs

  • Mathematica
    v[n_] := Times @@ (Divisors[n] + 1); Select[Range[1000], Divisible[v[#], #] &]
  • PARI
    f(n) = my(d = divisors(n)); prod(i=1, #d, d[i]+1); \\ A020696
isok(k) = !(f(k) % k); \\ Michel Marcus, Jun 30 2022
  • Python
    from itertools import count, islice
from functools import reduce
from sympy import divisors
def A355331_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:reduce(lambda a,b:a*b%n,(d+1 for d in divisors(n,generator=True)))%n==0,count(max(startvalue,1)))
A355331_list = list(islice(A355331_gen(),30)) # Chai Wah Wu, Jun 30 2022

A377949 Numbers k such that k | A377484(k) and (k+1) | A377484(k+1).

Original entry on oeis.org

156519, 245024, 310155, 524799, 638000, 893024, 1079000, 2055780, 2095975, 2203200, 2566025, 2592512, 2853135, 2934063, 3213375, 3294719, 4056975, 4322240, 4471935, 5746455, 6515145, 7289919, 7316000, 7329608, 7866495, 8459360, 8555624, 8934464, 9035415, 11291091
Offset: 1

Views

Author

Amiram Eldar, Nov 12 2024

Keywords

Comments

Numbers k such that k and k+1 are both terms in A056954.

Examples

			156519 is a term since A377484(156519) is divisible by 156519 and A377484(156520) is divisible by 156520.

Links

Crossrefs

Cf. A377484.
Subsequence of A056954.
Similar sequences: A355332, A377951, A377953.

Programs

  • Mathematica
    q[n_] := q[n] = Divisible[Times @@ (Rest@ Divisors[n] - 1), n]; Select[Range[10^6], q[#] && q[#+1] &]
  • PARI
    is1(k) = if(k == 1, 1, my(d = divisors(k)); !(prod(i = 2, #d, d[i]-1) % k));
lista(kmax) = {my(q1 = is1(1), q2); for(k = 2, kmax, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

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

Views

Author

Amiram Eldar, Nov 12 2024

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Views

Author

Amiram Eldar, Nov 12 2024

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Views

Author

Amiram Eldar, Nov 15 2024

Keywords

Crossrefs

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

Programs

  • Mathematica
    q[n_] := q[n] = And @@ Divisible[{Times @@ ((d = Divisors[n]) + 1), Times @@ (Rest@d - 1)}, n]; Select[Range[2*10^7], q[#] && q[# + 1] &]
  • PARI
    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);}
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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