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 1-6 of 6 results.

A341409 a(n) = (Sum_{k=1..3} k^n) mod n.

Original entry on oeis.org

0, 0, 0, 2, 1, 2, 6, 2, 0, 4, 6, 2, 6, 0, 6, 2, 6, 2, 6, 18, 15, 14, 6, 2, 1, 14, 0, 14, 6, 14, 6, 2, 3, 14, 31, 2, 6, 14, 36, 18, 6, 38, 6, 10, 36, 14, 6, 2, 13, 24, 36, 46, 6, 2, 1, 42, 36, 14, 6, 38, 6, 14, 36, 2, 16, 2, 6, 30, 36, 14, 6, 2, 6, 14, 51, 22, 17, 14, 6, 18, 0, 14, 6, 38, 21
Offset: 1

Views

Author

Seiichi Manyama, Feb 11 2021

Keywords

Links

Crossrefs

(Sum_{k=1..m} k^n) mod n: A096196 (m=2), this sequence (m=3), A341410 (m=4), A341411 (m=5), A341412 (m=6), A341413 (m=7).
Cf. A001550, A045576, A056645, A220235.

Programs

  • Maple
    a:= n-> add(i&^n, i=1..3) mod n:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 11 2021
  • Mathematica
    a[n_] := Mod[Sum[k^n, {k, 1, 3}], n]; Array[a, 100] (* Amiram Eldar, Feb 11 2021 *)
  • PARI
    a(n) = sum(k=1, 3, k^n)%n;

Formula

a(n) = A001550(n) mod n.
a(A056645(n)) = 0.

A220235 a(n) = (2^n + 3^n) modulo n.

Original entry on oeis.org

0, 1, 2, 1, 0, 1, 5, 1, 8, 3, 5, 1, 5, 13, 5, 1, 5, 1, 5, 17, 14, 13, 5, 1, 0, 13, 26, 13, 5, 13, 5, 1, 2, 13, 30, 1, 5, 13, 35, 17, 5, 37, 5, 9, 35, 13, 5, 1, 12, 23, 35, 45, 5, 1, 0, 41, 35, 13, 5, 37, 5, 13, 35, 1, 15, 1, 5, 29, 35, 13, 5, 1, 5, 13, 50
Offset: 1

Views

Author

Zak Seidov, Dec 08 2012

Keywords

Comments

a(n) = (A015910(n) + A066601(n)) mod n.
a(n) = 0 at n = 1, 5, 25, 55, 125, 275, 605, 625, ... (A045576).

Links

Crossrefs

Cf. A015910 (2^n mod n), A066601 (3^n mod n), A045576 (n|(2^n + 3^n)).

Programs

  • Mathematica
    Table[Mod[2^n + 3^n, n],{n,100}]

A220170 Numbers k that divide 5^k + 3^k + 2^k.

Original entry on oeis.org

1, 2, 5, 25, 38, 85, 125, 625, 722, 2225, 3125, 5825, 13718, 15625, 16502, 51325, 53125, 60625, 78125, 155125, 260642, 313538, 315514, 390625, 739925, 1953125, 4097125, 4952198, 5260202, 5265625, 5931914, 5957222, 9603322, 9765625
Offset: 1

Views

Author

Zak Seidov, Dec 06 2012

Keywords

Links

Crossrefs

Cf. A045576 (numbers k that divide 3^k + 2^k).

Programs

  • Mathematica
    Select[Range[1000000], Mod[PowerMod[5, #, #] + PowerMod[3, #, #] + PowerMod[2, #, #], #] == 0 &] (* T. D. Noe, Dec 06 2012 *)

A289259 Numbers k such that k^2 divides 2^k + 3^k.

Original entry on oeis.org

1, 5, 55, 1971145, 3061355, 109715901845, 340799222665
Offset: 1

Views

Author

Robert Israel, Jun 29 2017

Keywords

Comments

If k is in the sequence and p is a prime factor, coprime to k, of 2^k + 3^k, then k*p is in the sequence.
55 = 5 * 11
1971145 = 5 * 11 * 35839
3061355 = 5 * 11 * 55661
109715901845 = 5 * 11 * 35839 * 55661
340799222665 = 5 * 11 * 55661 * 111323
See Known Terms link for additional terms.
From Felix Fröhlich, Jun 29 2017: (Start)
For k in the sequence, A220235(k) = 0.
Subsequence of A045576. (End)

Examples

			2^5 + 3^5 = 275 is divisible by 5^2, so 5 is in the sequence.

Links

Crossrefs

Cf. A007689, A045576, A220235.

Programs

  • Maple
    select(t -> 2&^t + 3&^t mod t^2 = 0, [$1..10^6]);
  • PARI
    is(n) = Mod(2, n^2)^n==-3^n \\ Felix Fröhlich, Jun 29 2017
  • PARI
    is(n) = Mod(2,n^2)^n+Mod(3,n^2)^n==0 \\ Charles R Greathouse IV, Jun 29 2017

Extensions

a(6)-a(7) confirmed as next terms by Ray Chandler, Jul 02 2017
Known terms updated and moved to a-file by Ray Chandler, Jul 03 2017

A123049 Numbers n such that (n+2) | (2^n+3^n).

Original entry on oeis.org

0, 3, 33, 75, 2385, 6345, 6963, 11625, 18555, 57825, 89505, 92475, 265995, 473625, 575265, 1254363, 1720035, 3930705, 4295763, 4638603, 5686875, 6662115, 8731875, 8782515, 13964025, 14951385, 17714475, 18979035, 21868875, 26854155, 45546345
Offset: 1

Views

Author

Zak Seidov, Sep 25 2006

Keywords

Comments

All terms are multiples of 3. - Robert G. Wilson v, Sep 29 2006
Entries not congruent to 0 (modulo 5): 3, 33, 6963, 1254363, 4295763, 4638603, 50045553, 69151563, 114829611, 121716633, 208974987, 249618633, 292403403, ..., . - Robert G. Wilson v, Sep 29 2006

Crossrefs

Cf. A045576.

Programs

  • Mathematica
    Do[m = n; If[ Mod[ PowerMod[2, n, n + 2] + PowerMod[3, n, n + 2], n + 2] == 0, Print@n], {n, 0, 45546345}] (* Robert G. Wilson v, Sep 29 2006 *)

Extensions

More terms from Robert G. Wilson v, Sep 29 2006

A343977 Numbers k such that k | 11^k + 7^k + 5^k + 3^k + 2^k.

Original entry on oeis.org

1, 2, 4, 7, 26, 49, 338, 343, 2401, 4394, 7076, 15043, 16807, 17764, 57122, 117649, 226723, 241484, 295687, 742586, 818974, 823543, 826973, 1456511, 2040506, 2806769, 3472189, 5764801, 6321233, 9653618, 32036249, 40353607, 89758063, 107884133, 125497034, 126090551, 132590423
Offset: 1

Views

Author

Zak Seidov, May 06 2021

Keywords

Comments

This sequence is infinite as it contains 7^n. - David A. Corneth, May 06 2021

Crossrefs

Cf. A000420, A045576, A220170.

Programs

  • Maple
    q:= k-> is(0=11&^k+7&^k+5&^k+3&^k+2&^k mod k):
select(q, [$1..100000])[];  # Alois P. Heinz, May 06 2021
  • Mathematica
    q[k_] := Divisible[Plus @@ (PowerMod[#, k, k] & /@ {2, 3, 5, 7, 11}), k]; Select[Range[10^6], q] (* Amiram Eldar, May 06 2021 *)
  • PARI
    is(n) = lift(Mod(11, n)^n + Mod(7, n)^n + Mod(5, n)^n + Mod(3, n)^n + Mod(2, n)^n) == 0 \\ David A. Corneth, May 06 2021

Extensions

a(24)-a(37) from Alois P. Heinz, May 06 2021
Showing 1-6 of 6 results.

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