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 1-10 of 18 results. Next

A361438 Triangle T(n,k), n >= 1, 1 <= k <= A046801(n), read by rows, where T(n,k) is k-th smallest divisor of 2^n-1.

Original entry on oeis.org

1, 1, 3, 1, 7, 1, 3, 5, 15, 1, 31, 1, 3, 7, 9, 21, 63, 1, 127, 1, 3, 5, 15, 17, 51, 85, 255, 1, 7, 73, 511, 1, 3, 11, 31, 33, 93, 341, 1023, 1, 23, 89, 2047, 1, 3, 5, 7, 9, 13, 15, 21, 35, 39, 45, 63, 65, 91, 105, 117, 195, 273, 315, 455, 585, 819, 1365, 4095, 1, 8191, 1, 3, 43, 127, 129, 381, 5461, 16383
Offset: 1

Views

Author

Seiichi Manyama, Mar 12 2023

Keywords

Examples

			Triangle begins:
  1;
  1,   3;
  1,   7;
  1,   3,  5,   15;
  1,  31;
  1,   3,  7,    9, 21, 63;
  1, 127;
  1,   3,  5,   15, 17, 51,  85,  255;
  1,   7, 73,  511;
  1,   3, 11,   31, 33, 93, 341, 1023;
  1,  23, 89, 2047;

Links

Crossrefs

Subsequence of A027750.
Cf. A000225, A049479 (2nd column), A075708 (row sums).
Cf. A374237 (analogous for 2^n + 1).

Programs

  • Maple
    T:= n-> sort([numtheory[divisors](2^n-1)[]])[]:
seq(T(n), n=1..12);  # Alois P. Heinz, Oct 20 2024
  • Mathematica
    Divisors[2^Range[15] - 1] (* Paolo Xausa, Jul 02 2024 *)

A366622 Sum of the divisors of 6^n-1.

Original entry on oeis.org

6, 48, 264, 1824, 9672, 67584, 335928, 2367552, 13031040, 94708224, 454285152, 3523559424, 15677418768, 113738502240, 599516366592, 4210539708672, 20465720064000, 154928015278080, 735060126170880, 5906693566844928, 26937015875831424, 188358079273592832
Offset: 1

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Examples

			a(4)=1824 because 6^4-1 has divisors {1, 5, 7, 35, 37, 185, 259, 1295}.

Links

Crossrefs

Cf. A024062, A000203, A057955, A059888, A274907, A366620, A366621, A366623, A366629.
Cf. A075708, A366576, A366603, A366613, A366634, A366653, A366662, A102146, A366684, A366710.

Programs

  • Maple
    a:=n->numtheory[sigma](6^n-1):
seq(a(n), n=1..100);
  • Mathematica
    DivisorSigma[1, 6^Range[30]-1]

Formula

a(n) = sigma(6^n-1) = A000203(A024062(n)).

A366653 Sum of the divisors of 8^n-1.

Original entry on oeis.org

8, 104, 592, 8736, 38912, 473600, 2466048, 38054016, 155493536, 2015330304, 10359014400, 166290432000, 636328345600, 7645340651520, 42424026529792, 648494317126656, 2599936977797120, 32817383473149440, 164708609085669376, 3010983668199456768
Offset: 1

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Examples

			a(5)=38912 because 8^5-1 has divisors {1, 7, 31, 151, 217, 1057, 4681, 32767}.

Links

Crossrefs

Cf. A024088, A000203, A057953, A059890, A366651, A366652, A366654.
Cf. A075708, A366576, A366603, A366613, A366622, A366634, A366662, A102146, A366684, A366710.

Programs

  • Maple
    a:=n->numtheory[sigma](8^n-1):
seq(a(n), n=1..100);
  • Mathematica
    DivisorSigma[1, 8^Range[30]-1]
  • SageMath
    [sigma(8**n-1, 1) for n in range(1, 21)] # Stefano Spezia, Aug 02 2025

Formula

a(n) = sigma(8^n-1) = A000203(A024088(n)).
a(n) = A075708(3*n). - Max Alekseyev, Jan 09 2024

A366662 Sum of the divisors of 9^n-1.

Original entry on oeis.org

15, 186, 1680, 15876, 123690, 1541568, 8992680, 111757968, 967814400, 9366647892, 62424587520, 852903426816, 4766016364260, 55176998178240, 550081165885440, 4829754617483040, 31725040326819840, 471309320999516160, 2535353780263288800, 33995669076586206864
Offset: 1

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Examples

			a(2)=186 because 9^2-1 has divisors {1, 2, 4, 5, 8, 10, 16, 20, 40, 80}.

Links

Crossrefs

Cf. A024101, A000203, A366660, A366661, A366663.
Cf. A075708, A366576, A366603, A366613, A366622, A366634, A366653, A102146, A366684, A366710.

Programs

  • Maple
    a:=n->numtheory[sigma](9^n-1):
seq(a(n), n=1..30);
  • Mathematica
    DivisorSigma[1, 9^Range[30]-1]

Formula

a(n) = sigma(9^n-1) = A000203(A024101(n)).
a(n) = A366576(2*n) = A366576(n) * A366578(n) * (2^(4 + A007814(n)) - 1) / (2^(3 + A007814(n)) - 1) / 3. - Max Alekseyev, Jan 07 2024

A366578 Sum of the divisors of 3^n+1.

Original entry on oeis.org

3, 7, 18, 56, 126, 434, 1332, 3836, 10476, 42560, 109926, 315112, 816732, 2790074, 8906760, 30220288, 64570086, 229156928, 706911048, 2034690952, 5357742012, 21838961760, 56496274632, 164750562956, 456919958880, 1517043139136, 4661686010664, 16489453890560
Offset: 0

Views

Author

Sean A. Irvine, Oct 13 2023

Keywords

Examples

			a(4)=126 because 3^4+1 has divisors {1, 2, 41, 82}.

Links

Crossrefs

Cf. A000203, A002592, A024023, A057935, A057941, A074476, A075708, A274909, A366576, A366577, A366579, A366580

Programs

  • Maple
    a:=n->numtheory[sigma](3^n+1):
seq(a(n), n=0..100);
  • Mathematica
    DivisorSigma[1,3^Range[0,30]+1] (* Paolo Xausa, Oct 15 2023 *)

Formula

a(n) = sigma(3^n+1) = A000203(A034472(n)).

A366603 Sum of the divisors of 4^n-1.

Original entry on oeis.org

4, 24, 104, 432, 1536, 8736, 22528, 111456, 473600, 1999872, 5909760, 38054016, 89522176, 462274560, 2015330304, 7304603328, 22907191296, 166290432000, 366506672128, 2220409884672, 7645340651520, 29833839544320, 95821839806976, 648494317126656
Offset: 1

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Examples

			a(4)=432 because 4^4-1 has divisors {1, 3, 5, 15, 17, 51, 85, 255}.

Links

Crossrefs

Cf. A024036, A000203, A057957, A069061, A075708, A295501, A366602, A366604.
Cf. A075708, A366576, A366613, A366622, A366634, A366653, A366662, A102146, A366684, A366710.

Programs

  • Maple
    a:=n->numtheory[sigma](4^n-1):
seq(a(n), n=1..100);
  • Mathematica
    DivisorSigma[1,4^Range[30]-1] (* Paolo Xausa, Oct 14 2023 *)

Formula

a(n) = sigma(4^n-1) = A000203(A024036(n)).
a(n) = A069061(n) * A075708(n). - Robert Israel, Nov 22 2023

A366613 Sum of the divisors of 5^n-1.

Original entry on oeis.org

7, 60, 224, 1736, 6048, 49920, 136724, 1107792, 3718400, 27060480, 85449224, 869499904, 2136230474, 15820920000, 61359427584, 461863805760, 1338408456700, 13177159680000, 33558717136896, 301282248701952, 863701914880000, 6313641012910080, 20863951122979048
Offset: 1

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Examples

			a(3)=224 because 5^3-1 has divisors {1, 2, 4, 31, 62, 124}.

Links

Crossrefs

Cf. A024049, A000203, A057956, A059887, A074479, A143665, A218357, A295502, A366611, A366612.
Cf. A075708, A366576, A366603, A366622, A366634, A366653, A366662, A102146, A366684, A366710.

Programs

  • Maple
    a:=n->numtheory[sigma](5^n-1):
seq(a(n), n=1..100);
  • Mathematica
    DivisorSigma[1, 5^Range[30]-1]

Formula

a(n) = sigma(5^n-1) = A000203(A024049(n)).

A366634 Sum of the divisors of 7^n-1.

Original entry on oeis.org

12, 124, 780, 7812, 33624, 354640, 1704240, 18929096, 97036800, 800520192, 3958188480, 56928231360, 193778020824, 1830926384640, 11181115146240, 115997032277280, 465294239722800, 5175558387507200, 22852200371636160, 287850454432579584, 1318081737957660000
Offset: 1

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Examples

			a(5)=33624 because 7^5-1 has divisors {1, 2, 3, 6, 2801, 5602, 8403, 16806}.

Links

Crossrefs

Cf. A024075, A000203, A057954, A059887, A074249, A218358, A366632, A366633, A366635.
Cf. A075708, A366576, A366603, A366613, A366622, A366653, A366662, A102146, A366684, A366710.

Programs

  • Maple
    a:=n->numtheory[sigma](7^n-1):
seq(a(n), n=1..100);
  • Mathematica
    DivisorSigma[1, 7^Range[30]-1]

Formula

a(n) = sigma(7^n-1) = A000203(A024075(n)).

A366576 Sum of the divisors of 3^n-1.

Original entry on oeis.org

3, 15, 42, 186, 399, 1680, 3282, 15876, 31836, 123690, 277344, 1541568, 2391486, 8992680, 25483332, 111757968, 193819392, 967814400, 1744488660, 9366647892, 16912999320, 62424587520, 144219337920, 852903426816, 1397135488896, 4766016364260, 12477973754400
Offset: 1

Views

Author

Sean A. Irvine, Oct 13 2023

Keywords

Examples

			a(4)=186 because 3^4-1 has divisors {1, 2, 4, 5, 8, 10, 16, 20, 40, 80}.

Links

Crossrefs

Cf. A024023, A000203, A075708, A133801, A295500, A366575.
Cf. A075708, A366603, A366613, A366622, A366634, A366653, A366662, A102146, A366684, A366710.

Programs

  • Maple
    a:=n->numtheory[sigma](3^n-1):
seq(a(n), n=1..100);
  • Mathematica
    DivisorSigma[1,3^Range[30]-1] (* Paolo Xausa, Oct 15 2023 *)

Formula

a(n) = sigma(3^n-1) = A000203(A024023).

A246601 Sum of divisors d of n with property that the binary representation of d can be obtained from the binary representation of n by changing any number of 1's to 0's.

Original entry on oeis.org

1, 2, 4, 4, 6, 8, 8, 8, 10, 12, 12, 16, 14, 16, 24, 16, 18, 20, 20, 24, 22, 24, 24, 32, 26, 28, 40, 32, 30, 48, 32, 32, 34, 36, 36, 40, 38, 40, 43, 48, 42, 44, 44, 48, 60, 48, 48, 64, 50, 52, 72, 56, 54, 80, 61, 64, 58, 60, 60, 96, 62, 64, 104, 64, 66, 68, 68, 72, 70, 72
Offset: 1

Views

Author

N. J. A. Sloane, Sep 06 2014

Keywords

Comments

Equivalently, the sum of the divisors d of n such that the bitwise OR of d and n is equal to n. - Chai Wah Wu, Sep 06 2014
Equivalently, the sum of the divisors d of n such that the bitwise AND of n and d is equal to d. - Amiram Eldar, Dec 15 2022

Examples

			12 = 1100_2; only the divisors 4 = 0100_2 and 12 = 1100_2 satisfy the condition, so(12) = 4+12 = 16.
15 = 1111_2; all divisors 1,3,5,15 satisfy the condition, so a(15)=24.

Links

Crossrefs

Cf. A000005, A000203, A075708, A246600.

Programs

  • Maple
    with(numtheory);
sd:=proc(n) local a,d,s,t,i,sw;
s:=convert(n,base,2);
a:=0;
for d in divisors(n) do
sw:=-1;
t:=convert(d,base,2);
for i from 1 to nops(t) do if t[i]>s[i] then sw:=1; fi; od:
if sw<0 then a:=a+d; fi;
od;
a;
end;
[seq(sd(n),n=1..100)];
  • Mathematica
    a[n_] := DivisorSum[n, #*Boole[BitOr[#, n] == n] &]; Array[a, 100] (* Jean-François Alcover, Dec 02 2015, adapted from PARI *)
  • PARI
    a(n) = sumdiv(n, d, d*(bitor(n,d)==n)); \\ Michel Marcus, Sep 07 2014
  • Python
    from sympy import divisors
def A246601(n):
    return sum(d for d in divisors(n) if n|d == n)
# Chai Wah Wu, Sep 06 2014

Formula

a(2^i) = 2^i.
a(odd prime p) = p+1.
From Amiram Eldar, Dec 15 2022: (Start)
a(2*n) = 2*a(n), and therefore a(m*2^k) = 2^k*a(m) for m odd and k>=0.
a(2^n-1) = sigma(2^n-1) = A075708(n). (End)
a(n) = Sum_{d|n} d*(binomial(n,d) mod 2). - Ridouane Oudra, Apr 09 2025
Showing 1-10 of 18 results. Next

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

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