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.

Previous Showing 11-20 of 45 results. Next

A339686 a(n) = Sum_{d|n} 6^(d-1).

Original entry on oeis.org

1, 7, 37, 223, 1297, 7819, 46657, 280159, 1679653, 10078999, 60466177, 362805091, 2176782337, 13060740679, 78364165429, 470185264735, 2821109907457, 16926661132171, 101559956668417, 609359750089711, 3656158440109669, 21936950700844039, 131621703842267137
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 12 2020

Keywords

Links

Crossrefs

Column 6 of A308813.
Cf. A000400, A320071.
Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), this sequence (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).

Programs

  • Magma
    A339686:= func< n | (&+[6^(d-1): d in Divisors(n)]) >;
[A339686(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024
  • Mathematica
    Table[Sum[6^(d - 1), {d, Divisors[n]}], {n, 1, 23}]
nmax = 23; CoefficientList[Series[Sum[x^k/(1 - 6 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sumdiv(n, d, 6^(d-1)); \\ Michel Marcus, Dec 13 2020
  • SageMath
    def A339686(n): return sum(6^(k-1) for k in (1..n) if (k).divides(n))
[A339686(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

Formula

G.f.: Sum_{k>=1} x^k / (1 - 6*x^k).
G.f.: Sum_{k>=1} 6^(k-1) * x^k / (1 - x^k).
a(n) ~ 6^(n-1). - Vaclav Kotesovec, Jun 05 2021

A339687 a(n) = Sum_{d|n} 7^(d-1).

Original entry on oeis.org

1, 8, 50, 351, 2402, 16864, 117650, 823894, 5764851, 40356016, 282475250, 1977343950, 13841287202, 96889128064, 678223075300, 4747562333837, 33232930569602, 232630519768872, 1628413597910450, 11398895225729502, 79792266297729700, 558545864365759264
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 12 2020

Keywords

Links

Crossrefs

Column 7 of A308813.
Cf. A000420, A320072.
Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), this sequence (q=7), A339688 (q=8), A339689 (q=9).

Programs

  • Magma
    A339687:= func< n | (&+[7^(d-1): d in Divisors(n)]) >;
[A339687(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024
  • Mathematica
    Table[Sum[7^(d - 1), {d, Divisors[n]}], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[x^k/(1 - 7 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sumdiv(n, d, 7^(d-1)); \\ Michel Marcus, Dec 13 2020
  • SageMath
    def A339687(n): return sum(7^(k-1) for k in (1..n) if (k).divides(n))
[A339687(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

Formula

G.f.: Sum_{k>=1} x^k / (1 - 7*x^k).
G.f.: Sum_{k>=1} 7^(k-1) * x^k / (1 - x^k).
a(n) ~ 7^(n-1). - Vaclav Kotesovec, Jun 05 2021

A339688 a(n) = Sum_{d|n} 8^(d-1).

Original entry on oeis.org

1, 9, 65, 521, 4097, 32841, 262145, 2097673, 16777281, 134221833, 1073741825, 8589967945, 68719476737, 549756076041, 4398046515265, 35184374186505, 281474976710657, 2251799830495305, 18014398509481985, 144115188210078217, 1152921504607109185
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 12 2020

Keywords

Links

Crossrefs

Column 8 of A308813.
Cf. A001018, A320073.
Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), this sequence (q=8), A339689 (q=9).

Programs

  • Magma
    A339688:= func< n | (&+[8^(d-1): d in Divisors(n)]) >;
[A339688(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024
  • Mathematica
    Table[Sum[8^(d - 1), {d, Divisors[n]}], {n, 1, 21}]
nmax = 21; CoefficientList[Series[Sum[x^k/(1 - 8 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sumdiv(n, d, 8^(d-1)); \\ Michel Marcus, Dec 13 2020
  • SageMath
    def A339688(n): return sum(8^(k-1) for k in (1..n) if (k).divides(n))
[A339688(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

Formula

G.f.: Sum_{k>=1} x^k / (1 - 8*x^k).
G.f.: Sum_{k>=1} 8^(k-1) * x^k / (1 - x^k).
a(n) ~ 8^(n-1). - Vaclav Kotesovec, Jun 05 2021

A339689 a(n) = Sum_{d|n} 9^(d-1).

Original entry on oeis.org

1, 10, 82, 739, 6562, 59140, 531442, 4783708, 43046803, 387427060, 3486784402, 31381119478, 282429536482, 2541866359780, 22876792461604, 205891136878357, 1853020188851842, 16677181742772430, 150094635296999122, 1350851718060419878, 12157665459057460324
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 12 2020

Keywords

Links

Crossrefs

Column 9 of A308813.
Cf. A001019, A320074.
Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), this sequence (q=9).

Programs

  • Magma
    A339689:= func< n | (&+[9^(d-1): d in Divisors(n)]) >;
[A339689(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024
  • Mathematica
    Table[Sum[9^(d - 1), {d, Divisors[n]}], {n, 1, 21}]
nmax = 21; CoefficientList[Series[Sum[x^k/(1 - 9 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sumdiv(n, d, 9^(d-1)); \\ Michel Marcus, Dec 13 2020
  • SageMath
    def A339689(n): return sum(9^(k-1) for k in (1..n) if (k).divides(n))
[A339689(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

Formula

G.f.: Sum_{k>=1} x^k / (1 - 9*x^k).
G.f.: Sum_{k>=1} 9^(k-1) * x^k / (1 - x^k).
a(n) ~ 9^(n-1). - Vaclav Kotesovec, Jun 05 2021

A275700 a(n) = Product_{d|n} prime(d).

Original entry on oeis.org

2, 6, 10, 42, 22, 390, 34, 798, 230, 1914, 62, 101010, 82, 4386, 5170, 42294, 118, 547170, 134, 951258, 12410, 14694, 166, 170807910, 2134, 24846, 23690, 3285114, 218, 660741510, 254, 5540514, 42470, 49206, 55726, 21399271530, 314, 65526, 68470, 3126785046, 358
Offset: 1

Views

Author

Jaroslav Krizek, Aug 05 2016

Keywords

Comments

a(n) mod n = 0 for n: 1, 2, 6, 30, 78, 330, 390, 870, 1410, 3198, ...

Examples

			a(4) = 42 because the divisors of 4 are: 1, 2 and 4; and prime(1) * prime(2) * prime(4) = 2 * 3 * 7 = 42.

Crossrefs

Cf. A007445 (Sum_{d|n} prime(d)).
A version for binary indices is A034729.
Partitions of this type are counted by A054973, strict case of A371284.
The sorted version is A371283, squarefree case of A371288.
These numbers have products A371286, unsorted version A371285.
A000005 counts divisors, row-lengths of A027750.
A027746 lists prime factors, indices A112798, length A001222.
Cf. A000720, A001221, A005179, A007416.

Programs

  • Magma
    [(&*[NthPrime(d): d in Divisors(n)]): n in [1..100]]
  • Mathematica
    Table[Times@@(Prime[#]&/@Divisors[n]),{n,50}] (* Harvey P. Dale, Jun 16 2017 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(i=1, #d, prime(d[i])) \\ Felix Fröhlich, Aug 05 2016
  • Perl
    use ntheory ":all"; sub a275700 { vecprod(map { nth_prime($) } divisors($[0])); } # Dana Jacobsen, Aug 09 2016

A289271 A bijective binary representation of the prime factorization of a number, shown in decimal (see Comments for precise definition).

Original entry on oeis.org

0, 1, 2, 4, 8, 3, 16, 32, 64, 5, 128, 6, 256, 9, 10, 512, 1024, 17, 2048, 12, 18, 33, 4096, 34, 8192, 65, 16384, 20, 32768, 7, 65536, 131072, 66, 129, 24, 36, 262144, 257, 130, 40, 524288, 11, 1048576, 68, 72, 513, 2097152, 258, 4194304, 1025, 514, 132
Offset: 1

Views

Author

Rémy Sigrist, Jun 30 2017

Keywords

Comments

For n > 0, with prime factorization Product_{i=1..k} p_i ^ e_i (all p_i distinct and all e_i > 0):
- let S_n = A000961 \ { p_i ^ (e_i + j) with i=1..k and j > 0 },
- a(n) = Sum_{i=1..k} 2^#{ s in S_n with 1 < s < p_i ^ e_i }.
In an informal way, we encode the prime powers > 1 that are unitary divisors of n as 1's in binary, while discarding the 0's corresponding to their "proper" multiples.
a(A002110(n)) = 2^n-1 for any n >= 0.
a(A000961(n+1)) = 2^(n-1) for any n > 0.
A000120(a(n)) = A001221(n) for any n > 0 (each prime divisor p of n (alongside the p-adic valuation of n) is encoded as a single 1 bit in the base-2 representation of a(n)).
A000961(2+A007814(a(n))) = A034684(n) for any n > 1 (the least significant bit of a(n) encodes the smallest unitary divisor of n that is larger than 1).
This sequence establishes a bijection between the positive numbers and the nonnegative numbers; see A289272 for the inverse of this sequence.
The numbers 4, 36, 40 and 532 equal their image; are there other such numbers?
This sequence has connections with A034729 (which encodes the divisors of a number, and is not surjective) and A087207 (which encodes the prime divisors of a number, and is not injective).

Examples

			For n = 204 = 2^2 * 3 * 17:
- S_204 = A000961 \ { 2^3, 2^4, ..., 3^2, ... }
        = { 1, 2, 3, 4, 5, 7, 11, 13, 17, ... },
- a(204) = 2^#{ 2, 3 } + 2^#{ 2 } + 2^#{ 2, 3, 4, 5, 7, 11, 13 }
         = 2^2 + 2^1 + 2^7
         = 134.
See also the illustration of the first terms in Links section.

Links

Crossrefs

Cf. A000120, A000961, A001221, A002110, A007814, A034684, A034729, A087207, A289272 (inverse), A322988, A322990, A322991, A322992, A322995.
Cf. also A156552, A052331 for similar constructions.

Programs

  • PARI
    See Links section.
  • PARI
    A289271(n) = { my(f = factor(n), pps = vecsort(vector(#f~, i, f[i, 1]^f[i, 2])), s=0, x=1, pp=1, k=-1); for(i=1,#f~, while(pp < pps[i], pp++; while(!isprimepower(pp)||(gcd(pp,x)>1), pp++); k++); s += 2^k; x *= pp); (s); }; \\ Antti Karttunen, Jan 01 2019

A308813 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where A(n,k) is Sum_{d|n} k^(d-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 5, 3, 1, 1, 5, 10, 11, 2, 1, 1, 6, 17, 31, 17, 4, 1, 1, 7, 26, 69, 82, 39, 2, 1, 1, 8, 37, 131, 257, 256, 65, 4, 1, 1, 9, 50, 223, 626, 1045, 730, 139, 3, 1, 1, 10, 65, 351, 1297, 3156, 4097, 2218, 261, 4, 1
Offset: 1

Views

Author

Seiichi Manyama, Jun 26 2019

Keywords

Examples

			Square array, A(n,k), begins:
  1, 1,  1,   1,    1,     1,     1, ...
  1, 2,  3,   4,    5,     6,     7, ...
  1, 2,  5,  10,   17,    26,    37, ...
  1, 3, 11,  31,   69,   131,   223, ...
  1, 2, 17,  82,  257,   626,  1297, ...
  1, 4, 39, 256, 1045,  3156,  7819, ...
  1, 2, 65, 730, 4097, 15626, 46657, ...
Antidiagonal triangle, T(n,k), begins as:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  2,   1;
  1,  4,  5,   3,    1;
  1,  5, 10,  11,    2,    1;
  1,  6, 17,  31,   17,    4,    1;
  1,  7, 26,  69,   82,   39,    2,    1;
  1,  8, 37, 131,  257,  256,   65,    4,   1;
  1,  9, 50, 223,  626, 1045,  730,  139,   3,   1;
  1, 10, 65, 351, 1297, 3156, 4097, 2218, 261,   4,   1;

Links

Crossrefs

Columns k=0..10 give A000012, A000005, A034729, A034730, A339684, A339685, A339686, A339687, A339688, A339689, A113999.
Row n=1..3 give A000012, A000027(k+1), A002522.
A(n,n) gives A308814.

Programs

  • Magma
    A:= func< n,k | (&+[k^(d-1): d in Divisors(n)]) >;
A308813:= func< n,k | A(k+1,n-k-1) >;
[A308813(n,k): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Jun 26 2024
  • Mathematica
    A[n_, k_] := DivisorSum[n, If[k == # - 1 == 0, 1, k^(# - 1)] &];
Table[A[k + 1, n - k - 1], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* Amiram Eldar, May 07 2021 *)
  • SageMath
    def A(n,k): return sum(k^(j-1) for j in (1..n) if (j).divides(n))
def A308813(n,k): return A(k+1,n-k-1)
flatten([[A308813(n,k) for k in range(n)] for n in range(1,13)]) # G. C. Greubel, Jun 26 2024

Formula

G.f. of column k: Sum_{j>=1} x^j/(1 - k*x^j).
T(n, k) = Sum_{d|(k+1)} (n-k-1)^(d-1), with T(n, n) = 1. - G. C. Greubel, Jun 26 2024

A323774 Number of multiset partitions, whose parts are constant and all have the same sum, of integer partitions of n.

Original entry on oeis.org

1, 1, 3, 3, 7, 3, 12, 3, 16, 8, 14, 3, 39, 3, 16, 15, 40, 3, 50, 3, 54, 17, 20, 3, 135, 10, 22, 25, 73, 3, 129, 3, 119, 21, 26, 19, 273, 3, 28, 23, 217, 3, 203, 3, 123, 74, 32, 3, 590, 12, 106, 27, 154, 3, 370, 23, 343, 29, 38, 3, 963, 3, 40, 95, 450, 25, 467, 3
Offset: 0

Views

Author

Gus Wiseman, Jan 27 2019

Keywords

Comments

An unlabeled version of A279789.

Examples

			The a(1) = 1 through a(6) = 12 multiset partitions:
  (1)  (2)     (3)        (4)           (5)              (6)
       (11)    (111)      (22)          (11111)          (33)
       (1)(1)  (1)(1)(1)  (1111)        (1)(1)(1)(1)(1)  (222)
                          (2)(2)                         (3)(3)
                          (2)(11)                        (111111)
                          (11)(11)                       (3)(111)
                          (1)(1)(1)(1)                   (2)(2)(2)
                                                         (111)(111)
                                                         (2)(2)(11)
                                                         (2)(11)(11)
                                                         (11)(11)(11)
                                                         (1)(1)(1)(1)(1)(1)

Links

Crossrefs

Cf. A001970, A006171 (constant parts), A007716, A034729, A047966 (uniform partitions), A047968, A279787, A279789 (twice-partitions version), A305551 (equal part-sums), A306017, A319056, A323766, A323775, A323776.

Programs

  • Mathematica
    Table[Length[Join@@Table[Union[Sort/@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@ptn]],{ptn,Select[IntegerPartitions[n],SameQ@@#&]}]],{n,30}]
  • PARI
    a(n) = if (n==0, 1, sumdiv(n, d, binomial(numdiv(d) + n/d - 1, n/d))); \\ Michel Marcus, Jan 28 2019

Formula

a(0) = 1; a(n) = Sum_{d|n} binomial(tau(d) + n/d - 1, n/d), where tau = A000005.

A081295 a(n) = (-1)^(n+1) * coefficient of x^n in Sum_{k>=1} x^k/(1+2*x^k).

Original entry on oeis.org

1, 1, 5, 9, 17, 29, 65, 137, 261, 497, 1025, 2085, 4097, 8129, 16405, 32905, 65537, 130845, 262145, 524793, 1048645, 2096129, 4194305, 8390821, 16777233, 33550337, 67109125, 134225865, 268435457, 536855053, 1073741825, 2147516553, 4294968325, 8589869057
Offset: 1

Views

Author

Benoit Cloitre, Apr 20 2003

Keywords

Links

Crossrefs

Cf. A034729, A128315.

Programs

  • Magma
    A081295:= func< n | (-1)^(n+1)*(&+[(-2)^(d-1): d in Divisors(n)]) >;
[A081295(n): n in [1..40]]; // G. C. Greubel, Jun 22 2024
  • Maple
    f:= n -> (-1)^(n+1)*add((-2)^(d-1),d=numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Jun 04 2018
  • Mathematica
    A081295[n_]:= (-1)^(n+1)*DivisorSum[n, (-2)^(#-1) &];
Table[A081295[n], {n, 40}] (* G. C. Greubel, Jun 22 2024 *)
  • PARI
    a(n) =if(n<1, 0, (-1)^(n+1)*polcoeff(sum(k=1, n, x^k/(1+2*x^k), x*O(x^n)), n))
  • SageMath
    def A081295(n): return (-1)^(n+1)*sum((-2)^(k-1) for k in (1..n) if (k).divides(n))
[A081295(n) for n in range(1,41)] # G. C. Greubel, Jun 22 2024

Formula

a(n) = (-1)^(n+1) * [x^n]( Sum_{k>=1} x^k/(1+2*x^k) ).
a(p) = 2^(p-1) - 1, for p prime.
a(n) = (-1)^(n+1) * Sum_{d|n} (-2)^(d-1). - Robert Israel, Jun 04 2018
a(n) = (-1)^(n-1)*Sum_{k=1..n} (-1)^(k-1)*A128315(n, k). - G. C. Greubel, Jun 22 2024

A371288 Numbers whose distinct prime indices form the set of divisors of some positive integer.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 32, 34, 36, 40, 42, 44, 48, 50, 54, 62, 64, 68, 72, 80, 82, 84, 88, 96, 100, 108, 118, 124, 126, 128, 134, 136, 144, 160, 162, 164, 166, 168, 176, 192, 200, 216, 218, 230, 236, 242, 248, 250, 252, 254, 256, 268, 272, 288
Offset: 1

Views

Author

Gus Wiseman, Mar 22 2024

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The prime indices of 694782 are {1,2,2,5,5,5,10} with distinct elements {1,2,5,10}, which form the set of divisors of 10, so 694782 is in the sequence.
The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}

Links

Crossrefs

The squarefree case is A371283, unsorted version A275700.
Partitions of this type are counted by A371284, strict A054973.
Products of squarefree terms are A371286, unsorted version A371285.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, indices A112798, length A001222.
Cf. A000720, A003963, A005179, A007416, A034729, A370820, A371131, A371177.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Union[prix[#]]==Divisors[Max@@prix[#]]&]
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