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 11-17 of 17 results.

A325502 Heinz number of row n of Pascal's triangle A007318.

Original entry on oeis.org

2, 4, 12, 100, 2548, 407044, 106023164, 136765353124, 399090759725236, 4445098474836287524, 151287513513627682258436, 12698799587219706700017036196, 3463928752077516667634331415766516, 2591202267595530693505786197581910681796
Offset: 0

Views

Author

Gus Wiseman, May 06 2019

Keywords

Comments

The Heinz number of a positive integer sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Every odd-indexed term is a square of a squarefree number.

Examples

			Row n = 5 of Pascal's triangle is (1,5,10,10,5,1), with Heinz number prime(1)*prime(5)*prime(10)*prime(10)*prime(5)*prime(1) = 407044, so a(5) = 407044.
The sequence of terms together with their prime indices begins:
                    2: {1}
                    4: {1,1}
                   12: {1,1,2}
                  100: {1,1,3,3}
                 2548: {1,1,4,4,6}
               407044: {1,1,5,5,10,10}
            106023164: {1,1,6,6,15,15,20}
         136765353124: {1,1,7,7,21,21,35,35}
      399090759725236: {1,1,8,8,28,28,56,56,70}
  4445098474836287524: {1,1,9,9,36,36,84,84,126,126}

Links

Crossrefs

Cf. A000040, A001222, A001405, A007318, A056239, A112798, A145519, A215366, A325500, A325503, A325505, A325514.

Programs

  • Mathematica
    Times@@@Table[Prime[Binomial[n,k]],{n,0,5},{k,0,n}]

Formula

A061395(a(n)) = A001405(n).
A056239(a(n)) = A000079(n).
A181819(a(n)) = A038754(n + 1).

A378176 Sum over all multiplicative partitions mu of n (with factors > 1) of the encoding as Product_{j in mu} prime(j).

Original entry on oeis.org

1, 3, 5, 16, 11, 28, 17, 67, 48, 62, 31, 156, 41, 94, 102, 303, 59, 270, 67, 334, 158, 172, 83, 743, 218, 224, 343, 508, 109, 707, 127, 1173, 292, 316, 336, 1651, 157, 364, 372, 1587, 179, 1091, 191, 926, 960, 448, 211, 3468, 516, 1202, 528, 1198, 241, 2209
Offset: 1

Views

Author

Alois P. Heinz, Nov 18 2024

Keywords

Examples

			The multiplicative partitions of n=8 are {[8], [4,2], [2,2,2]}, encodings give {prime(8), prime(4)*prime(2), prime(2)^3} = {19, 7*3, 3^3} = {19, 21, 27}; the sum gives a(8) = 67.

Links

Crossrefs

Row sums of A378175.
Cf. A000040, A001055, A006450, A145519, A377853.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=1, {1}, {seq(map(x-> x*
      ithprime(d), b(n/d))[], d=numtheory[divisors](n) minus {1})})
    end:
a:= n-> add(i, i=b(n)):
seq(a(n), n=1..54);

Formula

a(prime(n)) = a(A000040(n)) = A006450(n).

A305881 Expansion of Product_{k>=1} 1/(1 + prime(k)*x^k).

Original entry on oeis.org

1, -2, 1, -7, 16, -28, 62, -118, 303, -630, 1152, -2426, 5315, -10718, 20482, -43449, 91111, -179254, 358910, -727829, 1484601, -2995681, 5924606, -11935441, 24382120, -48702245, 96682698, -195063604, 392983826, -784903199, 1569490057, -3146479152, 6317124649, -12652202092
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 13 2018

Keywords

Comments

Convolution inverse of A147655.

Links

Crossrefs

Cf. A000040, A002099, A061151, A145519, A147655, A298160, A304791, A305882.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, b(n-i, i-1)*ithprime(i))))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(b(n-i$2)*a(i$2), i=0..n-1))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 13 2018
  • Mathematica
    nmax = 33; CoefficientList[Series[Product[1/(1 + Prime[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[Exp[Sum[Sum[(-1)^k Prime[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (-Prime[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*prime(j)^k*x^(j*k)/k).

A325514 Heinz number of row n of the triangle of partition numbers A008284.

Original entry on oeis.org

2, 2, 4, 8, 24, 72, 600, 4200, 101640, 2042040, 107869080, 6435365640, 644779672680, 62219208188280, 14408598135902520, 3195700205016233640, 1246437353286578234760, 527744165981695537415640, 417665868515500206974318760, 314096677106179199154141208440
Offset: 0

Views

Author

Gus Wiseman, May 07 2019

Keywords

Comments

The Heinz number of a positive integer sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The sequence of terms together with their prime indices begins:
             2: {1}
             2: {1}
             4: {1,1}
             8: {1,1,1}
            24: {1,1,1,2}
            72: {1,1,1,2,2}
           600: {1,1,1,2,3,3}
          4200: {1,1,1,2,3,3,4}
        101640: {1,1,1,2,3,4,5,5}
       2042040: {1,1,1,2,3,4,5,6,7}
     107869080: {1,1,1,2,3,5,5,7,8,9}
    6435365640: {1,1,1,2,3,5,5,7,10,10,11}
  644779672680: {1,1,1,2,3,5,6,7,11,12,13,15}

Links

Crossrefs

Cf. A000041, A002569, A008284, A056239, A112798, A145519, A215366, A325500, A325502, A325503, A325512, A325514.

Programs

  • Mathematica
    Times@@@Table[If[n>0&&k==0,1,Prime[Length[IntegerPartitions[n,{k}]]]],{n,0,20},{k,0,n}]

Formula

A001221(a(n)) = A325512(n).
A061395(a(n)) = A002569(n).
A056239(a(n)) = A000041(n).

A304791 Expansion of Product_{k>=1} (1 - prime(k)*x^k).

Original entry on oeis.org

1, -2, -3, 1, 3, 18, 0, 35, -27, -85, -91, -109, -366, 118, 942, -957, 2791, 2091, 4855, -1157, -6903, 3341, 3162, -37034, -46480, -89890, 581, 131275, -296935, 167543, 108671, 801491, 616017, 2441581, -307733, -1864550, 4495872, 1158228, -2589768, -767646, -21062537
Offset: 0

Views

Author

Ilya Gutkovskiy, May 18 2018

Keywords

Comments

Convolution inverse of A145519.

Links

Crossrefs

Cf. A000040, A007441, A145519, A147655, A298159.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1 - Prime[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[-Sum[d Prime[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 40}]

Formula

G.f.: Product_{k>=1} (1 - A000040(k)*x^k).

A325536 Sum of sums of omegas of parts over all integer partitions of n.

Original entry on oeis.org

0, 0, 1, 2, 6, 9, 19, 28, 51, 75, 119, 170, 261, 362, 525, 723, 1019, 1373, 1890, 2512, 3386, 4452, 5893, 7658, 10017, 12881, 16627, 21210, 27097, 34266, 43392, 54462, 68399, 85285, 106305, 131712, 163132, 200936, 247332, 303066, 370989, 452296, 550875, 668495
Offset: 0

Views

Author

Gus Wiseman, May 08 2019

Keywords

Comments

Also omega of the product of products of parts over all integer partitions of n.
The omega of n is A001222(n), the number of prime factors of n counted with multiplicity.

Examples

			The integer partitions of 5 are {(5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1)} with products {5,4,6,3,4,2,1} with product 2880 with omega 9, so a(5) = 9.

Crossrefs

Cf. A001222, A006128, A006906, A007870, A066186, A066633, A145519, A215366, A325507.

Programs

  • Mathematica
    Table[Plus@@PrimeOmega/@Join@@IntegerPartitions[n],{n,0,30}]

Formula

a(n) = A001222(A007870(n)).

A302837 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^prime(k)).

Original entry on oeis.org

1, 0, 2, 3, 16, 30, 252, 399, 4672, 8100, 111200, 193006, 3279888, 5631093, 114137072, 196030350, 4600308224, 7873377297, 210745840824, 359490499457, 10808589776800, 18389567969010, 613522956193532, 1041400845949218, 38179982045681856, 64675343605547500, 2584515863960831060
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 13 2018

Keywords

Links

Crossrefs

Cf. A000607, A002098, A124577, A145519, A301971.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 - n x^Prime[k]), {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Previous Showing 11-17 of 17 results.

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