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.

A305630 Expansion of Product_{r = 1 or not a perfect power} 1/(1 - x^r).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 12, 16, 21, 28, 36, 48, 61, 78, 99, 124, 156, 195, 241, 299, 367, 450, 549, 670, 811, 982, 1183, 1422, 1704, 2040, 2431, 2894, 3435, 4070, 4811, 5679, 6684, 7858, 9217, 10797, 12623, 14738, 17174, 19988, 23225, 26951, 31227, 36141, 41759
Offset: 0

Views

Author

Gus Wiseman, Jun 07 2018

Keywords

Comments

a(n) is the number of integer partitions of n such that each part is either 1 or not a perfect power (A001597, A007916).

Examples

			The a(5) = 6 integer partitions whose parts are 1's or not perfect powers are (5), (32), (311), (221), (2111), (11111).

Links

Crossrefs

Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305631-A305635.

Programs

  • Maple
    q:= n-> is(n=1 or 1=igcd(map(i-> i[2], ifactors(n)[2])[])):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(q(d), d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 07 2018
  • Mathematica
    nn=20;
radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
ser=Product[1/(1-x^p),{p,Select[Range[nn],radQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

A305631 Expansion of Product_{r not a perfect power} 1/(1 - x^r).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 4, 5, 7, 8, 12, 13, 17, 21, 25, 32, 39, 46, 58, 68, 83, 99, 121, 141, 171, 201, 239, 282, 336, 391, 463, 541, 635, 741, 868, 1005, 1174, 1359, 1580, 1826, 2115, 2436, 2814, 3237, 3726, 4276, 4914, 5618, 6445, 7359, 8414, 9594, 10947, 12453
Offset: 0

Views

Author

Gus Wiseman, Jun 07 2018

Keywords

Comments

a(n) is the number of integer partitions of n whose parts are not perfect powers (A001597, A007916).

Examples

			The a(9) = 5 integer partitions whose parts are not perfect powers are (72), (63), (522), (333), (3222).

Links

Crossrefs

Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305630-A305635.

Programs

  • Maple
    q:= n-> is(1=igcd(map(i-> i[2], ifactors(n)[2])[])):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(q(d), d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 07 2018
  • Mathematica
    nn=100;
wadQ[n_]:=n>1&&GCD@@FactorInteger[n][[All,2]]==1;
ser=Product[1/(1-x^p),{p,Select[Range[nn],wadQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

A305632 Expansion of Product_{r = 1 or not a perfect power} 1/(1 + (-x)^r).

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 1, 2, 4, 3, 2, 4, 6, 5, 4, 7, 10, 8, 7, 11, 15, 13, 12, 17, 22, 19, 18, 25, 30, 28, 26, 35, 42, 39, 38, 49, 59, 56, 54, 69, 81, 77, 76, 94, 110, 105, 105, 127, 147, 141, 142, 171, 195, 189, 190, 227, 257, 250, 254, 299, 335, 328, 334, 390, 432
Offset: 0

Views

Author

Gus Wiseman, Jun 07 2018

Keywords

Examples

			O.g.f.: 1/((1 - x)(1 + x^2)(1 - x^3)(1 - x^5)(1 + x^6)(1 - x^7)(1 + x^10)...).

Crossrefs

Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305630-A305635.

Programs

  • Mathematica
    nn=20;
radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
ser=Product[1/(1+(-x)^p),{p,Select[Range[nn],radQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

A305633 Expansion of Sum_{r not a perfect power} x^r/(1 + x^r).

Original entry on oeis.org

0, 0, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -2, 1, 1, 3, -1, 1, 2, 1, -2, 3, 1, 1, -3, 1, 1, 1, -2, 1, 1, 1, -1, 3, 1, 3, -3, 1, 1, 3, -3, 1, 1, 1, -2, 4, 1, 1, -4, 1, 2, 3, -2, 1, 3, 3, -3, 3, 1, 1, -4, 1, 1, 4, -1, 3, 1, 1, -2, 3, 1, 1, -3, 1, 1, 4, -2, 3, 1, 1, -4
Offset: 0

Views

Author

Gus Wiseman, Jun 07 2018

Keywords

Crossrefs

Cf. A000607, A001597, A007916, A048165, A081362, A091050, A303707, A304779, A304817, A305614, A305630-A305635.

Programs

  • Mathematica
    nn=100;
wadQ[n_]:=n>1&&GCD@@FactorInteger[n][[All,2]]==1;
ser=Sum[x^p/(1+x^p),{p,Select[Range[nn],wadQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

A305634 Even numbers that are not perfect powers.

Original entry on oeis.org

2, 6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30, 34, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 130, 132, 134, 136, 138
Offset: 1

Views

Author

Gus Wiseman, Jun 07 2018

Keywords

Comments

Perfect powers are of the form m^k where m > 0 and k > 1 (A001597).

Examples

			10 is in the sequence since it is even and is not a power of an integer.  17 is not in the sequence since it is odd, and 36 is not in the sequence since it is a power of an integer (36 = 6^2).

Links

Crossrefs

Cf. A001597, A005117, A007916, A039956.
Cf. A305630, A305631, A305632, A305633, A305635.
Cf. A005843, A001597.

Programs

  • Maple
    N:= 1000:
S:={seq(i,i=2..N,2)} minus {seq(seq(e^m,m=2..floor(log[e](N))),e=2..floor(sqrt(N)),2)}:
sort(convert(S,list)); # Robert Israel, Jan 24 2019
  • Mathematica
    radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
Select[Range[200],EvenQ[#]&&radQ[#]&]
  • PARI
    isok(n) = !(n % 2) && !ispower(n); \\ Michel Marcus, Jun 08 2018

Formula

A005843 \ A001597. - Eric Chen, Jun 14 2018
Showing 1-5 of 5 results.

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