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.

A319238 Positions of zeros in A114592, the list of coefficients in the expansion of Product_{n > 1} (1 - 1/n^s).

Original entry on oeis.org

6, 8, 10, 14, 15, 16, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 64, 65, 69, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 96, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 144, 145, 146, 155, 158, 159, 160, 161, 166
Offset: 1

Views

Author

Gus Wiseman, Sep 15 2018

Keywords

Comments

From Tian Vlasic, Jan 01 2022: (Start)
Numbers that have an equal number of even- and odd-length unordered factorizations into distinct factors.
For prime p, by the pentagonal number theorem, p^k is a term if and only if k is in A090864.
For primes p and q, p*q^k is a term if and only if k = A000326(m)+N with 0 <= N < m. (End)

Examples

			16 = 2*8 = 4*4 = 2*2*4 = 2*2*2*2 has an equal number of even-length factorizations and odd-length factorizations into distinct factors (1). - _Tian Vlasic_, Dec 31 2021

Links

Crossrefs

Complement of A319237.
Cf. A001055, A045778, A114592, A162247, A190938, A281116, A281118, A303386, A316441, A319240.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Join@@Position[Table[Sum[(-1)^Length[f],{f,Select[facs[n],UnsameQ@@#&]}],{n,100}],0]

A331051 Numbers whose number of factorizations into factors > 1 (A001055) is even.

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 25, 26, 28, 33, 34, 35, 38, 39, 44, 45, 46, 48, 49, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 108, 111, 112, 115, 116, 117, 118, 119, 121, 122
Offset: 1

Views

Author

Gus Wiseman, Jan 10 2020

Keywords

Comments

First differs from A319240 in having 256.

Crossrefs

Complement of A331050.
The version for powers of two (instead of evens) is A330977.
The version for primes (instead of evens) is A330991.
Cf. A001055, A001221, A001222, A002033, A005117, A045778, A045782, A045783, A330972, A330973.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[300],EvenQ[Length[facs[#]]]&]

A319239 Positions of nonzero terms in A316441, the list of coefficients in the expansion of Product_{n > 1} 1/(1 + 1/n^s).

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 13, 16, 17, 19, 23, 24, 27, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 47, 53, 54, 56, 59, 60, 61, 64, 66, 67, 70, 71, 73, 78, 79, 81, 83, 84, 88, 89, 90, 96, 97, 100, 101, 102, 103, 104, 105, 107, 109, 110, 113, 114, 120, 125, 126, 127, 128
Offset: 1

Views

Author

Gus Wiseman, Sep 15 2018

Keywords

Crossrefs

Complement of A319240.
Cf. A001055, A045778, A114592, A162247, A190938, A281116, A281118, A303386, A316441, A319237.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Join@@Position[Table[Sum[(-1)^Length[f],{f,facs[n]}],{n,100}],_Integer?(Abs[#]>0&)]

A349931 Numbers that have an equal number of factorizations of even and odd length in both unordered and ordered manners.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 63, 68, 72, 75, 76, 80, 92, 98, 99, 108, 112, 116, 117, 121, 124, 147, 148, 153, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 200, 207, 208, 212, 236, 240, 242, 244, 245, 252, 261, 268, 272, 275, 279, 284, 289, 292, 300
Offset: 1

Views

Author

Tian Vlasic, Dec 05 2021

Keywords

Comments

Intersection of A319240 and A013929, i.e., terms of A319240 that are not squarefree.
A319240 lists the numbers that have an equal number of factorizations of even and odd length in an unordered manner.
A013929 lists the numbers that have an equal number of factorizations of even and odd length in an ordered manner.
There are infinitely many terms in this sequence since p^2 is always such a number for prime p.
Out of all numbers of the form p^k with p prime (listed in A000961), only the numbers of the form p^2 (A001248) are terms.
Out of all numbers of the form p*q^k, p and q prime, only the numbers of the form p*q (A006881), p*q^2 (A054753), p*q^4 (A178739) and p*q^6 (A189987) are terms.
Similar methods can be applied to all prime signatures.
Wilf's conjecture is equivalent to the statement that this sequence is the set difference of A319240 and A006881.

Examples

			12 = 2*6 = 3*4 = 2*2*3 (unordered) has an equal number (2) of even-length factorizations and odd-length factorizations, and 12 = 2*6 = 6*2 = 3*4 = 4*3 = 2*2*3 = 2*3*2 = 3*2*2 (ordered) has an equal number (4) of even-length factorizations and odd-length factorizations.

References

  • Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005.

Links

  • Valerio De Angelis and Dominic Marcello, Wilf's Conjecture, The American Mathematical Monthly 123.6 (2016): 557-573.
  • S. de Wannemacker, T. Laffey and R. Osburn, On a conjecture of Wilf, arXiv:math/0608085 [math.NT], 2006-2007.

Crossrefs

Cf. A006881, A013929, A074206, A319240, A316441, A000587.

Programs

  • PARI
    f(n, m=n, k=0) = if(1==n, (-1)^k, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += f(n/d, d, k+1))); (s)); \\ A316441
isok(m) = (f(m) == 0) && ! issquarefree(m); \\ Michel Marcus, Dec 09 2021

A350486 Numbers that have an equal number of even- and odd-length unordered factorizations and also an equal number of even- and odd-length unordered factorizations into distinct factors.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 180, 183, 185, 187, 192, 194
Offset: 1

Views

Author

Tian Vlasic, Jan 01 2022

Keywords

Comments

First differs from A006881 at a(53) = 180.
By length, we mean the number of factors in a particular factorization.
Intersection of A319240 (factors are not necessarily distinct) and A319238 (factors are distinct).
Numbers k such that A316441(k) = A114592(k) = 0.
There are infinitely many terms in this sequence since all squarefree semiprimes (listed in A006881) are always such numbers.
There are no terms of the form p^k with p prime (listed in A000961).
Out of all numbers of the form p*q^k, p and q prime, only the numbers of the form p*q (A006881) and p*q^6 (A189987) are terms.
Similar methods can be applied to all prime signatures.

Examples

			6=2*3 (unrestricted) has an equal number (1) of even-length factorizations and odd-length factorizations, and 6=2*3 (distinct) has an equal number (1) of even-length factorizations and odd-length factorizations.

Links

Crossrefs

Cf. A006881, A114592, A189987, A316441, A319240, A319238.

Programs

  • Mathematica
    facs[n_] := If[n <= 1, {{}}, Join @@ Table[Map[Prepend[#, d] &, Select[facs[n/d], Min @@ # >= d &]], {d, Rest[Divisors[n]]}]]; Intersection @@ First@Flatten[Position[#, 0] & /@ Transpose@Table[Sum[(-1)^Length[f], {f, #}] & /@ {facs[n], Select[facs[n], UnsameQ @@ # &]}, {n, #1, #2}], {3}]&[1,194] (* Robert P. P. McKone, Jan 05 2022, from Gus Wiseman in A319238 and A319240 *)
Showing 1-5 of 5 results.

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