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 1-6 of 6 results.

A325614 Unsorted q-signature of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 1, 2, 2, 1, 4, 2, 1, 1, 3, 2, 3, 1, 3, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 2, 1, 4, 1, 2, 2, 2, 3, 1, 1, 3, 3, 4, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 2, 2, 1, 1, 3, 1, 1, 3, 1, 1
Offset: 1

Views

Author

Gus Wiseman, May 12 2019

Keywords

Comments

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
11 = q(1) q(2) q(3) q(5)
50 = q(1)^3 q(2)^2 q(3)^2
360 = q(1)^6 q(2)^3 q(3)
Row n lists the nonzero multiplicities in the q-factorization of n, in order of q-index. For example, row 11 is (1,1,1,1) and row 360 is (6,3,1).

Examples

			Triangle begins:
  {}
  1
  1 1
  2
  1 1 1
  2 1
  2 1
  3
  2 2
  2 1 1
  1 1 1 1
  3 1
  2 1 1
  3 1
  2 2 1
  4
  2 1 1
  3 2
  3 1
  3 1 1

Crossrefs

Row lengths are A324923.
Row sums are A196050.
Row-maxima are A109129.
Cf. A118914, A324922, A324924, A324931, A324934, A325608, A325609, A325613, A325615, A325660.

Programs

  • Mathematica
    difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[Length/@Split[difac[n]],{n,30}]

A325615 Sorted q-signature of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 3, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 3, 1, 2, 2, 4, 1, 1, 2, 2, 3, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 2, 2, 1, 4, 2, 2, 2, 1, 1, 3, 3, 3, 1, 4, 1, 1, 1, 2, 1, 2, 3, 1, 1, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1
Offset: 1

Views

Author

Gus Wiseman, May 12 2019

Keywords

Comments

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
11 = q(1) q(2) q(3) q(5)
50 = q(1)^3 q(2)^2 q(3)^2
360 = q(1)^6 q(2)^3 q(3)
Row n is the multiset of nonzero multiplicities in the q-factorization of n. For example, row 11 is (1,1,1,1) and row 360 is (1,3,6).

Examples

			Triangle begins:
  {}
  1
  1 1
  2
  1 1 1
  1 2
  1 2
  3
  2 2
  1 1 2
  1 1 1 1
  1 3
  1 1 2
  1 3
  1 2 2
  4
  1 1 2
  2 3
  1 3
  1 1 3

Crossrefs

Row lengths are A324923.
Row sums are A196050.
Row-maxima are A109129.
Cf. A118914, A324922, A324924, A324931, A324934, A325608, A325609, A325613, A325614, A325660.

Programs

  • Mathematica
    difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[Sort[Length/@Split[difac[n]]],{n,30}]

A325661 q-powerful numbers. Numbers whose factorization into factors prime(i)/i has no factor of multiplicity 1.

Original entry on oeis.org

1, 4, 8, 9, 16, 18, 25, 27, 32, 36, 49, 50, 54, 64, 72, 75, 81, 98, 100, 108, 121, 125, 128, 144, 150, 162, 169, 196, 200, 216, 225, 242, 243, 250, 256, 288, 289, 300, 324, 338, 343, 361, 363, 375, 392, 400, 432, 441, 450, 484, 486, 500, 507, 512, 529, 576
Offset: 1

Views

Author

Gus Wiseman, May 13 2019

Keywords

Comments

First differs from A070003 in having 1 and lacking 147.
Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
11 = q(1) q(2) q(3) q(5)
50 = q(1)^3 q(2)^2 q(3)^2
360 = q(1)^6 q(2)^3 q(3)
Also Matula-Goebel numbers of rooted trees with no terminal subtree appearing at only one place in the tree.

Examples

			The sequence of terms together with their q-signatures begins:
    1: {}
    4: {2}
    8: {3}
    9: {2,2}
   16: {4}
   18: {3,2}
   25: {2,2,2}
   27: {3,3}
   32: {5}
   36: {4,2}
   49: {4,2}
   50: {3,2,2}
   54: {4,3}
   64: {6}
   72: {5,2}
   75: {3,3,2}
   81: {4,4}
   98: {5,2}
  100: {4,2,2}

Links

Crossrefs

Cf. A001222, A001221, A001694, A056239, A112798, A124010.
Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713.
q-factorization: A324922, A324923, A324924, A325615, A325660.

Programs

  • Mathematica
    difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Select[Range[100],Count[Length/@Split[difac[#]],1]==0&]

A325613 Full q-signature of n. Irregular triangle read by rows where T(n,k) is the multiplicity of q(k) in the q-factorization of n.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, May 12 2019

Keywords

Comments

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
11 = q(1) q(2) q(3) q(5)
50 = q(1)^3 q(2)^2 q(3)^2
360 = q(1)^6 q(2)^3 q(3)
Also the number of terminal subtrees with Matula-Goebel number k of the rooted tree with Matula-Goebel number n.

Examples

			Triangle begins:
  {}
  1
  1 1
  2
  1 1 1
  2 1
  2 0 0 1
  3
  2 2
  2 1 1
  1 1 1 0 1
  3 1
  2 1 0 0 0 1
  3 0 0 1
  2 2 1
  4
  2 0 0 1 0 0 1
  3 2
  3 0 0 0 0 0 0 1
  3 1 1

Crossrefs

Row lengths are A061395.
Row sums are A196050.
Row-maxima are A109129.
The number whose full prime signature is the n-th row is A324922(n).
Cf. A067255.
Matula-Goebel numbers: A007097, A061775, A109082, A317713.
q-factorization: A324923, A324924, A325613, A325614, A325615, A325660.

Programs

  • Mathematica
    difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
qsig[n_]:=If[n==1,{},With[{ms=difac[n]},Table[Count[ms,i],{i,Max@@ms}]]];
Table[qsig[n],{n,30}]

A325609 Unsorted q-signature of n!. Irregular triangle read by rows where T(n,k) is the multiplicity of q(k) in the factorization of n! into factors q(i) = prime(i)/i.

Original entry on oeis.org

1, 2, 1, 4, 1, 5, 2, 1, 7, 3, 1, 9, 3, 1, 1, 12, 3, 1, 1, 14, 5, 1, 1, 16, 6, 2, 1, 17, 7, 3, 1, 1, 20, 8, 3, 1, 1, 22, 9, 3, 1, 1, 1, 25, 9, 3, 2, 1, 1, 27, 11, 4, 2, 1, 1, 31, 11, 4, 2, 1, 1, 33, 11, 4, 3, 1, 1, 1, 36, 13, 4, 3, 1, 1, 1, 39, 13, 4, 3, 1, 1, 1, 1
Offset: 1

Views

Author

Gus Wiseman, May 12 2019

Keywords

Comments

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
11 = q(1) q(2) q(3) q(5)
50 = q(1)^3 q(2)^2 q(3)^2
360 = q(1)^6 q(2)^3 q(3)
Row n is the sequence of nonzero exponents in the q-factorization of n!.
Also the number of terminal subtrees with Matula-Goebel number k of the rooted tree with Matula-Goebel number n!.

Examples

			We have 10! = q(1)^16 q(2)^6 q(3)^2 q(4), so row n = 10 is (16,6,2,1).
Triangle begins:
  {}
   1
   2  1
   4  1
   5  2  1
   7  3  1
   9  3  1  1
  12  3  1  1
  14  5  1  1
  16  6  2  1
  17  7  3  1  1
  20  8  3  1  1
  22  9  3  1  1  1
  25  9  3  2  1  1
  27 11  4  2  1  1
  31 11  4  2  1  1
  33 11  4  3  1  1  1
  36 13  4  3  1  1  1
  39 13  4  3  1  1  1  1
  42 14  5  3  1  1  1  1

Crossrefs

Row lengths are A000720.
Row sums are A325544(n) - 1.
Column k = 1 is A325543.
Cf. A056239, A067255, A112798, A118914, A124010.
Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713, A324935.
Factorial numbers: A000142, A011371, A022559, A071626, A115627, A325276.
q-factorization: A324922, A324923, A324924, A325614, A325615, A325660.

Programs

  • Mathematica
    difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[Length/@Split[difac[n!]],{n,20}]

A325608 Numbers whose factorization into factors prime(i)/i does not have weakly decreasing nonzero multiplicities.

Original entry on oeis.org

147, 245, 294, 357, 490, 511, 539, 588, 595, 637, 681, 714, 735, 845, 847, 853, 867, 903, 980, 1022, 1029, 1043, 1078, 1083, 1135, 1176, 1183, 1190, 1239, 1241, 1267, 1274, 1309, 1362, 1421, 1428, 1445, 1470, 1505, 1519, 1547, 1553, 1563, 1617, 1631, 1690
Offset: 1

Views

Author

Gus Wiseman, May 12 2019

Keywords

Comments

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example, 147 = q(1)^5 q(2) q(4)^2 has multiplicities (5,1,2), which are not weakly decreasing, so 147 belongs to the sequence.

Crossrefs

Cf. A001222, A056239, A112798, A118914.
Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713, A324935.
q-factorization: A324922, A324923, A324924, A324931, A325613, A325614, A325615, A325660, A325662.

Programs

  • Mathematica
    difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Select[Range[1000],!GreaterEqual@@Length/@Split[difac[#]]&]
Showing 1-6 of 6 results.

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

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