A325614 -id:A325614 - cp's OEIS frontend

A325660 Number of ones in the q-signature of n.

0, 1, 2, 0, 3, 1, 1, 0, 0, 2, 4, 1, 2, 1, 1, 0, 2, 0, 1, 2, 2, 3, 1, 1, 0, 2, 0, 1, 3, 1, 5, 0, 2, 2, 3, 0, 2, 1, 1, 2, 3, 2, 2, 3, 1, 1, 2, 1, 0, 0, 3, 2, 1, 0, 1, 1, 2, 3, 3, 1, 1, 4, 1, 0, 2, 2, 2, 2, 1, 3, 3, 0, 3, 2, 0, 1, 4, 1, 4, 2, 0, 3, 2, 2, 4, 2, 2

Offset: 1

Gus Wiseman, May 13 2019

nonn

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)
Then a(n) is the number of factors of multiplicity one in the q-factorization of n.
Also the number of rooted trees appearing only once in the multiset of terminal subtrees of the rooted tree with Matula-Goebel number n.

Cf. A001694, A055231, A056169, A056239, A112798, A118914, A124010.
Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713, A324968.
q-factorization: A324922, A324923, A324924, A325614, A325661.

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[Count[Length/@Split[difac[n]],1],{n,100}]

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

Gus Wiseman, May 12 2019

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).

			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

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

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}]

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

Gus Wiseman, May 12 2019

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.

			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

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.

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

Gus Wiseman, May 12 2019

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!.

			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

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.

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}]

A325662 Matula-Goebel numbers of regular rooted stars.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 11, 16, 25, 27, 31, 32, 64, 81, 121, 125, 127, 128, 243, 256, 512, 625, 709, 729, 961, 1024, 1331, 2048, 2187, 3125, 4096, 5381, 6561, 8192, 14641, 15625, 16129, 16384, 19683, 29791, 32768, 52711, 59049, 65536, 78125, 131072, 161051

Offset: 1

Gus Wiseman, May 13 2019

nonn

Powers of members of A007097.
A regular rooted star is a rooted tree whose branches are all rooted paths of equal length.
The number of terms <= 10^k, k=0,1,2,...: 1, 7, 15, 26, 35, 46, 56, 67, 76, 87, 98, 109, 121, 131, 142, 154, 163, 175, 185, 198, 208, 220, 231, 241, 254, 265, 275, etc. - Robert G. Wilson v, May 13 2019

			The sequence of regular rooted stars together with their Matula-Goebel numbers begins:
    1: o
    2: (o)
    3: ((o))
    4: (oo)
    5: (((o)))
    8: (ooo)
    9: ((o)(o))
   11: ((((o))))
   16: (oooo)
   25: (((o))((o)))
   27: ((o)(o)(o))
   31: (((((o)))))
   32: (ooooo)
   64: (oooooo)
   81: ((o)(o)(o)(o))
  121: ((((o)))(((o))))
  125: (((o))((o))((o)))
  127: ((((((o))))))
  128: (ooooooo)

Robert G. Wilson v, Table of n, a(n) for n = 1..275 (terms 1..48 from _Gus Wiseman_)

Cf. A007097, A056239, A061775, A109082, A109129, A112798, A196050, A324924, A325614, A325661, A325663.

rpQ[n_]:=n==1||PrimeQ[n]&&rpQ[PrimePi[n]];
Select[Range[100],#==1||PrimePowerQ[#]&&rpQ[FactorInteger[#][[1,1]]]&]
(* generates terms <= A007097(max) *) seq[max_] := Module[{ps = NestList[Prime@# &, 1, max], psmax, s = {1}, emax, s1}, pmax = Max[ps]; Do[p = ps[[k]]; emax = Floor[Log[p, pmax]]; s1 = p^Range[emax]; s = Union[s, s1], {k, 2, Length[ps]}]; s]; seq[10] (* Amiram Eldar, Jul 26 2024 *)

Sum_{n>=1} 1/a(n) = 1 + Product_{k>=1} 1/(A007097(k)-1) = 2.8928887669834086909... - Amiram Eldar, Jul 26 2024

A325663 Matula-Goebel numbers of not necessarily regular rooted stars.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 18, 20, 22, 24, 25, 27, 30, 31, 32, 33, 36, 40, 44, 45, 48, 50, 54, 55, 60, 62, 64, 66, 72, 75, 80, 81, 88, 90, 93, 96, 99, 100, 108, 110, 120, 121, 124, 125, 127, 128, 132, 135, 144, 150, 155, 160, 162, 165, 176

Offset: 1

Gus Wiseman, May 13 2019

nonn

Products of members of A007097.

A rooted star is a rooted tree whose branches are all rooted paths.

			The sequence of rooted stars together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   6: (o(o))
   8: (ooo)
   9: ((o)(o))
  10: (o((o)))
  11: ((((o))))
  12: (oo(o))
  15: ((o)((o)))
  16: (oooo)
  18: (o(o)(o))
  20: (oo((o)))
  22: (o(((o))))
  24: (ooo(o))
  25: (((o))((o)))
  27: ((o)(o)(o))
  30: (o(o)((o)))

Amiram Eldar, Table of n, a(n) for n = 1..10538 (terms up to A007097(12))

Cf. A007097, A056239, A061775, A109082, A109129, A112798, A196050, A324924, A325614, A325661, A325662.

rpQ[n_]:=n==1||PrimeQ[n]&&rpQ[PrimePi[n]];
Select[Range[100],And@@rpQ/@First/@FactorInteger[#]&]
(* generates terms <= A007097(max) *) seq[max_] := Module[{ps = NestList[Prime@# &, 1, max], psmax, s = {1}, emax, s1, s2}, pmax = Max[ps]; Do[p = ps[[k]]; emax = Floor[Log[p, pmax]]; s1 = p^Range[0, emax]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= pmax &]; s = Union[s, s2], {k, 2, Length[ps]}]; s]; seq[7] (* Amiram Eldar, Jul 26 2024 *)

Sum_{n>=1} 1/a(n) = Product_{k>=1} A007097(k)/(A007097(k)-1) = 4.30328607286382284593... . - Amiram Eldar, Jul 26 2024

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

Gus Wiseman, May 12 2019

nonn

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.

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.

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[#]]&]

cp's OEIS Frontend

A325660 Number of ones in the q-signature of n.

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A325615 Sorted q-signature of n.

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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.

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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.

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A325662 Matula-Goebel numbers of regular rooted stars.

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A325663 Matula-Goebel numbers of not necessarily regular rooted stars.

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

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