A049076 -id:A049076 - cp's OEIS frontend

A318046 a(n) is the number of initial subtrees (subtrees emanating from the root) of the unlabeled rooted tree with Matula-Goebel number n.

1, 2, 3, 2, 4, 3, 3, 2, 5, 4, 5, 3, 4, 3, 7, 2, 4, 5, 3, 4, 5, 5, 6, 3, 10, 4, 9, 3, 5, 7, 6, 2, 9, 4, 7, 5, 4, 3, 7, 4, 5, 5, 4, 5, 13, 6, 8, 3, 5, 10, 7, 4, 3, 9, 13, 3, 5, 5, 5, 7, 6, 6, 9, 2, 10, 9, 4, 4, 11, 7, 5, 5, 6, 4, 19, 3, 9, 7, 6, 4, 17, 5, 7, 5

Offset: 1

Gus Wiseman, Aug 13 2018

nonn

We require that an initial subtree contain either all or none of the branchings under any given node.

			70 is the Matula-Goebel number of the tree (o((o))(oo)), which has 7 distinct initial subtrees: {o, (ooo), (oo(oo)), (o(o)o), (o(o)(oo)), (o((o))o), (o((o))(oo))}. So a(70) = 7.

Index entries for sequences related to Matula-Goebel numbers

Cf. A000081, A007097, A007853, A049076, A061773, A061775, A076146, A109082, A109129, A206491, A303431, A316476, A317713.

si[n_]:=If[n==1,1,1+Product[si[PrimePi[b[[1]]]]^b[[2]],{b,FactorInteger[n]}]];
Array[si,100]

a(1) = 1 and if n > 1 has prime factorization n = prime(x_1)^y_1 * ... * prime(x_k)^y_k then a(n) = 1 + a(x_1)^y_1 * ... * a(x_k)^y_k.

A322027 Maximum order of primeness among the prime factors of n; a(1) = 0.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 24 2018

nonn

The order of primeness (A078442) of a prime number p is the number of times one must apply A000720 to obtain a nonprime number.

			a(105) = 3 because the prime factor of 105 = 3*5*7 with maximum order of primeness is 5, with order 3.

Alois P. Heinz, Table of n, a(n) for n = 1..65536
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included at A006450 with permission of the author]

Cf. A000720, A006450, A007097, A007821, A049076, A076610, A078442, A109082, A114537, A184155, A276625, A279065, A322028, A322030.

with(numtheory):
p:= proc(n) option remember;
      `if`(isprime(n), 1+p(pi(n)), 0)
    end:
a:= n-> max(0, map(p, factorset(n))):
seq(a(n), n=1..120);  # Alois P. Heinz, Nov 24 2018

Table[If[n==1,0,Max@@(Length[NestWhileList[PrimePi,PrimePi[#],PrimeQ]]&/@FactorInteger[n][[All,1]])],{n,100}]

A050440 Sixth-order composites.

Original entry on oeis.org

56, 69, 77, 78, 84, 94, 100, 105, 106, 115, 124, 125, 126, 133, 140, 141, 145, 152, 156, 162, 164, 165, 170, 174, 183, 184, 188, 198, 202, 203, 206, 209, 212, 213, 218, 222, 231, 235, 236, 242, 243, 253, 256, 258, 259, 262, 264, 266, 270, 272, 278, 284

Offset: 1

Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999

			C(C(C(C(C(C(1)))))) = C(C(C(C(C(4))))) = C(C(C(C(9)))) = C(C(C(16))) = C(C(26)) = C(39) = 56. So 56 is in the sequence. So 77 is in the sequence.

N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076-A049081, A006450, A050435, A050436, A050438, A050439.

C := remove(isprime,[$4..1000]): seq(C[C[C[C[C[C[n]]]]]],n=1..100);

Let C(n) be the n-th composite number, with C(1)=4. Then these are numbers C(C(C(C(C(C(n)))))).

More terms from Asher Auel Dec 15 2000

A138947 Square array T[i+1,j] = prime(T[i,j]), T[1,j] = j-th nonprime = A018252(j); read by upward antidiagonals.

Original entry on oeis.org

1, 4, 2, 6, 7, 3, 8, 13, 17, 5, 9, 19, 41, 59, 11, 10, 23, 67, 179, 277, 31, 12, 29, 83, 331, 1063, 1787, 127, 14, 37, 109, 431, 2221, 8527, 15299, 709, 15, 43, 157, 599, 3001, 19577, 87803

Offset: 1

M. F. Hasler, Apr 28 2008

For i>1, T[i,j] = A018252(j)-th number among those not occurring in rows < i.
A permutation of the integers > 0.
Transpose of A114537. See that sequence and the link for more information and references.

			The first row (1,4,6,8,9,10...) of the array gives the nonprime numbers A018252.
The 2nd row (2,7,13,19,23,29,37,...) of the array gives the primes with nonprime index, A000040(A018252(j)) = A007821(j).
The i-th row is { A000040(k) | A049076(k)=i-1 } = A078442^{-1}(i-1).
Column j is the sequence b(n+1)=prime(b(n)) starting with b(j)=A018252(j): A007097, A057450, A057451, A057452, A057453, A057456, A057457, ...

Alexandrov, Lubomir. "On the nonasymptotic prime number distribution." arXiv preprint math/9811096 (1998). (See Appendix.)

N. Fernandez, An order of primeness, F(p).
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A018252, A007821, A006450, A049076, A007097.

If the antidiagonals are read in the opposite direction we get A114537.

t[1, 1] = 1; t[1, 2] = 4; t[1, k_] := (p = t[1, k-1]; If[ PrimeQ[p+1], p+2, p+1]); t[n_ /; n > 1, k_] := Prime[t[n-1, k]]; Flatten[ Table[ t[n, k-n+1], {k, 1, 9}, {n, 1, k}]] (* Jean-François Alcover, Oct 03 2011 *)

p=c=0; T=matrix( 10,10, i,j, if( i==1, while( isprime(c++),); p=c, p=prime(p))); A138947=concat( vector( vecmin( matsize( T )),i, vector( i,j, T[ j,i+1-j ])))

T[i,j] = j-th number for which A078442 equals i-1.

A322028 Number of distinct orders of primeness among the prime factors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2

Offset: 1

Gus Wiseman, Nov 24 2018

nonn

The order of primeness (A078442) of a prime number p is the number of times one must apply A000720 to obtain a nonprime number.

			a(105) = 3 because the prime factors of 105 = 3*5*7 have 3 different orders of primeness, namely 2, 3, and 1 respectively.

Alois P. Heinz, Table of n, a(n) for n = 1..65536
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included at A006450 with permission of the author]

Cf. A000720, A006450, A007097, A007821, A049076, A076610, A078442, A109082, A114537, A184155, A276625, A279065, A322027, A322030.

with(numtheory):
p:= proc(n) option remember;
      `if`(isprime(n), 1+p(pi(n)), 0)
    end:
a:= n-> nops(map(p, factorset(n))):
seq(a(n), n=1..120);  # Alois P. Heinz, Nov 24 2018

Table[If[n==1,0,Length[Union[Length[NestWhileList[PrimePi,PrimePi[#],PrimeQ]]&/@FactorInteger[n][[All,1]]]]],{n,100}]

A322030 Numbers whose prime factors all have the same order of primeness.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 16, 17, 19, 23, 25, 26, 27, 28, 29, 31, 32, 37, 38, 41, 43, 46, 47, 49, 51, 52, 53, 56, 58, 59, 61, 64, 67, 71, 73, 74, 76, 79, 81, 83, 86, 89, 91, 92, 94, 97, 98, 101, 103, 104, 106, 107, 109, 112, 113, 116, 121, 122, 123

Offset: 1

Gus Wiseman, Nov 24 2018

nonn

The order of primeness (A078442) of a prime number p is the number of times one must apply A000720 to obtain a nonprime number.

			182 is in the sequence because its prime factors 2, 7, 13 all have order of primeness 1.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included at A006450 with permission of the author]

Cf. A000720, A006450, A007097, A007821, A049076, A076610, A078442, A109082, A114537, A184155, A276625, A322027, A322028.

with(numtheory):
p:= proc(n) option remember;
      `if`(isprime(n), 1+p(pi(n)), 0)
    end:
a:= proc(n) option remember; local k; for k from 1+`if`(n=1,
      0, a(n-1)) while nops(map(p, factorset(k)))>1 do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 24 2018

ordpri[n_]:=If[!PrimeQ[n],0,Length[NestWhileList[PrimePi,PrimePi[n],PrimeQ]]];
Select[Range[200],SameQ@@ordpri/@FactorInteger[#][[All,1]]&]

A323161 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n<=3) = -n, f(n) = 0 if n-1 is an odd prime, and f(n) = floor((n-1)/2) for all other numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 6, 4, 7, 7, 8, 4, 9, 4, 10, 10, 11, 4, 12, 4, 13, 13, 14, 4, 15, 15, 16, 16, 17, 4, 18, 4, 19, 19, 20, 20, 21, 4, 22, 22, 23, 4, 24, 4, 25, 25, 26, 4, 27, 27, 28, 28, 29, 4, 30, 30, 31, 31, 32, 4, 33, 4, 34, 34, 35, 35, 36, 4, 37, 37, 38, 4, 39, 4, 40, 40, 41, 41, 42, 4, 43, 43, 44, 4, 45, 45, 46, 46, 47, 4, 48, 48, 49, 49, 50, 50, 51, 4, 52, 52, 53, 4, 54, 4

Offset: 1

Antti Karttunen, Jan 06 2019

nonn

For all i, j: A322809(i) = A322809(j) <=> a(i+1) = a(j+1).
For all i, j: a(i) = a(j) => b(i) = b(j), where b can be, but is not limited to, any of the following sequences: A029834, A049084, A062590, A063377, A064891, A078442 (A049076), A175663, A175682, A269668, A292936, A323162, many of which are related to counting primes in certain kinds of chains or iterations.
Why does this work? Consider the function f given in the definition: based on its properties, we can deduce from the value of f(n) the following information about n:
  (A) If f(n) = -2, then n is 2, the only even prime,
  (B) If f(n) = -3, then n is 3, the first odd prime,
  (C) If f(n) is zero, then n is an even composite preceded by a prime, but we don't know which even composite exactly,
  (D) If f(n) > 0 and f(1+2*f(n)) = f(2+2*f(n)), then n is either (D1) an odd composite number, or (D2) an even composite number preceded by an odd composite number, and the said composite number in both cases is 1 + 2*f(n),
  (E) If f(n) > 0 and f(1+2*f(n)) <> f(2+2*f(n)), then n is an odd prime > 3, specifically, 1 + 2*f(n).
As this sequence is a restricted growth sequence transform of the said function f, we have a(i) = a(j) <=> f(i) = f(j) for all i, j, thus, even without knowing the value of n, but just a(n), we can find the value of f(n) by searching for the minimal k such that a(k) = a(n), then compute f(k) with that k. Furthermore, any function g defined as g(n) = h(f(n)) [where h is any function], clearly satisfies
  a(i) = a(j) => g(i) = g(j), for all i, j. [*]
For instances of such functions g, we can consider many sequences like those sequences b(n) listed above, that have g(n) = 0 for all composite numbers, and g(p) > 0 for all primes p. This is usually the pattern, but there are exceptions, like A323162, which is the characteristic function of A005381, composites n such that n-1 is also composite. These are precisely the numbers that occur twice in this sequence, while all other numbers (including primes), occur just once, that is, reside in their own singular equivalence classes. Thus, it is not guaranteed that all sequences g matching to this sequence (i.e. those satisfying the implication *), even if not false positives in strict sense, would necessarily have some consistent relation to primes, instead, they might contain any random values at the positions given by A093515. However, in the current OEIS, such sequences are exceedingly rare.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A305801, A305900, A322809.
Cf. A005381 (numbers that occur twice in this sequence), A093515 (numbers > 1 that occur just once).
Cf. A010051, A029834, A049076, A049084, A062590, A063377, A064891, A078442, A175663, A175682, A269668, A292936, A323162 (some of the matched sequences).

up_to = 10000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A323161aux(n) = if(n<=3,-n,if(isprime(n-1),0,((n-1)>>1))); \\ This implements the function f of the definition.
v323161 = rgs_transform(vector(up_to,n,A323161aux(n)));
A323161(n) = v323161[n];

a(1) = 1; for n > 1, a(n) = 1 + A322809(n-1).

A064960 The prime then composite recurrence; a(2n) = a(2n-1)-th prime and a(2n+1) = a(2n)-th composite and a(1) = 1.

Original entry on oeis.org

1, 2, 6, 13, 22, 79, 108, 593, 722, 5471, 6290, 62653, 69558, 876329, 951338, 14679751, 15692307, 289078661, 305618710, 6588286337, 6908033000, 171482959009, 178668550322, 5040266614919, 5225256019175, 165678678591359, 171068472492228, 6039923990345039

Offset: 1

Robert G. Wilson v, Oct 29 2001

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..31
Andrew R. Booker, The Nth Prime Page
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A007097, A006508 & A064961, see also A057450, A057451, A057452, A057453, A057456 & A057457 and A049076, A049077, A049078, A049079, A049080 & A049081.

Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; a = {1}; b = 1; Do[ If[ !PrimeQ[b], b = Prime[b], b = Composite[b]]; a = Append[a, b], {n, 1, 23}]; a

from functools import cache
from sympy import prime, composite
@cache
def A064960(n): return 1 if n == 1 else composite(A064960(n-1)) if n % 2 else prime(A064960(n-1)) # Chai Wah Wu, Jan 01 2022

a(26)-a(28) from Chai Wah Wu, May 07 2018

A064961 Composite-then-prime recurrence; a(2n) = a(2n-1)-th composite and a(2n+1) = a(2n)-th prime and a(1) = 1.

Original entry on oeis.org

1, 4, 7, 14, 43, 62, 293, 366, 2473, 2892, 26317, 29522, 344249, 376259, 5429539, 5831545, 101291779, 107457490, 2198218819, 2310909505, 54720307351, 57128530327, 1543908890351, 1603146693999, 48871886538151, 50527531769529, 1720466016680911, 1772475453490311

Offset: 1

Robert G. Wilson v, Oct 29 2001

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..32
Andrew R. Booker, The Nth Prime Page
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A007097, A006508 & A064960, see also A057450, A057451, A057452, A057453, A057456 & A057457 and A049076, A049077, A049078, A049079, A049080 & A049081.

Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; a = {1, 4}; b = 4; Do[ If[ !PrimeQ[b], b = Prime[b], b = Composite[b]]; a = Append[a, b], {n, 1, 23}]; a

a(24)-a(26) corrected and a(27)-a(28) added by Chai Wah Wu, Aug 22 2018

A318048 Size of the span of the unlabeled rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 2, 3, 2, 4, 4, 4, 2, 6, 6, 5, 4, 6, 3, 9, 2, 6, 6, 4, 6, 6, 8, 10, 4, 12, 6, 10, 4, 9, 9, 6, 2, 12, 6, 9, 6, 6, 4, 9, 6, 9, 7, 6, 8, 15, 10, 15, 4, 5, 12, 9, 7, 4, 10, 16, 4, 7, 9, 8, 9, 10, 10, 11, 2, 13, 12, 6, 7, 14, 10, 9, 6, 10, 7, 21, 3, 12, 10, 12, 6

Offset: 1

Gus Wiseman, Aug 13 2018

nonn

The span of a tree is defined to be the set of possible terminal subtrees of initial subtrees, or, which is the same, the set of possible initial subtrees of terminal subtrees.

			42 is the Matula-Goebel number of (o(o)(oo)), which has span {o, (o), (oo), (ooo), (oo(oo)), (o(o)o), (o(o)(oo))}, so a(42) = 7.

Index entries for sequences related to Matula-Goebel numbers

Cf. A000081, A007097, A007853, A049076, A061773, A061775, A109082, A109129, A206491, A317713, A318046.

ext[c_,{}]:=c;ext[c_,s:{}]:=Extract[c,s];rpp[c_,v_,{}]:=v;rpp[c_,v_,s:{}]:=ReplacePart[c,v,s];
RLO[ear_,rue:{}]:=Union@@(Function[x,rpp[ear,x,#2]]/@ReplaceList[ext[ear,#2],#1]&@@@Select[Tuples[{rue,Position[ear,_]}],MatchQ[ext[ear,#[[2]]],#[[1,1]]]&]);
RL[ear_,rue:{}]:=FixedPoint[Function[keeps,Union[keeps,Join@@(RLO[#,rue]&/@keeps)]],{ear}];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
MGTree[n_]:=If[n==1,{},MGTree/@primeMS[n]];
Table[Length[Union[Cases[RL[MGTree[n],{List[__List]:>List[]}],_List,{1,Infinity}]]],{n,100}]

cp's OEIS Frontend

A318046 a(n) is the number of initial subtrees (subtrees emanating from the root) of the unlabeled rooted tree with Matula-Goebel number n.

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A322027 Maximum order of primeness among the prime factors of n; a(1) = 0.

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Maple

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A050440 Sixth-order composites.

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A138947 Square array T[i+1,j] = prime(T[i,j]), T[1,j] = j-th nonprime = A018252(j); read by upward antidiagonals.

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A322028 Number of distinct orders of primeness among the prime factors of n.

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Maple

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A322030 Numbers whose prime factors all have the same order of primeness.

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A323161 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n<=3) = -n, f(n) = 0 if n-1 is an odd prime, and f(n) = floor((n-1)/2) for all other numbers.

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A064960 The prime then composite recurrence; a(2n) = a(2n-1)-th prime and a(2n+1) = a(2n)-th composite and a(1) = 1.