A106490 -id:A106490 - cp's OEIS frontend

A276230 a(n) is the A106490 index where n first occurs.

1, 2, 4, 12, 36, 144, 720, 3600, 25200, 176400, 1587600, 17463600, 192099600, 2497294800, 32464832400, 551902150800, 9382336563600, 178264394708400, 3387023499459600, 77901540487570800, 1791735431214128400, 44793385780353210000, 1299008187630243090000

Offset: 0

Hans Havermann, Aug 24 2016

nonn

			0, 1, 2, and 3 first appear in A106490 at positions 1, 2, 4, and 12, respectively.

Cf. A106490.

a[n_] := a[n] = If[n == 1, 0, Sum[1 + a[i[[2]]], {i, FactorInteger[n]}]];
Flatten@ Table[FirstPosition[#, k_ /; k == n], {n, 0, 9}] &@ Table[a[n], {n, 10^6}] (* Michael De Vlieger, Aug 25 2016, Version 10, after Jean-François Alcover at A106490 *)

More terms from Alois P. Heinz, Aug 25 2016

A277562 Numbers of the form c(x_1)^c(x_2)^...^c(x_k) where each c(i) = A007916(i) is a non-perfect-power, k >= 2, and the exponents are nested from the right.

Original entry on oeis.org

16, 81, 256, 512, 625, 1296, 2401, 6561, 10000, 14641, 19683, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, 456976, 614656, 707281, 810000, 923521, 1185921, 1336336, 1500625, 1679616, 1874161, 1953125, 2085136, 2313441, 2560000, 2825761, 3111696, 3418801

Offset: 1

Gus Wiseman, Oct 19 2016

nonn

Non-perfect-powers, or NPPs (A007916), are numbers whose prime multiplicities are relatively prime. As discussed in A007916, the expansion of a positive integer into a tower of NPPs is unique and always possible. 65536=2^2^2^2 is the smallest number that requires a tower of height more than 3.

			       16 = 2^2^2        81 = 3^2^2       256 = 2^2^3       512 = 2^3^2
      625 = 5^2^2      1296 = 6^2^2      2401 = 7^2^2      6561 = 3^2^3
    10000 = 10^2^2    14641 = 11^2^2    19683 = 3^3^2     20736 = 12^2^2
    28561 = 13^2^2    38416 = 14^2^2    50625 = 15^2^2
    65536 = 2^2^2^2   83521 = 17^2^2   104976 = 18^2^2   130321 = 19^2^2
   160000 = 20^2^2   194481 = 21^2^2   234256 = 22^2^2   279841 = 23^2^2
   331776 = 24^2^2   390625 = 5^2^3    456976 = 26^2^2   614656 = 28^2^2
   707281 = 29^2^2   810000 = 30^2^2   923521 = 31^2^2  1185921 = 33^2^2
  1336336 = 34^2^2  1500625 = 35^2^2  1679616 = 6^2^3   1874161 = 37^2^2
  1953125 = 5^3^2   2085136 = 38^2^2  2313441 = 39^2^2  2560000 = 40^2^2
  2825761 = 41^2^2  3111696 = 42^2^2  3418801 = 43^2^2  3748096 = 44^2^2
  4100625 = 45^2^2  4477456 = 46^2^2  4879681 = 47^2^2  5308416 = 48^2^2
  5764801 = 7^2^3   6250000 = 50^2^2  6765201 = 51^2^2  7311616 = 52^2^2
  7890481 = 53^2^2  8503056 = 54^2^2  9150625 = 55^2^2  9834496 = 56^2^2

David A. Corneth, Table of n, a(n) for n = 1..10025 (terms <= 10^16)
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy)

Cf. A007916, A001597, A164336, A164337, A106490 (Quetian Superfactorization).

radicalQ[1]:=False;
radicalQ[n_]:=SameQ[GCD@@FactorInteger[n][[All,2]],1];
hyperfactor[1]:={};
hyperfactor[n_?radicalQ]:={n};
hyperfactor[n_]:=With[{g=GCD@@FactorInteger[n][[All,2]]},Prepend[hyperfactor[g],Product[Apply[Power[#1,#2/g]&,r],{r,FactorInteger[n]}]]];
Select[Range[10^6],Length[hyperfactor[#]]>2&]

Edited by N. J. A. Sloane, Nov 09 2016

Offset changed to 1 by David A. Corneth, Apr 30 2024

A279944 Number of positions in the free pure symmetric multifunction in one symbol with j-number n.

Original entry on oeis.org

1, 3, 5, 5, 7, 7, 9, 4, 7, 9, 11, 6, 9, 11, 13, 7, 8, 11, 13, 15, 9, 10, 13, 15, 9, 17, 6, 11, 12, 15, 17, 6, 11, 19, 8, 9, 13, 14, 17, 19, 8, 13, 21, 10, 11, 15, 16, 19, 11, 21, 10, 15, 23, 12, 13, 17, 18, 21, 13, 23, 12, 17, 25, 7, 14, 15, 19, 20, 23, 15, 25, 14, 19, 27, 9, 16, 17, 21, 22, 25, 9, 17, 27, 16, 21, 29, 11, 18, 19, 23, 24, 27, 11, 19, 29, 18, 23, 31, 13, 11

Offset: 1

Gus Wiseman, Dec 24 2016

nonn

A free pure symmetric multifunction in one symbol f in PSM(x) is either (case 1) f = the symbol x, or (case 2) f = an expression of the form h[g_1,...,g_k] where h is in PSM(x), each of the g_i for i=1..(k>0) is in PSM(x), and for i < j we have g_i <= g_j under a canonical total ordering of PSM(x), such as the Mathematica ordering of expressions. For a positive integer n we define a free pure symmetric multifunction j(n) by: j(1)=x; j(n>1) = j(h)[j(g_1),...,j(g_k)] where n = r(h)^(p(g_1)*...*p(g_k)-1). Here r(n) is the n-th number that is not a perfect power (A007916) and p(n) is the n-th prime number (A000040). See example. Then a(n) is the number of brackets [...] plus the number of x's in j(n).

			The first 20 free pure symmetric multifunctions in x are:
j(1)  = j(1)            = x
j(2)  = j(1)[j(1)]      = x[x]
j(3)  = j(2)[j(1)]      = x[x][x]
j(4)  = j(1)[j(2)]      = x[x[x]]
j(5)  = j(3)[j(1)]      = x[x][x][x]
j(6)  = j(4)[j(1)]      = x[x[x]][x]
j(7)  = j(5)[j(1)]      = x[x][x][x][x]
j(8)  = j(1)[j(1),j(1)] = x[x,x]
j(9)  = j(2)[j(2)]      = x[x][x[x]]
j(10) = j(6)[j(1)]      = x[x[x]][x][x]
j(11) = j(7)[j(1)]      = x[x][x][x][x][x]
j(12) = j(8)[j(1)]      = x[x,x][x]
j(13) = j(9)[j(1)]      = x[x][x[x]][x]
j(14) = j(10)[j(1)]     = x[x[x]][x][x][x]
j(15) = j(11)[j(1)]     = x[x][x][x][x][x][x]
j(16) = j(1)[j(3)]      = x[x[x][x]]
j(17) = j(12)[j(1)]     = x[x,x][x][x]
j(18) = j(13)[j(1)]     = x[x][x[x]][x][x]
j(19) = j(14)[j(1)]     = x[x[x]][x][x][x][x]
j(20) = j(15)[j(1)]     = x[x][x][x][x][x][x][x].

Cf. A005043, A007916, A106490, A277564, A277615, A277996, A278028, A280000.

Cf. A279984 (numbers j(n)[x]=j(prime(n))), A277576 (numbers j(n)=x[x][x][x]...), A058891 (numbers j(n)=x[x,...,x]), A279969 (numbers j(n)=x[x[...[x]]]).

nn=100;
radQ[n_]:=If[n===1,False,SameQ[GCD@@FactorInteger[n][[All,2]],1]];
rad[n_]:=rad[n]=If[n===0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
Set@@@Array[radPi[rad[#]]==#&,nn];
jfac[n_]:=With[{g=GCD@@FactorInteger[n+1][[All,2]]},JIX[radPi[Power[n+1,1/g]],Flatten[Cases[FactorInteger[g+1],{p_,k_}:>ConstantArray[PrimePi[p],k]]]]];
diwt[n_]:=If[n===1,1,Apply[1+diwt[#1]+Total[diwt/@#2]&,jfac[n-1]]];
Array[diwt,nn]

a(A007916(h)^(A000040(g_1)*...*A000040(g_k)-1)) = 1 + a(h) + a(g_1) + ... + a(g_k).

A064372 Additive function a(n) defined by the recursive formula a(1)=1 and a(p^k)=a(k) for any prime p.

Original entry on oeis.org

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

Offset: 1

Steven Finch, Sep 26 2001

That is, if i, j, k, ... are relatively prime, then a(i*j*k*...) = a(i) + a(j) + a(k) + ... - N. J. A. Sloane, Nov 20 2007
Starts almost the same as A001221 (the number of distinct primes dividing n): the first twelve terms which are different are a(1), a(64), a(192), a(320), a(448), a(576), a(704), a(729), a(832), a(960), a(1024) and a(1088), since the first non-unitary values of n are a(6) and(10). - Henry Bottomley, Sep 23 2002
a(A164336(n)) = 1. - Reinhard Zumkeller, Aug 27 2011

			a(30) = a(5^1 * 3^1 * 2^1) = a(1) + a(1) + a(1) = 3.

Cf. A001221, A079553, A106444, A106490, A106491, A106495, A124010, A164336.

a064372 1 = 1
a064372 n = sum $ map a064372 $ a124010_row n
-- Reinhard Zumkeller, Aug 27 2011

a:= proc(n) option remember; `if`(n=1, 1,
      add(a(i[2]), i=ifactors(n)[2]))
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Aug 23 2020

a[1] = 1; a[n_] := a[n] = Plus @@ a /@ FactorInteger[n][[All, 2]]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Sep 19 2012 *)

a(n) = A106491(n) - A106490(n) = A106495(A106444(n)). - Antti Karttunen, May 09 2005

a(1) = 1, a(n) = Sum_{k=1..A001221(n)} a(A124010(n,k)) for n > 1. - Reinhard Zumkeller, Aug 27 2011

A075167 Number of edges in each rooted plane tree produced with the unranking algorithm presented in A075166, which is based on prime factorization.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 5, 4, 6, 5, 4, 3, 7, 4, 8, 5, 5, 6, 9, 4, 4, 7, 4, 6, 10, 5, 11, 4, 6, 8, 5, 5, 12, 9, 7, 5, 13, 6, 14, 7, 5, 10, 15, 5, 5, 5, 8, 8, 16, 5, 6, 6, 9, 11, 17, 6, 18, 12, 6, 4, 7, 7, 19, 9, 10, 6, 20, 5, 21, 13, 5, 10, 6, 8, 22, 6, 4, 14, 23, 7, 8, 15, 11, 7, 24, 6, 7, 11

Offset: 1

Antti Karttunen, Sep 13 2002

nonn

Each n occurs A000108(n) times in total.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Permutation of A072643 and A106457.
A253782 gives the positions where this sequence differs from A252464 (first time at n=16).
Cf. A000108, A000120, A029837, A055642, A051119, A061395, A071178, A075165, A075166, A106442, A253783.
Cf. also A106490.

a(n) = A106457(A106442(n)). - Antti Karttunen, May 09 2005
From Antti Karttunen, Jan 16 2015: (Start)
a(1) = 0; for n>1: a(n) = a(A071178(n)) + (A061395(n) - A061395(A051119(n))) + A253783(A051119(n)).
Other identities.
For all n >= 2, a(n) = A055642(A075166(n))/2. [Half of the number of decimal digits in A075166(n).]
For all n >= 2, a(n) = A029837(1+A075165(n))/2. [Half of the binary width of A075165(n).]
For all n >= 1, a(n) = A000120(A075165(n)). [Thus also the binary weight of A075165(n), because half of the bits are zeros.]
(End)

More terms from Antti Karttunen, May 09 2005

A106444 Exponent-recursed cross-domain bijection from N to GF(2)[X]. Variant of A091202 and A106442.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 11, 8, 5, 14, 13, 12, 19, 22, 9, 16, 25, 10, 31, 28, 29, 26, 37, 24, 21, 38, 15, 44, 41, 18, 47, 128, 23, 50, 49, 20, 55, 62, 53, 56, 59, 58, 61, 52, 27, 74, 67, 48, 69, 42, 43, 76, 73, 30, 35, 88, 33, 82, 87, 36, 91, 94, 39, 64, 121, 46, 97, 100, 111

Offset: 0

Antti Karttunen, May 09 2005

nonn

This map from the multiplicative domain of N to that of GF(2)[X] preserves 'superfactorized' structures, e.g. A106490(n) = A106493(a(n)), A106491(n) = A106494(a(n)), A064372(n) = A106495(a(n)). Shares with A091202 and A106442 the property that maps A000040(n) to A014580(n). Differs from A091202 for the first time at n=32, where A091202(32)=32, while a(32)=128. Differs from A106442 for the first time at n=48, where A106442(48)=192, while a(48)=48. Differs from A106446 for the first time at n=11, where A106446(11)=25, while a(11)=13.

			a(5) = 7, as 5 is the 3rd prime and the third irreducible GF(2)[X] polynomial x^2+x+1 is encoded as A014580(3) = 7. a(32) = a(2^5) = A048723(A014580(1),a(5)) = A048723(2,7) = 128. a(48) = a(3 * 2^4) = 3 X A048723(2,a(4)) = 3 X A048723(2,4) = 3 X 16 = 48.

Inverse: A106445.

a(0)=0, a(1)=1, a(p_i) = A014580(i) for primes p_i with index i and for composites n = p_i^e_i * p_j^e_j * p_k^e_k * ..., a(n) = A048723(a(p_i), a(e_i)) X A048723(a(p_j), a(e_j)) X A048723(a(p_k), a(e_k)) X ..., where X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and A048723(n, y) raises the n-th GF(2)[X] polynomial to the y:th power.

A106445 Exponent-recursed cross-domain bijection from GF(2)[X] to N. Variant of A091203 and A106443.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 6, 5, 8, 15, 18, 7, 12, 11, 10, 27, 16, 81, 30, 13, 36, 25, 14, 33, 24, 17, 22, 45, 20, 21, 54, 19, 512, 57, 162, 55, 60, 23, 26, 63, 72, 29, 50, 51, 28, 135, 66, 31, 48, 35, 34, 19683, 44, 39, 90, 37, 40, 99, 42, 41, 108, 43, 38, 75, 64, 225, 114, 47

Offset: 0

Antti Karttunen, May 09 2005

nonn

This map from the multiplicative domain of GF(2)[X] to that of N preserves 'superfactorized' structures, e.g. A106493(n) = A106490(a(n)), A106494(n) = A106491(a(n)), A106495(n) = A064372(a(n)). Shares with A091203 and A106443 the property that maps A014580(n) to A000040(n). Differs from the plain variant A091203 for the first time at n=32, where A091203(32)=32, while a(32)=512. Differs from the variant A106443 for the first time at n=48, where A106443(48)=768, while a(48)=48. Differs from a yet deeper variant A106447 for the first time at n=13, where A106447(13)=23, while a(13)=11.

			a(5) = 9, as 5 encodes the GF(2)[X] polynomial x^2+1, which is the square of the second irreducible GF(2)[X] polynomial x+1 (encoded as 3) and the square of the second prime is 3^2=9. a(32) = a(A048723(2,5)) = 2^a(5) = 2^9 = 512. a(48) = a(3 X A048723(2,4)) = 3 * 2^a(4) = 3 * 2^4 = 3 * 16 = 48.

Inverse: A106444.

a(0)=0, a(1)=1. For irreducible GF(2)[X] polynomials ir_i with index i (i.e. A014580(i)), a(ir_i) = A000040(i) and for composite polynomials n = A048723(ir_i, e_i) X A048723(ir_j, e_j) X A048723(ir_k, e_k) X ..., a(n) = a(ir_i)^a(e_i) * a(ir_j)^a(e_j) * a(ir_k)^a(e_k) * ... = A000040(i)^a(e_i) * A000040(j)^a(e_j) * A000040(k)^a(e_k), where X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and A048723(n, y) raises the n-th GF(2)[X] polynomial to the y:th power, while * is the ordinary multiplication and ^ is the ordinary exponentiation.

A106491 Total number of bases and exponents in Quetian Superfactorization of n, including the unity-exponents at the tips of branches.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 5, 2, 4, 4, 4, 2, 5, 2, 5, 4, 4, 2, 5, 3, 4, 3, 5, 2, 6, 2, 3, 4, 4, 4, 6, 2, 4, 4, 5, 2, 6, 2, 5, 5, 4, 2, 6, 3, 5, 4, 5, 2, 5, 4, 5, 4, 4, 2, 7, 2, 4, 5, 5, 4, 6, 2, 5, 4, 6, 2, 6, 2, 4, 5, 5, 4, 6, 2, 6, 4, 4, 2, 7, 4, 4, 4, 5, 2, 7, 4, 5, 4, 4, 4, 5, 2, 5, 5, 6, 2, 6

Offset: 1

Antti Karttunen, May 09 2005 based on Leroy Quet's message ('Super-Factoring' An Integer) posted to SeqFan-mailing list on Dec 06 2003

nonn

			a(64) = 5, as 64 = 2^6 = 2^(2^1*3^1) and there are 5 nodes in that superfactorization. Similarly, for 360 = 2^(3^1) * 3^(2^1) * 5^1 we get a(360) = 8. See comments at A106490.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
A. Karttunen, Scheme-program for computing this sequence.
Index entries for sequences computed from exponents in factorization of n

Cf. A000961, A028234, A064372, A067029, A106444, A106490, A106492, A106494.

a:= proc(n) option remember; `if`(n=1, 1,
      add(1+a(i[2]), i=ifactors(n)[2]))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 07 2014

a[n_] := a[n] = If[n == 1, 1, Sum[1 + a[i[[2]]], {i, FactorInteger[n]}]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

A067029(n) = if(n<2, 0, factor(n)[1,2]);
A028234(n) = my(f = factor(n)); if (#f~, f[1, 1] = 1); factorback(f); /* after Michel Marcus */
a(n) = if(n<2, 1, if(A028234(n)==1, 1 + a(A067029(n)), 1 + a(A067029(n)) + a(A028234(n))));
for(n=1, 150, print1(a(n),", ")) \\ Indranil Ghosh, Mar 23 2017, after formula by Antti Karttunen

From Antti Karttunen, Mar 23 2017: (Start)
a(1) = 1, and for n > 1, if A028234(n) = 1, a(n) = 1 + a(A067029(n)), otherwise a(n) = 1 + a(A067029(n)) + a(A028234(n)).
If n is a prime power p^k (a term of A000961), a(n) = 1 + a(k).
(End)
Other identities. For all n >= 1:
a(n) = A106490(n) + A064372(n).
a(n) = A106494(A106444(n)).

A106493 Total number of bases and exponents in GF(2)[X] Superfactorization of n, excluding the unity-exponents at the tips of branches.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 09 2005

nonn

GF(2)[X] Superfactorization proceeds in a manner analogous to normal superfactorization explained in A106490, but using factorization in domain GF(2)[X], instead of normal integer factorization in N.

			a(64) = 3, as 64 = A048723(2,6) = A048723(2,(A048723(2,1) X A048723(3,1))) and there are three non-1 nodes in that superfactorization. Similarly, for 27 = 5x7 = A048723(3,2) X A048273(7,1) we get a(27) = 3. The operation X stands for GF(2)[X] multiplication defined in A048720, while A048723(n,y) raises the n-th GF(2)[X] polynomial to the y:th power.

A. Karttunen, Scheme-program for computing this sequence.

a(n) = A106490(A106445(n)). a(n) = A106494(n)-A106495(n).

A106492 Total sum of bases and exponents in Quetian Superfactorization of n, excluding the unity-exponents at the tips of branches.

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 5, 5, 7, 11, 7, 13, 9, 8, 6, 17, 7, 19, 9, 10, 13, 23, 8, 7, 15, 6, 11, 29, 10, 31, 7, 14, 19, 12, 9, 37, 21, 16, 10, 41, 12, 43, 15, 10, 25, 47, 9, 9, 9, 20, 17, 53, 8, 16, 12, 22, 31, 59, 12, 61, 33, 12, 7, 18, 16, 67, 21, 26, 14, 71, 10, 73, 39, 10, 23, 18

Offset: 1

Antti Karttunen, May 09 2005 based on Leroy Quet's message ('Super-Factoring' An Integer) posted to SeqFan-mailing list on Dec 06 2003

nonn

			a(64) = 7, as 64 = 2^6 = 2^(2^1*3^1) and 2+2+3=7. Similarly, for 360 = 2^(3^1) * 3^(2^1) * 5^1 we get a(360) = 2+3+3+2+5 = 15. See comments at A106490.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
A. Karttunen, Scheme-program for computing this sequence.

Cf. A106490-A106491.

a:= proc(n) option remember;
      add(i[1]+a(i[2]), i=ifactors(n)[2])
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 06 2014

a[1] = 0; a[n_] := a[n] = #[[1]] + a[#[[2]]]& /@ FactorInteger[n] // Total; Array[a, 100] (* Jean-François Alcover, Mar 03 2016 *)

Additive with a(p^e) = p + a(e).

cp's OEIS Frontend

A276230 a(n) is the A106490 index where n first occurs.

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A277562 Numbers of the form c(x_1)^c(x_2)^...^c(x_k) where each c(i) = A007916(i) is a non-perfect-power, k >= 2, and the exponents are nested from the right.

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Mathematica

Extensions

A279944 Number of positions in the free pure symmetric multifunction in one symbol with j-number n.

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Mathematica

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A064372 Additive function a(n) defined by the recursive formula a(1)=1 and a(p^k)=a(k) for any prime p.

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Haskell

Maple

Mathematica

Formula

A075167 Number of edges in each rooted plane tree produced with the unranking algorithm presented in A075166, which is based on prime factorization.

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Extensions

A106444 Exponent-recursed cross-domain bijection from N to GF(2)[X]. Variant of A091202 and A106442.

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A106445 Exponent-recursed cross-domain bijection from GF(2)[X] to N. Variant of A091203 and A106443.

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A106491 Total number of bases and exponents in Quetian Superfactorization of n, including the unity-exponents at the tips of branches.

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Maple