A106491 -id:A106491 - cp's OEIS frontend

A106490 Total number of bases and exponents in Quetian Superfactorization of n, excluding the unity-exponents at the tips of branches.

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

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

Quetian Superfactorization proceeds by factoring a natural number to its unique prime-exponent factorization (p1^e1 * p2^e2 * ... pj^ej) and then factoring recursively each of the (nonzero) exponents in similar manner, until unity-exponents are finally encountered.

			a(64) = 3, as 64 = 2^6 = 2^(2^1*3^1) and there are three non-1 nodes in that superfactorization. Similarly, for 360 = 2^(3^1) * 3^(2^1) * 5^1 we get a(360) = 5. a(65536) = a(2^(2^(2^(2^1)))) = 4.

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. A028234, A064372, A067029, A106444, A106491, A106492, A106493.
Cf. A276230 (gives first k such that a(k) = n, i.e., this sequence is a left inverse of A276230).
After n=1 differs from A038548 for the first time at n=24, where A038548(24)=4, while a(24)=3.

a:= proc(n) option remember; `if`(n=1, 0,
      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, 0, 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, 0, 1 + a(A067029(n)) + a(A028234(n)));
for(n=1, 150, print1(a(n),", ")) \\ Indranil Ghosh, Mar 23 2017, after formula by Antti Karttunen

Additive with a(p^e) = 1 + a(e).
a(1) = 0; for n > 1, a(n) = 1 + a(A067029(n)) + a(A028234(n)). - Antti Karttunen, Mar 23 2017
Other identities. For all n >= 1:
a(A276230(n)) = n.
a(n) = A106493(A106444(n)).
a(n) = A106491(n) - A064372(n).

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

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.

A106494 Total number of bases and exponents in GF(2)[X] Superfactorization of n, including the unity-exponents at the tips of branches.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 09 2005

nonn

See comments at A106493.

			a(64) = 5, as 64 = A048723(2,6) = A048723(2,(A048723(2,1) X A048723(3,1))) and there are five nodes in that superfactorization. Similarly, for 27 = 5x7 = A048723(3, A048723(2,1)) X A048273(7,1) we get a(27) = 5. 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) = A106491(A106445(n)). a(n) = A106493(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).

A106391 A "cosh transform" of binomial(2n,n-1).

Original entry on oeis.org

0, 1, 4, 18, 80, 365, 1692, 7945, 37648, 179595, 861020, 4143832, 20004096, 96810779, 469508340, 2281123530, 11100465216, 54093131147, 263929559436, 1289217255934, 6303934406640, 30853639964847, 151139139048084

Offset: 0

Paul Barry, May 01 2005

Mean of binomial and inverse binomial transform of A001791.

A106491 := proc(n)
    add(binomial(n, 2*k)*binomial(2*(n-2*k), n-2*k+1),k=0..floor(n/2)) ;
end proc: # R. J. Mathar, Feb 20 2015

E.g.f.: cosh(x)exp(2x)I_1(2x), where I_1 is Bessel function; a(n)=sum{k=0..floor(n/2), binomial(n, 2k)binomial(2(n-2k), n-2k+1)}.
Conjecture: -(n+1)*(n-2)*a(n) +4*n*(3*n-7)*a(n-1) +(-49*n^2+193*n-148)*a(n-2) +8*(9*n-19)*(n-4)*a(n-3) +(5*n^2+115*n-418)*a(n-4) -12*(7*n-19)*(n-4)*a(n-5) +45*(n-4)*(n-5)*a(n-6)=0. - R. J. Mathar, Feb 20 2015
Conjecture: -(n-2)*(n+1)*(4*n^2-16*n+19)*a(n) +8*n*(n-2)*(4*n^2-14*n+13)*a(n-1) +2*(-28*n^4+168*n^3-375*n^2+401*n-180)*a(n-2) -8*(n-2)*(4*n^3-18*n^2+27*n-10)*a(n-3) +15*(n-2)*(n-3)*(4*n^2-8*n+7)*a(n-4)=0. - R. J. Mathar, Feb 20 2015

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A106490 Total number of bases and exponents in Quetian Superfactorization of n, excluding the unity-exponents at the tips of branches.

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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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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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A106494 Total number of bases and exponents in GF(2)[X] Superfactorization of n, including the unity-exponents at the tips of branches.

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

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Author

Keywords

Examples

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Crossrefs

Programs

Maple

Mathematica

Formula

A106391 A "cosh transform" of binomial(2n,n-1).

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