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A133468 A080814 complemented, then interpreted as binary and then re-expressed in decimal form (e.g., "1221" = "0110"). Alternately, view as A080814 with "1" mapped to "1" and "2" mapped to "0".

1, 2, 9, 150, 38505, 2523490710, 10838310072981296745, 199931532107794273605284333428918544790, 68033174967769840440887906939858451149105560803546820641877549596291376780905

Offset: 0

Dan Reif (integer-sequences(AT)angelfaq.com), Nov 28 2007, corrected Nov 30 2007

It appears that a(n) is the least positive number with binary expansion Sum_{k = 0..w} b_k * 2^k such that the polynomial Sum_{k = 0..w} (X+k)^n * (-1)^b_k is constant. - Rémy Sigrist, Sep 15 2020

Cf. A080814, A080815, A048707.

FromDigits[#, 2] & /@ NestList[ Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {1}, 9] (* Robert G. Wilson v, Aug 16 2011 *)

function invert(string s) returns string { s.replace("0","2"); s.replace("1","0"); s.replace("2","1"); }
function f(int n) returns string { if (n==0) return "1"; return concat(f(n-1),invert(f(n-1))); } // Blatant opportunity for optimization
function a(int n) returns int { return f(n).InterpretAsBinary(); }

The "~" operator, as used here, represents bitwise complement. a(0) = 1. a(n) = a(n-1) followed by ~a(n-1).

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

Original entry on oeis.org

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).

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)).

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).

A080341 Sum of the first n terms that are congruent to 1, 4 or 5 mod 6 (A047259).

Original entry on oeis.org

1, 5, 10, 17, 27, 38, 51, 67, 84, 103, 125, 148, 173, 201, 230, 261, 295, 330, 367, 407, 448, 491, 537, 584, 633, 685, 738, 793, 851, 910, 971, 1035, 1100, 1167, 1237, 1308, 1381, 1457, 1534, 1613, 1695, 1778, 1863, 1951, 2040, 2131, 2225, 2320, 2417

Offset: 1

Christian Mercat (Integer.Sequence(AT)entrelacs.net), Mar 20 2003

Number of edges needed in a sector of a hexagon of size n paved by rhombi coming from triangular/hexagonal lattices.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A047259.

Accumulate[Select[Range[100],MemberQ[{1,4,5},Mod[#,6]]&]] (* Harvey P. Dale, Aug 16 2012 *)

a(n) = n^2+(n+1)/3 with integer division, that is n mod 3 = 0 : n^2+n/3 n mod 3 = 1 : n^2+(n-1)/3 n mod 3 = 2 : n^2+(n+1)/3.

G.f.: x*(1+3*x+x^2+x^3)/(1-x)^3/(1+x+x^2). [Colin Barker, Feb 12 2012]

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A133468 A080814 complemented, then interpreted as binary and then re-expressed in decimal form (e.g., "1221" = "0110"). Alternately, view as A080814 with "1" mapped to "1" and "2" mapped to "0".

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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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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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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A080341 Sum of the first n terms that are congruent to 1, 4 or 5 mod 6 (A047259).

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Programs

Mathematica

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