A280726 -id:A280726 - cp's OEIS frontend

A280799 a(n) = A049502(phi(n)).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 3, 0, 3, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3, 2, 0, 0, 4, 0, 6, 3, 0, 3, 4, 0, 6, 3, 0, 0, 3, 2, 4, 0, 3, 0, 2, 0, 0, 0, 3, 0, 0, 3, 2, 0, 4, 0, 3, 0, 4, 3, 4, 3, 0, 0, 4, 0, 3, 4, 7, 0, 0, 6, 0, 4, 5, 0, 4, 4, 0, 4, 4, 0, 0, 6, 0, 4, 3, 0, 3, 0, 0, 3, 6, 3, 4

Offset: 1

Indranil Ghosh, Jan 08 2017

nonn

a(n) = Major index (2nd definition) of the total numbers <=n and prime to n i.e., phi(n).

			For n = 11, phi(n) = 10 and major index (2nd definition) of 10 = 2. So a(n) = 2.

Indranil Ghosh, Table of n, a(n) for n = 1..10000
Indranil Ghosh, Python Program to generate the sequence

Cf. A000010, A049502, A279519, A280062, A280726.

a(n) = A049502(A000010(n)).

A280800 a(n) = A049501(cototient(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 1, 2, 0, 1, 1, 3, 1, 1, 0, 5, 0, 1, 2, 5, 1, 2, 0, 4, 0, 2, 0, 4, 0, 2, 4, 2, 0, 1, 0, 4, 1, 3, 0, 5, 0, 1, 4, 4, 0, 5, 0, 1, 2, 1, 1, 6, 0, 5, 2, 6, 0, 2, 0, 6, 1, 4, 1, 7, 0, 2, 2, 9, 0, 4, 4, 5, 0, 2, 0, 7, 1, 2, 1, 2, 1, 1, 0, 3, 1, 4, 0, 7, 0, 3, 3, 7, 0, 5, 0

Offset: 1

Indranil Ghosh, Jan 08 2017

nonn

a(n) = major index (1st definition) of cototient(n), where cototient(n) = n - phi(n).

			For n = 10, A051953(n) = 6 and major index (1st definition) of 6 is 2. So, a(10) = 2.

Indranil Ghosh, Table of n, a(n) for n = 1..10000
Indranil Ghosh, Python Program to generate the sequence

Cf. A049501, A051953, A280726, A280799.

a(n) = A049501(A051953(n)).

A284725 a(n) = (1/3) * smallest multiple of 3 missing from [A280864(1), ..., A280864(n-1)].

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 23, 23, 23, 27, 27, 27, 27, 27, 27, 27, 27

Offset: 1

N. J. A. Sloane, Apr 06 2017

nonn

For k >= 1, n >= 1, let B_k(n) = smallest multiple of k missing from [A280864(1), ..., A280864(n-1)]. Sequence gives values of B_3(n)/3.

The analogous sequences B_k(n) for the EKG sequence A064413 were important for the analysis of that sequence, so they may also be useful for studying A280864.

			The initial terms of A280864 are 1,2,4,3,6,8,... The smallest missing multiple of 3 in [1,2,4,3,6] is 9, so a(6) = 9/3 = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, arXiv:math/0204011 [math.NT], 2002.
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG Sequence, Exper. Math. 11 (2002), 437-446.

Cf. A280864, A064413, A284724, A280726.

mex := proc(L)
local k;
for k from 1 do
if not k in L then
return k;
end if;
end do:
end proc:
read b280864;
k:=3; a:=[1,1]; ML:=[]; B:=1;
for n from 2 to 120 do
t:=b280864[n];
if (t mod k) = 0 then
ML:=[op(ML),t/k];
B:=mex(ML);
a:=[op(a),B];
else
a:=[op(a),B];
fi;
od:
a;

terms = 85; rad[n_] := Times @@ FactorInteger[n][[All, 1]]; A280864 = Reap[present = 0; p = 1; pp = 1; Do[forbidden = GCD[p, pp]; mandatory = p/forbidden; a = mandatory; While[BitGet[present, a] > 0 || GCD[forbidden, a] > 1, a += mandatory]; Sow[a]; present += 2^a; pp = p; p = rad[a], terms]][[2, 1]];
Clear[a]; a[1] = 1; a[n_] := a[n] = For[b = 3 a[n - 1], True, b += 3, If[FreeQ[A280864[[1 ;; n - 1]], b], Return[b/3]]];
Array[a, terms] (* Jean-François Alcover, Nov 26 2017, after Rémy Sigrist's program for A280864 *)

A280815 a(n) = A049502(cototient(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 3, 0, 0, 1, 2, 2, 0, 0, 3, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 3, 0, 0, 4, 0, 0, 4, 0, 0, 2, 0, 1, 4, 0, 3, 1, 4, 0, 0, 0, 3, 2, 4, 1, 3, 0, 0, 2, 6, 0, 0, 4, 4, 0, 0, 0, 2, 2, 0, 1, 0, 3, 0, 0, 0, 3, 0, 0, 3, 0, 0, 1, 3, 0, 4, 0

Offset: 1

Indranil Ghosh, Jan 08 2017

nonn

a(n) = major index (2nd definition) of cototient(n), where cototient(n) = n - phi(n).

			For n = 30, cototient(n) = 22 and major index (2nd definition) of 22 is 3. So, a(n) = 3.

Indranil Ghosh, Table of n, a(n) for n = 1..10000
Indranil Ghosh, Python Program to generate the sequence

Cf. A049502, A051953, A280726, A280799, A280800.

a(n) = A049502(A051953(n)).

A280817 a(n) = A049501(sigma(n)).

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 1, 0, 2, 5, 2, 3, 3, 2, 2, 0, 5, 1, 4, 9, 1, 5, 2, 4, 0, 9, 4, 3, 4, 5, 1, 0, 2, 7, 2, 5, 6, 4, 3, 11, 9, 2, 5, 9, 7, 5, 2, 5, 3, 6, 5, 8, 7, 4, 5, 4, 4, 11, 4, 9, 5, 2, 6, 0, 9, 5, 6, 6, 2, 5, 5, 2, 11, 9, 5, 7, 2, 9, 4, 13, 4, 6, 9, 3, 7, 7, 4, 11, 11, 15, 3, 9, 1, 5, 4, 6, 8, 9, 7, 7, 8, 7, 6, 13, 2, 11, 7, 7, 8

Offset: 1

Indranil Ghosh, Jan 08 2017

nonn

a(n) = major index (1st definition) of the sum of the divisors of n, i.e., sigma(n).

			For n = 10, sigma(n) = 18 and major index (1st definition) of 18 is 5. So, a(10) = 5.

Indranil Ghosh, Table of n, a(n) for n = 1..10000
Indranil Ghosh, Python Program to generate the sequence

Cf. A049501, A000203, A280726, A280800.

Table[Total@ Flatten@ Position[Partition[IntegerDigits[DivisorSigma[1, n], 2], 2, 1], {1, 0}], {n, 120}] (* Michael De Vlieger, Jan 10 2017, after Harvey P. Dale at A049501 *)

a(n) = A049501(A000203(n)).

cp's OEIS Frontend

A280799 a(n) = A049502(phi(n)).

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A280800 a(n) = A049501(cototient(n)).

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A284725 a(n) = (1/3) * smallest multiple of 3 missing from [A280864(1), ..., A280864(n-1)].

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A280815 a(n) = A049502(cototient(n)).

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A280817 a(n) = A049501(sigma(n)).

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