A250469 -id:A250469 - cp's OEIS frontend

A280703 a(n) = A003961(n) / A280702(n) = A003961(n) / gcd(A003961(n),A250469(n)).

1, 1, 1, 1, 1, 1, 1, 9, 1, 7, 1, 15, 1, 11, 1, 9, 1, 25, 1, 21, 1, 13, 1, 45, 1, 17, 25, 11, 1, 35, 1, 81, 13, 19, 1, 15, 1, 23, 17, 21, 1, 55, 1, 39, 35, 29, 1, 135, 1, 1, 19, 1, 1, 125, 1, 9, 23, 31, 1, 105, 1, 37, 55, 27, 1, 1, 1, 57, 29, 77, 1, 225, 1, 41, 49, 23, 1, 85, 1, 189, 125, 43, 1, 165, 1, 47, 31, 39, 1, 175, 1, 87, 37

Offset: 1

Antti Karttunen, Mar 08 2017

nonn

If there are no such n that A250469(n) = k*A003961(n) for some integer k > 1, then A280693 gives the positions of ones in this sequence. Cf. also comment in A280704.

Antti Karttunen, Table of n, a(n) for n = 1..13330

Cf. A003961, A250469, A280702, A280704, A280693.

f[n_] := f[n] = Which[n == 1, 1, PrimeQ@ n, NextPrime@ n, True, Times @@ Replace[FactorInteger[n], {p_, e_} :> f[p]^e, 1]]; g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[ Function[p, p/GCD[Lookup[s, g@ First@ #2 + 1][[#1]] - Boole[First@ #2 == 1], p]]@ f@ First@ #2 &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 08 2017, Version 10 *)

(define (A280703 n) (/ (A003961 n) (A280702 n)))

a(n) = A003961(n) / A280702(n) = A003961(n) / gcd(A003961(n),A250469(n)).

A280704 a(n) = A250469(n) / A280702(n) = A250469(n) / gcd(A003961(n),A250469(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 7, 1, 9, 1, 11, 1, 13, 1, 5, 1, 17, 1, 19, 1, 21, 1, 23, 1, 25, 13, 9, 1, 29, 1, 31, 17, 33, 1, 7, 1, 37, 19, 13, 1, 41, 1, 43, 23, 45, 1, 47, 1, 1, 25, 1, 1, 53, 1, 5, 29, 57, 1, 59, 1, 61, 31, 7, 1, 1, 1, 67, 35, 69, 1, 71, 1, 73, 37, 25, 1, 77, 1, 79, 41, 81, 1, 83, 1, 85, 43, 29, 1, 89, 1, 91, 47, 93, 1, 19, 1, 97, 49, 33, 1

Offset: 1

Antti Karttunen, Mar 08 2017

nonn

Note: a(352) = 1 even though A280703(352) = 3 as A003961(352) = 3159 = 3^5 * 13, while A250469(352) = 1053 = 3^4 * 13. (Thus also A266645(352) = 176 = 352/2.) Question: Are there more n for which A003961(n) = k*A250469(n) for some integer k > 1 ? Cf. also comments in A280703.

Antti Karttunen, Table of n, a(n) for n = 1..13098

Cf. A003961, A250469, A266645, A280701, A280702, A280703.

f[n_] := f[n] = Which[n == 1, 1, PrimeQ@ n, NextPrime@ n, True, Times @@ Replace[FactorInteger[n], {p_, e_} :> f[p]^e, 1]]; g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[Function[t, t/GCD[t, f@ First@ #2]][Lookup[s, g[First@ #2] + 1][[#1]] - Boole[First@ #2 == 1]] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 08 2017, Version 10 *)

(define (A280704 n) (/ (A250469 n) (A280702 n)))

a(n) = A250469(n) / A280702(n) = A250469(n) / gcd(A003961(n),A250469(n)).

A280701(n) = n - a(n).

A283465 a(n) = A046523(A250469(n)).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 6, 4, 8, 2, 6, 2, 6, 6, 12, 2, 6, 2, 6, 6, 12, 2, 6, 4, 12, 6, 16, 2, 6, 2, 6, 6, 12, 6, 30, 2, 6, 6, 12, 2, 6, 2, 6, 6, 24, 2, 6, 4, 12, 8, 12, 2, 6, 6, 30, 6, 12, 2, 6, 2, 6, 6, 24, 6, 30, 2, 6, 12, 12, 2, 6, 2, 6, 6, 36, 6, 30, 2, 6, 6, 32, 2, 6, 6, 30, 6, 12, 2, 6, 6, 30, 6, 12, 6, 30, 2, 6, 12, 24, 2, 6, 2, 6, 6, 60

Offset: 1

Antti Karttunen, Mar 08 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..9420

Cf. A046523, A250469, A266645, A283463, A283466.

(define (A283465 n) (A046523 (A250469 n)))

a(n) = A046523(A250469(n)) = A046523(A266645(n)).

A346476 a(n) = 2*n - A250469(n).

Original entry on oeis.org

1, 1, 1, -1, 3, -3, 3, -5, -7, -7, 9, -9, 9, -11, -5, -13, 15, -15, 15, -17, -13, -19, 17, -21, 1, -23, -11, -25, 27, -27, 25, -29, -19, -31, -7, -33, 33, -35, -17, -37, 39, -39, 39, -41, -25, -43, 41, -45, -23, -47, -23, -49, 47, -51, 19, -53, -31, -55, 57, -57, 55, -59, -29, -61, 11, -63, 63, -65, -37

Offset: 1

Antti Karttunen, Jul 29 2021

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves

Cf. A000040, A033879, A062234, A250469, A252748, A280692, A346473, A346477 (Dirichlet inverse), A346478, A347378.

PARI
```
A346476(n) = (n+n-A250469(n));
```

a(n) = A280692(n) - A252748(n).
a(n) = A033879(n) - A346473(n).
a(n) = A346478(n) - A346477(n).
a(n) = n - A347378(n).
a(A000040(n)) = -A252748(A000040(n)) = -A346477(A000040(n)) = A062234(n).

A346480 Sum of A250469 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 9, 0, 30, 0, 27, 25, 42, 0, 45, 0, 66, 70, 45, 0, 75, 0, 99, 110, 78, 0, 3, 49, 102, 125, 135, 0, 60, 0, 81, 130, 114, 154, -39, 0, 138, 170, 15, 0, 60, 0, 261, 175, 174, 0, 117, 121, 231, 190, 297, 0, -225, 182, 3, 230, 186, 0, -381, 0, 222, 275, 189, 238, 360, 0, 423, 290, 216, 0, 381, 0, 246, 245, 459

Offset: 1

Antti Karttunen, Jul 30 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves

Cf. A250469, A346479.

Cf. also A346478.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA250469(n)));
A346479(n) = v346479[n];
A346480(n) = (A250469(n)+A346479(n));

A346480(n) = if(1==n, 2, -sumdiv(n,d,if((1==d)||n==d,0,A250469(d)*A346479(n/d)))); \\ (Demonstrates the convolution formula).

a(n) = A250469(n) + A346479(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A250469(d) * A346479(n/d).

A347376 Möbius transform of A250469.

Original entry on oeis.org

1, 2, 4, 6, 6, 8, 10, 12, 20, 18, 12, 12, 16, 26, 24, 24, 18, 16, 22, 24, 40, 48, 28, 24, 42, 56, 40, 36, 30, 24, 36, 48, 68, 78, 60, 36, 40, 86, 74, 48, 42, 32, 46, 60, 60, 104, 52, 48, 110, 78, 102, 72, 58, 68, 72, 72, 118, 138, 60, 48, 66, 144, 80, 96, 96, 52, 70, 96, 142, 84, 72, 72, 78, 176, 108, 108, 120, 70

Offset: 1

Antti Karttunen, Sep 01 2021

nonn

Question: Are all terms positive?

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves

Cf. A003972, A008683, A020639, A078898, A250469, A347377.

Cf. also A346479.

up_to = 10000;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A250469(n) = if(1==n,n,my(spn = nextprime(1+A020639(n)), c = A078898(n), k = 0); while(c, k++; if((1==k)||(A020639(k)>=spn),c -= 1)); (k*spn));
A347376(n) = sumdiv(n,d,moebius(n/d)*A250469(d));

a(n) = Sum_{d|n} A008683(n/d) * A250469(d).

a(n) = A003972(n) - A347377(n).

A269851 a(0) = 1, a(A087686(1+n)) = 2*a(n), a(A088359(n)) = A250469(a(n)), where A088359 and A087686 = numbers that occur only once (resp. more than once) in A004001.

Original entry on oeis.org

1, 2, 4, 3, 8, 9, 5, 6, 16, 21, 25, 7, 18, 15, 10, 12, 32, 45, 55, 49, 11, 42, 51, 35, 50, 27, 14, 36, 33, 30, 20, 24, 64, 93, 115, 91, 121, 13, 90, 123, 125, 77, 110, 147, 65, 98, 39, 22, 84, 105, 85, 102, 87, 70, 100, 57, 54, 28, 72, 69, 66, 60, 40, 48, 128, 189, 235, 203, 187, 169, 17, 186, 267, 305, 217, 143, 230

Offset: 0

Antti Karttunen, Mar 07 2016

nonn

Permutation of natural numbers obtained from the sieve of Eratosthenes, combined with the permutation obtained from Hofstadter-Conway $10000 sequence (A004001). Note the indexing: Domain starts from 0, range from 1.

Inverse: A269852.
Cf. A004001, A087686, A088359, A093879, A250469.
Related or similar permutations: A252755, A267111, A269855.

a(0) = 1, a(1) = 2, for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = 2*a(n - A004001(n)), otherwise [when n is in A088359], a(n) = A250469(a(A004001(n)-1)).
As a composition of related permutations:
a(n) = A252755(A267111(n)).
Other identities. For all n >= 0:
a(2^n) = 2^(n+1).

A269855 a(0) = 1, a(1) = 2, after which, a(nth_odious_number_larger_than_one(n)) = A250469(a(n)), a(nth_evil_number_larger_than_zero(n)) = 2*a(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 7, 10, 12, 15, 16, 21, 25, 18, 11, 14, 20, 27, 24, 33, 35, 30, 32, 45, 55, 42, 49, 50, 36, 51, 13, 22, 28, 39, 40, 57, 65, 54, 48, 69, 85, 66, 77, 70, 60, 87, 64, 93, 115, 90, 91, 110, 84, 123, 121, 98, 100, 147, 72, 105, 125, 102, 17, 26, 44, 63, 56, 81, 95, 78, 80, 117, 145, 114, 119, 130, 108, 159, 96

Offset: 0

Antti Karttunen, Mar 07 2016

nonn

Permutation of natural numbers obtained from the sieve of Eratosthenes, combined with the inverse of Gray code. Note the indexing: Domain starts from 0, range from 1.

Inverse: A269856.
Cf. A000069, A001969, A010060, A115384, A245710, A250469.
Related or similar permutations: A006068, A252755, A269851.

a(0) = 1, a(1) = 2, for n > 1, if A010060(n) = 1 [when n is one of the odious numbers A000069], a(n) = A250469(a(A115384(n)-1)), otherwise [when n is one of the evil numbers A001969], a(n) = 2*a(A245710(n)).
As a composition of other permutations:
a(n) = A252755(A006068(n)).

A269857 Permutation of natural numbers: a(1) = 1; if n is an odd prime, a(n) = A250469(a(A026233(n))), else a(n) = 2*(a(A026233(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 10, 12, 7, 18, 15, 16, 20, 24, 25, 14, 21, 36, 30, 32, 27, 40, 48, 50, 28, 42, 33, 72, 11, 60, 64, 54, 80, 96, 51, 100, 56, 84, 35, 66, 45, 144, 22, 120, 57, 128, 108, 160, 192, 102, 69, 200, 112, 168, 70, 132, 49, 90, 39, 288, 44, 240, 114, 256, 55, 216, 320, 384, 105, 204, 87, 138, 400

Offset: 1

Antti Karttunen, Mar 06 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2662
Index entries for sequences that are permutations of the natural numbers

Inverse: A269858.
Cf. A005843, A026233, A010051, A250469.
Related or similar permutations: A071574, A252755, A269847.
Differs from A269847 for the first time at n=19, where a(19)=21, while A269847(19)=27.

a(1) = 1; if n is an odd prime, a(n) = A250469(a(A026233(n))), else a(n) = 2*(a(A026233(n))).
As a composition of other permutations:
a(n) = A252755(A071574(n)).

A279348 a(1) = 1, for n > 1, if A079559(n) = 0, a(n) = 2*a(A256992(n)), otherwise a(n) = A250469(a(A256992(n))).

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 7, 9, 10, 15, 11, 8, 12, 14, 25, 27, 18, 35, 13, 20, 30, 21, 33, 22, 39, 49, 16, 24, 28, 50, 65, 51, 54, 77, 17, 36, 70, 57, 87, 26, 55, 85, 40, 60, 42, 63, 95, 66, 121, 45, 44, 78, 69, 81, 98, 147, 119, 32, 48, 56, 100, 130, 125, 159, 102, 143, 19, 108, 154, 105, 207, 34, 145, 215, 72, 140, 114, 75, 91, 174, 133, 117, 52

Offset: 1

Antti Karttunen, Dec 12 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A279349.
Cf. A005187, A055938, A079559, A250469, A256992.
Related or similar permutations: A250245, A252753, A252755, A279338, A279341, A279343.

(definec (A279348 n) (cond ((= 1 n) n) ((zero? (A079559 n)) (* 2 (A279348 (A256992 n)))) (else (A250469 (A279348 (A256992 n))))))

a(1) = 1, for n > 1, if A079559(n) = 0 [when n is a term of A055938], a(n) = 2*a(A256992(n)), otherwise a(n) = A250469(a(A256992(n))).
As a composition of other permutations:
a(n) = A250245(A279338(n)).
a(n) = A252753(A279343(n)).
a(n) = A252755(A279341(n)).

cp's OEIS Frontend

A280703 a(n) = A003961(n) / A280702(n) = A003961(n) / gcd(A003961(n),A250469(n)).

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A280704 a(n) = A250469(n) / A280702(n) = A250469(n) / gcd(A003961(n),A250469(n)).

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A283465 a(n) = A046523(A250469(n)).

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A346476 a(n) = 2*n - A250469(n).

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A346480 Sum of A250469 and its Dirichlet inverse.

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A347376 Möbius transform of A250469.

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A269851 a(0) = 1, a(A087686(1+n)) = 2*a(n), a(A088359(n)) = A250469(a(n)), where A088359 and A087686 = numbers that occur only once (resp. more than once) in A004001.

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A269855 a(0) = 1, a(1) = 2, after which, a(nth_odious_number_larger_than_one(n)) = A250469(a(n)), a(nth_evil_number_larger_than_zero(n)) = 2*a(n).

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A269857 Permutation of natural numbers: a(1) = 1; if n is an odd prime, a(n) = A250469(a(A026233(n))), else a(n) = 2*(a(A026233(n))).

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