A280692 -id:A280692 - cp's OEIS frontend

A347377 Möbius transform of A280692, A003961(n) - A250469(n).

0, 0, 0, 0, 0, 0, 0, 6, 0, -6, 0, 12, 0, -6, 0, 30, 0, 24, 0, 12, 0, -24, 0, 48, 0, -24, 60, 24, 0, 24, 0, 114, -20, -42, 0, 84, 0, -42, -10, 60, 0, 48, 0, 12, 60, -48, 0, 168, 0, 6, -30, 24, 0, 132, 0, 108, -30, -78, 0, 96, 0, -72, 120, 390, 0, 44, 0, 12, -30, 36, 0, 288, 0, -96, 60, 24, 0, 58, 0, 228, 360, -114

Offset: 1

Antti Karttunen, Sep 01 2021

sign

Cf. A003961, A003972, A008683, A250469, A280692, A347376.

Cf. also A346480.

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A280692(n) = (A003961(n) - A250469(n));
A347377(n) = sumdiv(n,d,moebius(n/d)*A280692(d));

a(n) = Sum_{d|n} A008683(n/d) * A280692(d).

a(n) = A003972(n) - A347376(n).

A250469 a(1) = 1; and for n > 1, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)+1, where A055396(n) is the index of smallest prime dividing n.

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 11, 21, 25, 27, 13, 33, 17, 39, 35, 45, 19, 51, 23, 57, 55, 63, 29, 69, 49, 75, 65, 81, 31, 87, 37, 93, 85, 99, 77, 105, 41, 111, 95, 117, 43, 123, 47, 129, 115, 135, 53, 141, 121, 147, 125, 153, 59, 159, 91, 165, 145, 171, 61, 177, 67, 183, 155, 189, 119, 195, 71, 201, 175, 207, 73, 213, 79, 219, 185, 225, 143, 231, 83, 237, 205, 243, 89, 249, 133, 255

Offset: 1

Antti Karttunen, Dec 06 2014

nonn

Permutation of odd numbers.
For n >= 2, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)+1. In other words, a(n) tells which number is located immediately below n in the sieve of Eratosthenes (see A083140, A083221) in the same column of the sieve that contains n.
A250471(n) = (a(n)+1)/2 is a permutation of natural numbers.
Coincides with A003961 in all terms which are primes. - M. F. Hasler, Sep 17 2016. Note: primes are a proper subset of A280693 which gives all n such that a(n) = A003961(n). - Antti Karttunen, Mar 08 2017

Antti Karttunen, Table of n, a(n) for n = 1..5002

Cf. A000040, A003961, A016945, A046523, A055396, A078898, A083140, A083221, A084967, A249744, A249810, A249820, A249817, A249818, A250471, A266645, A280692, A280693, A283465.

Cf. A250470, A268674 (left inverses, the latter is simpler).

a[1] = 1; a[n_] := If[PrimeQ[n], NextPrime[n], m1 = p1 = FactorInteger[n][[ 1, 1]]; For[k1 = 1, m1 <= n, m1 += p1; If[m1 == n, Break[]]; If[ FactorInteger[m1][[1, 1]] == p1, k1++]]; m2 = p2 = NextPrime[p1]; For[k2 = 1, True, m2 += p2, If[FactorInteger[m2][[1, 1]] == p2, k2++]; If[k1+2 == k2, Return[m2]]]]; Array[a, 100] (* Jean-François Alcover, Mar 08 2016 *)
g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[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 *)

a(1) = 1, a(n) = A083221(A055396(n)+1, A078898(n)).
a(n) = A249817(A003961(A249818(n))).
Other identities. For all n >= 1:
A250470(a(n)) = A268674(a(n)) = n. [A250470 and A268674 provide left inverses for this function.]
a(2n) = A016945(n-1). [Maps even numbers to the numbers of form 6n+3, in monotone order.]
a(A016945(n-1)) = A084967(n). [Which themselves are mapped to the terms of A084967, etc. Cf. the Example section of A083140.]
a(A000040(n)) = A000040(n+1). [Each prime is mapped to the next prime.]
For all n >= 2, A055396(a(n)) = A055396(n)+1. [A more general rule.]
A046523(a(n)) = A283465(n). - Antti Karttunen, Mar 08 2017

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

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 11, 3, 25, 3, 13, 3, 17, 3, 35, 9, 19, 3, 23, 3, 55, 3, 29, 3, 49, 3, 5, 9, 31, 3, 37, 3, 5, 3, 77, 15, 41, 3, 5, 9, 43, 3, 47, 3, 5, 3, 53, 3, 121, 147, 5, 153, 59, 3, 91, 33, 5, 3, 61, 3, 67, 3, 5, 27, 119, 195, 71, 3, 5, 3, 73, 3, 79, 3, 5, 9, 143, 3, 83, 3, 5, 3, 89, 3, 133, 3, 5, 9, 97, 3, 187, 3, 5, 3, 161

Offset: 1

Antti Karttunen, Mar 08 2017

nonn

For n > 1, a(n) > 1 because A020639(A003961(n)) = A020639(A250469(n)) = A003961(A020639(n)).

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

Cf. A003961, A020639, A250469, A280497, A280498, A280692, A280703, A280704, A283463, A283464.

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[ GCD[ Lookup[s, g[First@ #2] + 1][[#1]] - Boole[First@ #2 == 1], f@ First@ #2] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 08 2017 *)

(define (A280702 n) (gcd (A003961 n) (A250469 n)))

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

A280693 Numbers n such that A003961(n) = A250469(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 50, 52, 53, 55, 59, 61, 65, 66, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169, 173, 179, 181, 186, 187, 191, 193, 197, 199, 203

Offset: 1

Antti Karttunen, Mar 08 2017

nonn

Positions of zeros in A280692. Conjectured to be also the positions of ones in A280703.
For most terms k of this sequence A003961(k) is also included as a term. Exceptions are: 50, 52, 66, 186, 435, 1245, 1445, 2068, 2085, 11605, ... that seems to be a subsequence of all those terms that have more than two prime factors: 50, 52, 66, 186, 435, 1245, 1445, 2068, 2085, 8695, 11605, ...
Note how 8695 = 5*37*47 and A003961(8695) = 7*41*53 = 15211 = A003961(8695) = A250469(8695) (for no apparent reason?).

Antti Karttunen, Table of n, a(n) for n = 1..2001

Fixed points of permutations A266645 & A266646.
Cf. A001222, A003961, A250469, A280692, A280703.
Cf. A000040, A001248, A006094, A251728 (subsequences).
Cf. also arrays A083221 and A246278.

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]]]; With[{nn = 204}, Function[s, Function[t, Select[Range@ nn, f@ # == t[[#]] &]]@ MapIndexed[Lookup[s, g[First@ #2] + 1][[#1]] - Boole[First@ #2 == 1] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ nn]]@ PositionIndex@ Array[g, 10^4]] (* Michael De Vlieger, Mar 08 2017, Version 10 *)

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).

A346473 a(n) = A250469(n) - sigma(n).

Original entry on oeis.org

0, 0, 1, 2, 1, 3, 3, 6, 12, 9, 1, 5, 3, 15, 11, 14, 1, 12, 3, 15, 23, 27, 5, 9, 18, 33, 25, 25, 1, 15, 5, 30, 37, 45, 29, 14, 3, 51, 39, 27, 1, 27, 3, 45, 37, 63, 5, 17, 64, 54, 53, 55, 5, 39, 19, 45, 65, 81, 1, 9, 5, 87, 51, 62, 35, 51, 3, 75, 79, 63, 1, 18, 5, 105, 61, 85, 47, 63, 3, 51, 84, 117, 5, 25, 25, 123, 95

Offset: 1

Antti Karttunen, Jul 28 2021

The first negative term is a(120) = -3.

Cf. A000203, A001359, A250469, A280692, A286385.

Block[{g}, g[n_] := If[n == 1, 0, PrimePi@FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[Lookup[s, g[First@ #2] + 1][[#1]] - Boole[First@ #2 == 1] - DivisorSigma[1, First@ #2] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4]] (* Michael De Vlieger, Oct 18 2021 *)

PARI
```
A346473(n) = A250469(n)-sigma(n);
```

a(n) = A250469(n) - A000203(n).
a(n) = A286385(n) - A280692(n).
a(A001359(n)) = 1 for all n >= 1.

A347378 a(n) = A250469(n) - n.

Original entry on oeis.org

0, 1, 2, 5, 2, 9, 4, 13, 16, 17, 2, 21, 4, 25, 20, 29, 2, 33, 4, 37, 34, 41, 6, 45, 24, 49, 38, 53, 2, 57, 6, 61, 52, 65, 42, 69, 4, 73, 56, 77, 2, 81, 4, 85, 70, 89, 6, 93, 72, 97, 74, 101, 6, 105, 36, 109, 88, 113, 2, 117, 6, 121, 92, 125, 54, 129, 4, 133, 106, 137, 2, 141, 6, 145, 110, 149, 66, 153, 4, 157, 124

Offset: 1

Antti Karttunen, Sep 01 2021

nonn

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

Cf. A000040, A001223, A250469, A280692, A336853, A346476.

PARI
```
A347378(n) = (A250469(n)-n);
```

a(n) = A250469(n) - n.
a(n) = n - A346476(n).
a(n) = A336853(n) - A280692(n).
For all n >= 1, a(A000040(n)) = A001223(n).

cp's OEIS Frontend

A347377 Möbius transform of A280692, A003961(n) - A250469(n).

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A250469 a(1) = 1; and for n > 1, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)+1, where A055396(n) is the index of smallest prime dividing n.

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

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A280693 Numbers n such that A003961(n) = A250469(n).

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

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A346473 a(n) = A250469(n) - sigma(n).

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Formula

A347378 a(n) = A250469(n) - n.

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