A354366 -id:A354366 - cp's OEIS frontend

A354360 Positions of 1's in A354366.

1, 2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 30, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 169, 171, 172, 175

Offset: 1

Antti Karttunen, Jun 07 2022

nonn

Cf. A354365, A354366.

isA354360(n) = (1==A354366(n)); \\ Uses the program from A354366.

A319627 Primorial deflation of n (denominator): Let f be the completely multiplicative function over the positive rational numbers defined by f(p) = A034386(p) for any prime number p; f constitutes a permutation of the positive rational numbers; let g be the inverse of f; for any n > 0, a(n) is the denominator of g(n).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 5, 1, 4, 3, 7, 1, 11, 5, 2, 1, 13, 2, 17, 3, 10, 7, 19, 1, 9, 11, 8, 5, 23, 1, 29, 1, 14, 13, 3, 1, 31, 17, 22, 3, 37, 5, 41, 7, 4, 19, 43, 1, 25, 9, 26, 11, 47, 4, 21, 5, 34, 23, 53, 1, 59, 29, 20, 1, 33, 7, 61, 13, 38, 3, 67, 1, 71, 31, 6

Offset: 1

Rémy Sigrist, Sep 25 2018

See A319626 for the corresponding numerators and additional comments.

			f(21/5) = (2*3) * (2*3*5*7) / (2*3*5) = 42, hence g(42) = 21/5 and a(42) = 5.

Daniel Suteu, Table of n, a(n) for n = 1..10000

Cf. A025487 (positions of 1's), A064989, A329900, A358217 [= bigomega(a(n))].
Cf. A319626 (numerators, see comments there).
Cf. also A307035, A337377, A348990 [= a(A003961(n))], A349169 (odd numbers k such that A348993(k) = a(k)), A354365/A354366.

Array[#2/GCD[#1, #2] & @@ {#, Apply[Times, Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#]]]} &, 120] (* Michael De Vlieger, Aug 27 2020 *)

a(n) = my (f=factor(n)); denominator(prod(i=1, #f~, my (p=f[i,1]); (p/if (p>2, precprime(p-1), 1))^f[i,2]))

a(n) = A064989(n) / gcd(n, A064989(n)).

a(n) = 1 iff n belongs to A025487.

"Primorial deflation" prefixed to the name by Antti Karttunen, Apr 29 2022

A354365 Numerators of Dirichlet inverse of primorial deflation fraction A319626(n) / A319627(n).

Original entry on oeis.org

1, -2, -3, 0, -5, 3, -7, 0, 0, 10, -11, 0, -13, 14, 5, 0, -17, 0, -19, 0, 21, 22, -23, 0, 0, 26, 0, 0, -29, -5, -31, 0, 33, 34, 7, 0, -37, 38, 39, 0, -41, -21, -43, 0, 0, 46, -47, 0, 0, 0, 51, 0, -53, 0, 55, 0, 57, 58, -59, 0, -61, 62, 0, 0, 65, -33, -67, 0, 69, -14, -71, 0, -73, 74, 0, 0, 11, -39, -79, 0, 0, 82

Offset: 1

Antti Karttunen, Jun 07 2022

Because the ratio n / A064989(n) = A319626(n) / A319627(n) is multiplicative, so is also its Dirichlet inverse (which also is a sequence of rational numbers). This sequence gives the numerators when presented in its lowest terms, while A354366 gives the denominators. See the examples.

			The ratio a(n)/A354366(n) for n = 1..22: 1, -2, -3/2, 0, -5/3, 3, -7/5, 0, 0, 10/3, -11/7, 0, -13/11, 14/5, 5/2, 0, -17/13, 0, -19/17, 0, 21/10, 22/7.

Index entries for sequences computed from indices in prime factorization

Cf. A013929 (positions of 0's), A055615, A319626, A319627, A354350.
Cf. A354366 (denominators).
Cf. also A349629, A354351, A354827.

A064989(n) = { my(f = factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A354365(n) = numerator((moebius(n)*n)/A064989(n));

a(n) = A055615(n) / gcd(A055615(n), A064989(n)).

A354351 Dirichlet inverse of A108951, primorial inflation of n.

Original entry on oeis.org

1, -2, -6, 0, -30, 12, -210, 0, 0, 60, -2310, 0, -30030, 420, 180, 0, -510510, 0, -9699690, 0, 1260, 4620, -223092870, 0, 0, 60060, 0, 0, -6469693230, -360, -200560490130, 0, 13860, 1021020, 6300, 0, -7420738134810, 19399380, 180180, 0, -304250263527210, -2520, -13082761331670030, 0, 0, 446185740, -614889782588491410

Offset: 1

Antti Karttunen, Jun 05 2022

Multiplicative with a(p^e) = 0 if e > 1, and -A034386(p) otherwise.

Cf. A002110, A008683, A013929 (positions of 0's), A034386, A108951, A354352.

Cf. also A347379, A354186, A354349, A354359, A354365, A354366.

A002110(n) = prod(i=1,n,prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A002110(primepi(f[i, 1]))^f[i, 2]) }; \\ From A108951
A354351(n) = (moebius(n)*A108951(n));

a(n) = A008683(n) * A108951(n).
a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d < n} A108951(n/d) * a(d).
a(n) = A354352(n) - A108951(n).

A354828 Denominators of Dirichlet inverse of fraction A003961(n) / sigma(n).

Original entry on oeis.org

1, 1, 4, 7, 6, 4, 8, 35, 208, 6, 12, 14, 14, 8, 24, 7595, 18, 208, 20, 3, 32, 12, 24, 7, 1116, 14, 832, 28, 30, 24, 32, 7595, 48, 18, 48, 728, 38, 20, 56, 15, 42, 32, 44, 42, 416, 24, 48, 1519, 3648, 1116, 72, 49, 54, 832, 72, 35, 16, 30, 60, 12, 62, 32, 1664, 33759775, 12, 48, 68, 63, 96, 48, 72, 182, 74, 38, 4464

Offset: 1

Antti Karttunen, Jun 07 2022

Cf. A000203, A003961, A349161, A349162.
Cf. A354827 (denominators).
Cf. also A349628, A354366.

up_to = 65537;
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA354827(n) = (A003961(n)/sigma(n));
vDirInv = DirInverseCorrect(vector(up_to,n,AuxA354827(n)));
A354828(n) = denominator(vDirInv[n]);

A355693 Dirichlet inverse of A330749, gcd(n, A064989(n)), where A064989 shifts the prime factorization one step towards lower primes.

Original entry on oeis.org

1, -1, -1, 0, -1, 0, -1, 0, 0, 1, -1, 1, -1, 1, -1, 0, -1, 1, -1, 0, 1, 1, -1, 0, 0, 1, 0, 0, -1, 0, -1, 0, 1, 1, -3, -2, -1, 1, 1, 0, -1, 0, -1, 0, 2, 1, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1, 1, -1, 1, -1, 1, 0, 0, 1, 0, -1, 0, 1, 3, -1, 1, -1, 1, 2, 0, -5, 0, -1, 0, 0, 1, -1, -1, 1, 1, 1, 0, -1, 3, 1, 0, 1, 1, 1, 0

Offset: 1

Antti Karttunen, Jul 18 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A064989, A330749.

Cf. also A354365, A354366.

A330749(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); gcd(n, factorback(f)); };
memoA355693 = Map();
A355693(n) = if(1==n,1,my(v); if(mapisdefined(memoA355693,n,&v), v, v = -sumdiv(n,d,if(dA330749(n/d)*A355693(d),0)); mapput(memoA355693,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA330749(n/d) * a(d).

cp's OEIS Frontend

A354360 Positions of 1's in A354366.

Views

Author

Keywords

Crossrefs

Programs

PARI

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A354365 Numerators of Dirichlet inverse of primorial deflation fraction A319626(n) / A319627(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A354351 Dirichlet inverse of A108951, primorial inflation of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A354828 Denominators of Dirichlet inverse of fraction A003961(n) / sigma(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A355693 Dirichlet inverse of A330749, gcd(n, A064989(n)), where A064989 shifts the prime factorization one step towards lower primes.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula