cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A354350 a(n) = n + A354365(n).

Original entry on oeis.org

2, 0, 0, 4, 0, 9, 0, 8, 9, 20, 0, 12, 0, 28, 20, 16, 0, 18, 0, 20, 42, 44, 0, 24, 25, 52, 27, 28, 0, 25, 0, 32, 66, 68, 42, 36, 0, 76, 78, 40, 0, 21, 0, 44, 45, 92, 0, 48, 49, 50, 102, 52, 0, 54, 110, 56, 114, 116, 0, 60, 0, 124, 63, 64, 130, 33, 0, 68, 138, 56, 0, 72, 0, 148, 75, 76, 88, 39, 0, 80, 81, 164, 0, 84, 170
Offset: 1

Views

Author

Antti Karttunen, Jun 07 2022

Keywords

Links

Crossrefs

Cf. A008683, A055615, A064989, A354365.

Programs

  • PARI
    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); };
A354350(n) = (n+numerator((moebius(n)*n)/A064989(n)));

Formula

a(n) = n + A354365(n).

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

Views

Author

Rémy Sigrist, Sep 25 2018

Keywords

Comments

See A319626 for the corresponding numerators and additional comments.

Examples

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

Links

Crossrefs

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.

Programs

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

Formula

a(n) = A064989(n) / gcd(n, A064989(n)).
a(n) = 1 iff n belongs to A025487.

Extensions

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

A354366 Denominators of Dirichlet inverse of primorial deflation fraction A319626(n) / A319627(n).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 5, 1, 1, 3, 7, 1, 11, 5, 2, 1, 13, 1, 17, 1, 10, 7, 19, 1, 1, 11, 1, 1, 23, 1, 29, 1, 14, 13, 3, 1, 31, 17, 22, 1, 37, 5, 41, 1, 1, 19, 43, 1, 1, 1, 26, 1, 47, 1, 21, 1, 34, 23, 53, 1, 59, 29, 1, 1, 33, 7, 61, 1, 38, 3, 67, 1, 71, 31, 1, 1, 5, 11, 73, 1, 1, 37, 79, 1, 39, 41, 46, 1, 83, 1, 55, 1, 58
Offset: 1

Views

Author

Antti Karttunen, Jun 07 2022

Keywords

Comments

Equally, denominators of Dirichlet inverse of fraction n / A064989(n). See also comments in A354365.

Links

Crossrefs

Cf. A055615, A064989, A319626, A319627, A354360 (positions of 1's).
Cf. A354365 (numerators).
Cf. also A349630.

Programs

  • PARI
    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); };
A354366(n) = denominator((moebius(n)*n)/A064989(n));

Formula

a(n) = A064989(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

Views

Author

Antti Karttunen, Jun 05 2022

Keywords

Comments

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

Links

Crossrefs

Cf. A002110, A008683, A013929 (positions of 0's), A034386, A108951, A354352.
Cf. also A347379, A354186, A354349, A354359, A354365, A354366.

Programs

Formula

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).

A354827 Numerators of Dirichlet inverse of fraction A003961(n) / sigma(n).

Original entry on oeis.org

1, -1, -5, -2, -7, 5, -11, -8, -75, 7, -13, 5, -17, 11, 35, -1648, -19, 75, -23, 1, 55, 13, -29, 2, -245, 17, -225, 11, -31, -35, -37, -1664, 65, 19, 77, 75, -41, 23, 85, 4, -43, -55, -47, 13, 175, 29, -53, 412, -847, 245, 95, 17, -59, 225, 91, 11, 23, 31, -61, -5, -67, 37, 825, -7662464, 17, -65, -71, 19, 145, -77
Offset: 1

Views

Author

Antti Karttunen, Jun 07 2022

Keywords

Comments

Because the ratio A003961(n) / A000203(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 A354828 gives the denominators. See the examples.

Examples

			The ratio a(n)/A354828(n) for n = 1..21: 1, -1, -5/4, -2/7, -7/6, 5/4, -11/8, -8/35, -75/208, 7/6, -13/12, 5/14, -17/14, 11/8, 35/24, -1648/7595, -19/18, 75/208, -23/20, 1/3, 55/32.

Links

Crossrefs

Cf. A000203, A003961, A349161, A349162.
Cf. A354828 (denominators).
Cf. also A349627, A354365.

Programs

  • PARI
    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)));
A354827(n) = numerator(vDirInv[n]);
A354828(n) = denominator(vDirInv[n]);

A354360 Positions of 1's in A354366.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 07 2022

Keywords

Crossrefs

Cf. A354365, A354366.

Programs

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

Views

Author

Antti Karttunen, Jul 18 2022

Keywords

Links

Crossrefs

Cf. A064989, A330749.
Cf. also A354365, A354366.

Programs

  • PARI
    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)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA330749(n/d) * a(d).
Showing 1-7 of 7 results.

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