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 1-4 of 4 results.

A342671 a(n) = gcd(sigma(n), A003961(n)), where A003961 is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 21, 1, 3, 1, 15, 1, 3, 5, 1, 1, 3, 1, 9, 1, 3, 1, 1, 1, 3, 1, 9, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 15, 1, 3, 5, 3, 1, 21, 1, 3, 1, 1, 7, 3, 1, 9, 1, 3, 1, 15, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 5, 9, 1, 3, 1, 3, 1, 3, 1, 9, 1, 3, 13, 7, 1, 3, 1, 3, 1
Offset: 1

Views

Author

Antti Karttunen, Mar 20 2021

Keywords

Links

Crossrefs

Cf. A000203, A003961, A161942, A286385, A341529, A342672, A342673, A348992, A349161, A349162, A349163, A349164, A349165 (positions of 1's), A349166 (of terms > 1), A349167, A349756, A350071 [= a(n^2)], A355828 (Dirichlet inverse).
Cf. A349169, A349745, A355833, A355924 (applied onto prime shift array A246278).
Cf. also A336850, A354827/A354828, A355932.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));

Formula

a(n) = gcd(A000203(n), A003961(n)).
a(n) = gcd(A000203(n), A286385(n)) = gcd(A003961(n), A286385(n)).
a(n) = A341529(n) / A342672(n).
From Antti Karttunen, Jul 21 2022: (Start)
a(n) = A003961(n) / A349161(n).
a(n) = A000203(n) / A349162(n).
a(n) = A161942(n) / A348992(n).
a(n) = A003961(A349163(n)) = A003961(n/A349164(n)).
(End)

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

Views

Author

Antti Karttunen, Jun 07 2022

Keywords

Comments

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.

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.

Links

Crossrefs

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

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); };
A354365(n) = numerator((moebius(n)*n)/A064989(n));

Formula

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

Views

Author

Antti Karttunen, Jun 07 2022

Keywords

Links

Crossrefs

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

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)));
A354828(n) = denominator(vDirInv[n]);

A378659 Positions of records for ratio A003961(n)/sigma(n), where A003961 is fully multiplicative with a(p) = nextprime(p) and sigma is the sum of the divisors function.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 16, 27, 32, 64, 128, 243, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864
Offset: 1

Views

Author

Antti Karttunen, Dec 06 2024

Keywords

Examples

			  Term k       A003961(k)/A000203(k)    ratio
       1                1/1          =  1
       3                5/4          =  1.25
       4                9/7          =  1.2857143
       7               11/8          =  1.375
       8               27/15         =  1.8
       9               25/13         =  1.9230769
      16               81/31         =  2.6129032
      27              125/40         =  3.125
      32              243/63         =  3.8571429
      64              729/127        =  5.7401575
     128             2187/255        =  8.5764706
     243             3125/364        =  8.5851648
     256             6561/511        =  12.839530
     ...
33554432     847288609443/67108863   =  12625.584
67108864    2541865828329/134217727  =  18938.376

Links

Crossrefs

Cf. A000203, A003961.
Cf. A001359 (apparently gives the positions of successive minima of the ratio, for n > 2).
Cf. also A341529, A349627, A349628, A354827, A354828.
Showing 1-4 of 4 results.

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