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 1-8 of 8 results.

A346251 Positions of zeros in A346246.

Original entry on oeis.org

192, 1280, 4608, 11520, 28672, 180224, 274428, 288684, 2013440, 3407872
Offset: 1

Views

Author

Antti Karttunen, Jul 19 2021

Keywords

Comments

Applying prime shift (A003961) to these terms gives 3645, 45927, 492075, ..., a subsequence of the positions of zeros in A323910.

Crossrefs

Cf. A003961, A323910, A346246.
Cf. also A346256.

A323910 Dirichlet inverse of the deficiency of n, A033879.

Original entry on oeis.org

1, -1, -2, 0, -4, 4, -6, 0, -1, 6, -10, 2, -12, 8, 10, 0, -16, 1, -18, 2, 14, 12, -22, 4, -3, 14, -2, 2, -28, -16, -30, 0, 22, 18, 26, 4, -36, 20, 26, 4, -40, -24, -42, 2, 4, 24, -46, 8, -5, -1, 34, 2, -52, 0, 42, 4, 38, 30, -58, 2, -60, 32, 6, 0, 50, -40, -66, 2, 46, -40, -70, 12, -72, 38, 2, 2, 62, -48, -78, 8, -4, 42, -82, -2, 66, 44, 58, 4, -88, 2, 74, 2
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2019

Keywords

Links

Crossrefs

Cf. A033879, A323911, A323912, A359549 (parity of terms).
Sequences that appear in the convolution formulas: A002033, A008683, A023900, A055615, A046692, A067824, A074206, A174725, A191161, A327960, A328722, A330575, A345182, A349341, A346246, A349387.

Programs

  • Mathematica
    b[n_] := 2 n - DivisorSigma[1, n];
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 17 2020 *)
  • PARI
    up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA033879(n) = (2*n-sigma(n));
v323910 = DirInverse(vector(up_to,n,A033879(n)));
A323910(n) = v323910[n];

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA033879(n/d) * a(d).
From Antti Karttunen, Nov 14 2024: (Start)
Following convolution formulas have been conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly:
a(n) = Sum_{d|n} A046692(d)*A067824(n/d).
a(n) = Sum_{d|n} A055615(d)*A074206(n/d).
a(n) = Sum_{d|n} A023900(d)*A174725(n/d).
a(n) = Sum_{d|n} A008683(d)*A323912(n/d).
a(n) = Sum_{d|n} A191161(d)*A327960(n/d).
a(n) = Sum_{d|n} A328722(d)*A330575(n/d).
a(n) = Sum_{d|n} A345182(d)*A349341(n/d).
a(n) = Sum_{d|n} A346246(d)*A349387(n/d).
a(n) = Sum_{d|n} A002033(d-1)*A055615(n/d).
(End)

A346248 Dirichlet inverse of -A252748, 2*n - A003961(n).

Original entry on oeis.org

1, -1, -1, 2, -3, 5, -3, 8, 8, 7, -9, 10, -9, 11, 11, 32, -15, 16, -15, 6, 19, 13, -17, 48, 8, 17, 56, 18, -27, -3, -25, 128, 17, 19, 25, 104, -33, 23, 25, 32, -39, 9, -39, -6, 24, 29, -41, 224, 32, 16, 23, 6, -47, 144, 35, 88, 31, 31, -57, 78, -55, 37, 72, 512, 43, -33, -63, -18, 41, -13, -69, 512, -67, 41, 40, -6, 43
Offset: 1

Views

Author

Antti Karttunen, Jul 19 2021

Keywords

Comments

Zeros occur at n = 352, 26840, 34816, 3787168, ...

Links

Crossrefs

Cf. A003961, A252748, A346250.
Cf. also A323910, A346235, A346246, A346254.

Programs

  • PARI
    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(dA003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
v346248 = DirInverseCorrect(vector(up_to,n,(n+n)-A003961(n)));
A346248(n) = v346248[n];

Formula

a(n) = A346250(n) + A252748(n).

A344587 Deficiency of prime-shifted n: a(n) = 2*A003961(n) - sigma(A003961(n)).

Original entry on oeis.org

1, 2, 4, 5, 6, 6, 10, 14, 19, 10, 12, 12, 16, 18, 22, 41, 18, 26, 22, 22, 38, 22, 28, 30, 41, 30, 94, 42, 30, 18, 36, 122, 46, 34, 58, 47, 40, 42, 62, 58, 42, 42, 46, 52, 102, 54, 52, 84, 109, 66, 70, 72, 58, 126, 70, 114, 86, 58, 60, 6, 66, 70, 178, 365, 94, 54, 70, 82, 110, 78, 72, 110, 78, 78, 148, 102, 118, 78
Offset: 1

Views

Author

Antti Karttunen, May 28 2021

Keywords

Comments

First negative value occurs as a(120) = -30.
Questions: Which subsets of natural numbers generate the "cut sigmoid" graph(s) that cross the X-axis in the (lowermost) scatter plot?

Links

Crossrefs

Cf. A000203, A003961, A003973, A033879, A153881, A336851, A337386 (positions of terms <= 0), A346246 (Dirichlet inverse), A349387, A378216, A378231 [= a(n^2)].
Inverse Möbius transform of A337544.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A344587(n) = { my(u=A003961(n)); (u+u - sigma(u)); };

Formula

a(n) = A033879(A003961(n)) = 2*A003961(n) - A003973(n).
a(n) = Sum_{d|n} A337544(d).
From Antti Karttunen, Nov 23 2024: (Start)
a(n) = Sum_{d|n} A003961(d)*A153881(n/d) = A003961(n) - A336851(n).
a(n) = Sum_{d|n} A033879(d)*A349387(n/d).
a(n) = Sum_{d|n} A003972(d)*A378216(n/d).
(End)

A346235 Dirichlet inverse of A341530, gcd(n*sigma(A003961(n)), sigma(n)*A003961(n)).

Original entry on oeis.org

1, -1, -2, 0, -2, -32, -4, -4, 3, 2, -2, 34, -2, -16, -112, 8, -2, 125, -4, 0, 8, 0, -6, -128, 3, -14, -8, -124, -2, 8, -2, -10, -4, 2, -320, 920, -2, -4, 4, 8, -2, 64, -4, -358, 430, -12, -6, 528, -3, -5, -352, 16, -6, -368, -48, 224, 0, 2, -2, 104, -2, -12, -12, 28, -4, 16, -4, 0, -36, 400, -2, -440, -2, -2, 450, 8, -248, 128
Offset: 1

Views

Author

Antti Karttunen, Jul 11 2021

Keywords

Links

Crossrefs

Cf. A000203, A003961, A341530, A346236.
Cf. also A346246, A346248, A346254.

Programs

  • PARI
    up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341530(n) = { my(t=A003961(n), s=sigma(t)); gcd((n*s), sigma(n)*t); };
v346235 = DirInverseCorrect(vector(up_to,n,A341530(n)));
A346235(n) = v346235[n];

A346254 Dirichlet inverse of A336849.

Original entry on oeis.org

1, -3, -5, 0, -7, 25, -11, 0, 0, 21, -13, -30, -17, 55, 35, 0, -19, -100, -23, 0, 55, 39, -29, 36, 0, 85, 0, -66, -31, -175, -37, 0, 65, 57, 77, 400, -41, 115, 85, 0, -43, -495, -47, 108, 0, 145, -53, -216, 0, 98, 171, -68, -59, 500, 169, 0, 115, 93, -61, 210, -67, 111, 0, 0, 119, -325, -71, 0, 261, -385, -73, -120, -79, 205, 0, -138
Offset: 1

Views

Author

Antti Karttunen, Jul 19 2021

Keywords

Links

Crossrefs

Cf. A000203, A003961, A003973, A336849, A346255, A346256 (positions of zeros).
Cf. also A346235, A346246, A346248.

Programs

  • PARI
    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(dA003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A336849(n) = { my(u=A003961(n)); (u/gcd(u, sigma(u))); };
v346254 = DirInverseCorrect(vector(up_to,n,A336849(n)));
A346254(n) = v346254[n];

Formula

a(n) = A346255(n) - A336849(n).

A346247 Sum of A344587 (the deficiency of prime shifted n) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 4, 0, 16, 0, 12, 16, 24, 0, 16, 0, 40, 48, 37, 0, 28, 0, 28, 80, 48, 0, 36, 36, 64, 88, 52, 0, -48, 0, 114, 96, 72, 120, 54, 0, 88, 128, 68, 0, -64, 0, 64, 116, 112, 0, 92, 100, 68, 144, 88, 0, 124, 144, 132, 176, 120, 0, -12, 0, 144, 204, 349, 192, -72, 0, 100, 224, -72, 0, 128, 0, 160, 160, 124, 240, -88, 0, 182
Offset: 1

Views

Author

Antti Karttunen, Jul 19 2021

Keywords

Links

Crossrefs

Cf. A000203, A003961, A323911, A344587, A346246.
Cf. also A346250.

Programs

  • PARI
    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(dA003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A344587(n) = { my(u=A003961(n)); (u+u - sigma(u)); };
v346246 = DirInverseCorrect(vector(up_to,n,A344587(n)));
A346246(n) = v346246[n];
A346247(n) = (A344587(n)+A346246(n));

Formula

a(n) = A344587(n) + A346246(n).
a(n) = A323911(A003961(n)).

A378220 Dirichlet inverse of phi(A003961(n)), where A003961 is fully multiplicative function with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, -2, -4, -2, -6, 8, -10, -2, -4, 12, -12, 8, -16, 20, 24, -2, -18, 8, -22, 12, 40, 24, -28, 8, -6, 32, -4, 20, -30, -48, -36, -2, 48, 36, 60, 8, -40, 44, 64, 12, -42, -80, -46, 24, 24, 56, -52, 8, -10, 12, 72, 32, -58, 8, 72, 20, 88, 60, -60, -48, -66, 72, 40, -2, 96, -96, -70, 36, 112, -120, -72, 8, -78, 80, 24
Offset: 1

Views

Author

Antti Karttunen, Nov 23 2024

Keywords

Links

Crossrefs

Cf. A023900, A003961, A048673, A151800, A349385, A378216.
Dirichlet inverse of A003972.
Inverse Möbius transform of A346234.
After the initial term, A349385 doubled.
Cf. A346246, A378216.

Programs

  • Mathematica
    f[p_, e_] := 1 - NextPrime[p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 13 2025 *)
  • PARI
    A378220(n) = factorback(apply(p -> 1-nextprime(1+p), factor(n)[, 1]));
  • Python
    from math import prod
from sympy import nextprime, primefactors
def A378220(n): return prod(1-nextprime(p) for p in primefactors(n)) # Chai Wah Wu, Nov 23 2024

Formula

Multiplicative with a(p^e) = (1-q), where q = A151800(p), i.e., the least prime > p.
a(n) = A023900(A003961(n)).
For n > 1, a(n) = 2*A349385(n).
a(n) = Sum_{d|n} A346234(d).
a(n) = Sum_{d|n} A346246(d)*A378216(n/d).
Showing 1-8 of 8 results.

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