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.

A378750 Dirichlet inverse of A377984, where A377984(n) = 2*sigma(n) - A003961(n).

Original entry on oeis.org

1, -3, -3, 4, -5, 9, -5, 0, 8, 15, -11, -8, -11, 15, 17, 8, -17, -24, -17, -16, 21, 33, -19, 0, 12, 33, 24, -8, -29, -51, -27, 24, 35, 51, 31, 56, -35, 51, 39, 0, -41, -63, -41, -40, -28, 57, -43, 0, 32, -36, 53, -32, -49, -72, 57, 0, 57, 87, -59, 48, -57, 81, -4, 88, 61, -105, -65, -64, 67, -93, -71, 0, -69, 105
Offset: 1

Views

Author

Antti Karttunen, Dec 09 2024

Keywords

Links

Crossrefs

Cf. A000203, A003961, A377984.
Cf. also A378749.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A377984(n) = (2*sigma(n) - A003961(n));
memoA378750 = Map();
A378750(n) = if(1==n,1,my(v); if(mapisdefined(memoA378750,n,&v), v, v = -sumdiv(n,d,if(dA377984(n/d)*A378750(d),0)); mapput(memoA378750,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA377984(n/d) * a(d).

A337378 Numbers k for which A003961(k) > 2*sigma(k).

Original entry on oeis.org

16, 24, 27, 32, 36, 40, 45, 48, 49, 54, 56, 63, 64, 72, 80, 81, 84, 90, 96, 98, 99, 100, 104, 105, 108, 112, 117, 120, 125, 126, 128, 135, 140, 144, 147, 152, 153, 160, 162, 168, 171, 175, 176, 180, 184, 189, 192, 196, 198, 200, 207, 208, 210, 216, 224, 225, 234, 240, 243, 245, 248, 250, 252, 256, 264, 270, 272, 273
Offset: 1

Views

Author

Antti Karttunen, Aug 27 2020

Keywords

Comments

Note that A003961(n) >= sigma(n) for all n. See A286385.

Links

Crossrefs

Subsequence of A246282 and of A337381.
Positions of negative terms in A377984, and in A378751.
Cf. A000203, A003961, A286385.
Cf. A337379 (complement), A337380 (characteristic function).

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337378(n) = (A003961(n)>2*sigma(n));

A377985 a(n) = 2*n - phi(A003961(n)), where phi is Euler totient function and A003961 is fully multiplicative function with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 4, -2, -2, 8, 10, 0, 10, 8, 6, -22, 16, -4, 16, 4, 2, 20, 18, -24, 8, 20, -46, -4, 28, 12, 26, -98, 18, 32, 10, -48, 34, 32, 14, -28, 40, 4, 40, 16, -30, 36, 42, -120, -12, 16, 30, 8, 48, -92, 38, -68, 26, 56, 58, -24, 56, 52, -74, -358, 34, 36, 64, 28, 26, 20, 70, -216, 68, 68, -18, 20, 34, 28
Offset: 1

Views

Author

Antti Karttunen, Nov 16 2024

Keywords

Links

Crossrefs

Cf. A000010, A003961, A003972, A008683.
Möbius transform of A377984.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A377985(n) = (2*n - eulerphi(A003961(n)));

Formula

a(n) = 2*n - A003972(n).
a(n) = Sum_{d|n} A008683(n/d) * A377984(d).

A378751 a(n) = A000203(n) - A048673(n).

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 2, 1, 0, 7, 5, 5, 5, 7, 6, -10, 8, 1, 8, 10, 4, 16, 9, -8, 6, 16, -23, 6, 14, 19, 13, -59, 15, 25, 9, -22, 17, 25, 13, -5, 20, 13, 20, 25, -10, 28, 21, -79, -4, 19, 24, 21, 24, -68, 26, -29, 22, 43, 29, 10, 28, 40, -34, -238, 24, 46, 32, 40, 23, 28, 35, -143, 34, 52, 1, 36, 24, 40, 38, -98, -192
Offset: 1

Views

Author

Antti Karttunen, Dec 09 2024

Keywords

Links

Crossrefs

Cf. A000203, A048673, A377984, A378747.
Cf. A337378 (positions of negative terms), A337379 (of terms >= 0).

Programs

  • PARI
    A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A378751(n) = (sigma(n)-A048673(n));

Formula

a(n) = n - A378747(n).
a(n) = (1/2) * (A377984(n)-1).
Showing 1-4 of 4 results.

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