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 1-6 of 6 results.

A354202 Fully multiplicative with a(p^e) = A354200(A000720(p))^e.

Original entry on oeis.org

1, 5, 7, 25, 13, 35, 11, 125, 49, 65, 19, 175, 17, 55, 91, 625, 29, 245, 23, 325, 77, 95, 31, 875, 169, 85, 343, 275, 37, 455, 43, 3125, 133, 145, 143, 1225, 41, 115, 119, 1625, 53, 385, 47, 475, 637, 155, 59, 4375, 121, 845, 203, 425, 61, 1715, 247, 1375, 161, 185, 67, 2275, 73, 215, 539, 15625, 221, 665, 71, 725
Offset: 1

Views

Author

Antti Karttunen, May 23 2022

Keywords

Comments

Permutation of A007310. Preserves the prime signature.

Links

Crossrefs

Cf. A007310 (terms sorted into ascending order), A354200, A354203 (left inverse), A354204 (Möbius transform), A354205 (inverse Möbius transform).
Cf. also A003961, A108548, A267099, A332818, A348746, A354091 for similar constructions.

Programs

  • PARI
    A354200(n) = if(1==n,5,my(p=prime(n), m=p%4); forprime(q=1+p,,if(m==(q%4),return(q))));
A354202(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); factorback(f); };

A354201 Inverse prime map of A354200.

Original entry on oeis.org

1, 1, 2, 3, 7, 5, 13, 11, 19, 17, 23, 29, 37, 31, 43, 41, 47, 53, 59, 67, 61, 71, 79, 73, 89, 97, 83, 103, 101, 109, 107, 127, 113, 131, 137, 139, 149, 151, 163, 157, 167, 173, 179, 181, 193, 191, 199, 211, 223, 197, 229, 227, 233, 239, 241, 251, 257, 263, 269, 277, 271, 281, 283, 307, 293, 313, 311, 317, 331, 337
Offset: 1

Views

Author

Antti Karttunen, May 23 2022

Keywords

Crossrefs

Cf. A000040, A000720, A354200, A354203.

Programs

  • PARI
    A354201(n) = if(n<=3,(n+1)\2,my(m=prime(n)%4); forstep(i=n-1,0,-1,if(m==(prime(i)%4),return(prime(i)))));

Formula

For all n >= 1, a(A000720(A354200(n))) = A000040(n).

A354205 a(n) = sigma(A354202(n)), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)).

Original entry on oeis.org

1, 6, 8, 31, 14, 48, 12, 156, 57, 84, 20, 248, 18, 72, 112, 781, 30, 342, 24, 434, 96, 120, 32, 1248, 183, 108, 400, 372, 38, 672, 44, 3906, 160, 180, 168, 1767, 42, 144, 144, 2184, 54, 576, 48, 620, 798, 192, 60, 6248, 133, 1098, 240, 558, 62, 2400, 280, 1872, 192, 228, 68, 3472, 74, 264, 684, 19531, 252, 960, 72
Offset: 1

Views

Author

Antti Karttunen, May 23 2022

Keywords

Links

Crossrefs

Cf. A000203, A000290 (positions of odd terms), A000720, A354200, A354202, A354204, A354206.
Cf. A003973, A354089, A354093 for variants.

Programs

  • PARI
    A354200(n) = if(1==n,5,my(p=prime(n), m=p%4); forprime(q=1+p,,if(m==(q%4),return(q))));
A354205(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); sigma(factorback(f)); };
\\ Alternatively:
A354205(n) = sumdiv(n,d,A354202(d));

Formula

Multiplicative with a(p^e) = (q^(e+1)-1)/(q-1) where q = A354200(A000720(p)).
a(n) = A000203(A354202(n)).
a(n) = Sum_{d|n} A354202(d).

A354206 a(n) = A354203(sigma(A354202(n))), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)), and A354203 is its left inverse.

Original entry on oeis.org

1, 1, 1, 23, 3, 1, 1, 5, 11, 3, 2, 23, 1, 1, 3, 469, 2, 11, 1, 69, 1, 2, 1, 5, 53, 1, 4, 23, 11, 3, 7, 69, 2, 2, 3, 253, 3, 1, 1, 15, 1, 1, 1, 46, 33, 1, 2, 469, 33, 53, 2, 23, 23, 4, 6, 5, 1, 11, 13, 69, 29, 7, 11, 19507, 3, 2, 1, 46, 1, 3, 2, 55, 2, 3, 53, 23, 2, 1, 3, 1407, 2797, 1, 5, 23, 6, 1, 11, 10, 9, 33
Offset: 1

Views

Author

Antti Karttunen, May 23 2022

Keywords

Links

Crossrefs

Cf. A000203, A000720, A354200, A354201, A354202, A354203, A354205, A354207.
Cf. A354361 (positions of 1's).
Cf. also A326042, A348750, A354088, A354096 for similar constructions.

Programs

  • PARI
    A354200(n) = if(1==n,5,my(p=prime(n), m=p%4); forprime(q=1+p,,if(m==(q%4),return(q))));
A354201(n) = if(n<=3,(n+1)\2,my(m=prime(n)%4); forstep(i=n-1,0,-1,if(m==(prime(i)%4),return(prime(i)))));
A354202(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); factorback(f); };
A354203(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354201(primepi(f[k,1]))); factorback(f); };
A354206(n) = A354203(sigma(A354202(n)));

Formula

Multiplicative with a(p^e) = A354203((q^(e+1)-1)/(q-1)) where q = A354200(A000720(p)).
a(n) = A354203(A354205(n)) = A354203(sigma(A354202(n))).
a(n) = n - A354207(n).

A354204 a(n) = phi(A354202(n)), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)).

Original entry on oeis.org

1, 4, 6, 20, 12, 24, 10, 100, 42, 48, 18, 120, 16, 40, 72, 500, 28, 168, 22, 240, 60, 72, 30, 600, 156, 64, 294, 200, 36, 288, 42, 2500, 108, 112, 120, 840, 40, 88, 96, 1200, 52, 240, 46, 360, 504, 120, 58, 3000, 110, 624, 168, 320, 60, 1176, 216, 1000, 132, 144, 66, 1440, 72, 168, 420, 12500, 192, 432, 70, 560, 180
Offset: 1

Views

Author

Antti Karttunen, May 23 2022

Keywords

Links

Crossrefs

Möbius transform of A354202.
Cf. A000010, A008683, A354200, A354205.

Programs

  • PARI
    A354200(n) = if(1==n,5,my(p=prime(n), m=p%4); forprime(q=1+p,,if(m==(q%4),return(q))));
A354204(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); eulerphi(factorback(f)); };
\\ Alternatively:
A354204v2(n) = { my(f=factor(n),q); prod(k=1,#f~,q = A354200(primepi(f[k,1])); (q-1)*(q^(f[k,2]-1))); };

Formula

Multiplicative with a(p^e) = (q-1) * q^(e-1), where q = A354200(A000720(p)).
a(n) = A000010(A354202(n)).
a(n) = Sum_{d|n} A008683(n/d) * A354202(d).

A354207 a(n) = n - A354203(sigma(A354202(n))), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)), and A354203 is its left inverse.

Original entry on oeis.org

0, 1, 2, -19, 2, 5, 6, 3, -2, 7, 9, -11, 12, 13, 12, -453, 15, 7, 18, -49, 20, 20, 22, 19, -28, 25, 23, 5, 18, 27, 24, -37, 31, 32, 32, -217, 34, 37, 38, 25, 40, 41, 42, -2, 12, 45, 45, -421, 16, -3, 49, 29, 30, 50, 49, 51, 56, 47, 46, -9, 32, 55, 52, -19443, 62, 64, 66, 22, 68, 67, 69, 17, 71, 71, 22, 53, 75, 77, 76
Offset: 1

Views

Author

Antti Karttunen, May 23 2022

Keywords

Links

Crossrefs

Cf. A000203, A354200, A354201, A354202, A354203, A354205, A354206.
Cf. also A348736.

Programs

  • PARI
    A354200(n) = if(1==n,5,my(p=prime(n), m=p%4); forprime(q=1+p,,if(m==(q%4),return(q))));
A354201(n) = if(n<=3,(n+1)\2,my(m=prime(n)%4); forstep(i=n-1,0,-1,if(m==(prime(i)%4),return(prime(i)))));
A354202(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); factorback(f); };
A354203(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354201(primepi(f[k,1]))); factorback(f); };
A354207(n) = (n-A354203(sigma(A354202(n))));

Formula

a(n) = n - A354206(n).
Showing 1-6 of 6 results.

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