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.

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).

A354361 Numbers k such that A354203(sigma(A354202(k))) = 1.

Original entry on oeis.org

1, 2, 3, 6, 7, 13, 14, 19, 21, 23, 26, 38, 39, 41, 42, 43, 46, 57, 67, 69, 78, 82, 86, 91, 103, 107, 114, 123, 129, 133, 134, 138, 161, 179, 182, 201, 206, 214, 246, 247, 258, 266, 273, 287, 299, 301, 309, 321, 322, 358, 379, 399, 402, 419, 437, 469, 483, 494, 533, 537, 546, 559, 574, 598, 602, 618, 642, 643, 721
Offset: 1

Views

Author

Antti Karttunen, May 24 2022

Keywords

Comments

Applying A354202 and sorting into ascending order gives A354357.

Crossrefs

Positions of ones in A354206.
Cf. A000203, A354202, A354203, A354357.

Programs

A354200 Prime map that sends 2 to 5, and each 4k+1 and 4k+3 prime to the next larger prime of the same type.

Original entry on oeis.org

5, 7, 13, 11, 19, 17, 29, 23, 31, 37, 43, 41, 53, 47, 59, 61, 67, 73, 71, 79, 89, 83, 103, 97, 101, 109, 107, 127, 113, 137, 131, 139, 149, 151, 157, 163, 173, 167, 179, 181, 191, 193, 199, 197, 229, 211, 223, 227, 239, 233, 241, 251, 257, 263, 269, 271, 277, 283, 281, 293, 307, 313, 311, 331, 317, 337, 347, 349, 359
Offset: 1

Views

Author

Antti Karttunen, May 23 2022

Keywords

Examples

			A000040(2) = 3, and the next larger prime of the form 4k+3 is 7, therefore a(2) = 7.
A000040(3) = 5, and the next larger prime of the form 4k+1 is 13, therefore a(3) = 13.
A000040(4) = 7, and the next larger prime of the form 4k+3 is 11, therefore a(4) = 11.

Crossrefs

Cf. A000040, A000720, A354201, A354202.

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))));

Formula

For all n >= 1, A354201(A000720(a(n))) = A000040(n).

A354203 Fully multiplicative with a(p^e) = A354201(A000720(p))^e.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 7, 1, 5, 3, 2, 1, 13, 1, 11, 2, 3, 7, 19, 1, 4, 5, 1, 3, 17, 2, 23, 1, 7, 13, 6, 1, 29, 11, 5, 2, 37, 3, 31, 7, 2, 19, 43, 1, 9, 4, 13, 5, 41, 1, 14, 3, 11, 17, 47, 2, 53, 23, 3, 1, 10, 7, 59, 13, 19, 6, 67, 1, 61, 29, 4, 11, 21, 5, 71, 2, 1, 37, 79, 3, 26, 31, 17, 7, 73, 2, 15, 19, 23
Offset: 1

Views

Author

Antti Karttunen, May 23 2022

Keywords

Links

Crossrefs

Left inverse of A354202.
Cf. A354201, A354206.

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)))));
A354203(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354201(primepi(f[k,1]))); factorback(f); };

A354357 Numbers k, not divisible by 2 or 3, such that sigma(k) is 3-smooth (has no larger prime factors than 3).

Original entry on oeis.org

1, 5, 7, 11, 17, 23, 31, 35, 47, 53, 55, 71, 77, 85, 107, 115, 119, 127, 155, 161, 187, 191, 217, 235, 253, 265, 329, 341, 355, 371, 383, 385, 391, 431, 497, 517, 527, 535, 583, 595, 635, 647, 713, 749, 781, 799, 805, 863, 889, 901, 935, 955, 971, 1081, 1085, 1151, 1177, 1207, 1219, 1265, 1309, 1337, 1397, 1457, 1633
Offset: 1

Views

Author

Antti Karttunen, May 24 2022

Keywords

Links

Crossrefs

Intersection of A007310 and A354356.
Sequence A354202(A354361(n)), n>=1, sorted into ascending order.
Cf. A000203, A065333.

Programs

  • Mathematica
    Select[Flatten @ Outer[Plus, 6 * Range[0, 300], {1, 5}], Max @ FactorInteger[DivisorSigma[1, #]][[;;, 1]] <= 3 &] (* Amiram Eldar, May 25 2022 *)
Select[Range[1,1701,2],Mod[#,3]!=0&&Max[FactorInteger[DivisorSigma[1,#]][[;;,1]]]<4&] (* Harvey P. Dale, Dec 17 2023 *)
  • PARI
    A065333(n) = ((3^valuation(n, 3)<A065333
A354355(n) = A065333(sigma(n));
isA354357(n) = ((n%2)&&(n%3)&&A354355(n));
Showing 1-8 of 8 results.

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