A354206 -id:A354206 - cp's OEIS frontend

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

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

Antti Karttunen, May 23 2022

Index entries for sequences computed from indices in prime factorization

Left inverse of A354202.

Cf. A354201, A354206.

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); };

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

Antti Karttunen, May 23 2022

Cf. A000203, A000290 (positions of odd terms), A000720, A354200, A354202, A354204, A354206.

Cf. A003973, A354089, A354093 for variants.

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));

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).

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

Antti Karttunen, May 23 2022

sign

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

Cf. also A348736.

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))));

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

Antti Karttunen, May 24 2022

nonn

Applying A354202 and sorting into ascending order gives A354357.

Positions of ones in A354206.

Cf. A000203, A354202, A354203, A354357.

isA354361(n) = (1==A354206(n)); \\ Uses the program given in A354206.

cp's OEIS Frontend

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

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

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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.

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Formula

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

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