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 1-7 of 7 results.

A267099 Fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes: a(1) = 1; a(prime(k)) = A267101(k), a(x*y) = a(x)*a(y) for x, y > 1.

Original entry on oeis.org

1, 2, 5, 4, 3, 10, 13, 8, 25, 6, 17, 20, 7, 26, 15, 16, 11, 50, 29, 12, 65, 34, 37, 40, 9, 14, 125, 52, 19, 30, 41, 32, 85, 22, 39, 100, 23, 58, 35, 24, 31, 130, 53, 68, 75, 74, 61, 80, 169, 18, 55, 28, 43, 250, 51, 104, 145, 38, 73, 60, 47, 82, 325, 64, 21, 170, 89, 44, 185, 78, 97, 200, 59, 46, 45, 116, 221, 70, 101
Offset: 1

Views

Author

Antti Karttunen, Feb 01 2016

Keywords

Comments

Lexicographically earliest self-inverse permutation of natural numbers where each prime of the form 4k+1 is replaced by a prime of the form 4k+3 and vice versa, with the composite numbers determined by multiplicativity.
Fully multiplicative with a(p_n) = p_{A267100(n)} = A267101(n).
Maps each term of A004613 to some term of A004614, each (nonzero) term of A001481 to some term of A268377 and each term of A004431 to some term of A268378 and vice versa.
Sequences A072202 and A078613 are closed with respect to this permutation.

Links

Crossrefs

Cf. A000035, A000040, A000720, A010051, A020639, A032742, A267100, A267101, A354102 (Möbius transform), A354103 (inverse Möbius transform), A354192 (fixed points).
Cf. A002144, A002145, A005094, A065338, A072202, A078613, A079635.
Cf. A001481, A268377, A004431, A268378, A004613, A004614.
Cf. also A108548.

Programs

  • PARI
    up_to = 2^16;
A267097list(up_to) = { my(v=vector(up_to),i=0,c=0); forprime(p=2,prime(up_to), if(1==(p%4), c++); i++; v[i] = c); (v); };
v267097 = A267097list(up_to);
A267097(n) = v267097[n];
A267098(n) = ((n-1)-A267097(n));
list_primes_of_the_form(up_to,m,k) = { my(v=vector(up_to),i=0); forprime(p=2,, if(k==(p%m), i++; v[i] = p; if(i==up_to,return(v)))); };
v002144 = list_primes_of_the_form(2*up_to,4,1);
A002144(n) = v002144[n];
v002145 = list_primes_of_the_form(2*up_to,4,3);
A002145(n) = v002145[n];
A267101(n) = if(1==n,2,if(1==(prime(n)%4),A002145(A267097(n)),A002144(A267098(n))));
A267099(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A267101(primepi(f[k,1]))); factorback(f); }; \\ Antti Karttunen, May 18 2022
(Scheme, with memoization-macro definec)
(definec (A267099 n) (cond ((<= n 1) n) ((= 1 (A010051 n)) (A267101 (A000720 n))) (else (* (A267099 (A020639 n)) (A267099 (A032742 n))))))

Formula

a(1) = 1; after which, if n is k-th prime [= A000040(k)], then a(n) = A267101(k), otherwise a(A020639(n)) * a(A032742(n)).
Other identities. For all n >= 1:
a(A000040(n)) = A267101(n).
a(2*n) = 2*a(n).
a(3*n) = 5*a(n).
a(5*n) = 3*a(n).
a(7*n) = 13*a(n).
a(11*n) = 17*a(n).
etc. See examples in A267101.
A000035(n) = A000035(a(n)). [Preserves the parity of n.]
A005094(a(n)) = -A005094(n).
A079635(a(n)) = -A079635(n).

Extensions

Verbal description prefixed to the name by Antti Karttunen, May 19 2022

A108546 Lexicographically earliest permutation of primes such that for n>1 forms 4*k+1 and 4*k+3 alternate.

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Jun 10 2005

Keywords

Links

Crossrefs

Cf. A000040, A002144, A002145, A102261, A108547 (fixed points), A108548, A111745, A332806 (inverse), A332807.
Cf. also A267101, A332211.

Programs

  • Haskell
    import Data.List (transpose)
a108546 n = a108546_list !! (n-1)
a108546_list =  2 : concat
   (transpose [a002145_list, a002144_list])
-- Reinhard Zumkeller, Nov 13 2014, Feb 22 2011
  • Mathematica
    terms = 60; A111745 = Module[{prs = Prime[Range[2terms]], m3, m1, min}, m3 = Select[prs, Mod[#, 4] == 3&]; m1 = Select[prs, Mod[#, 4] == 1&]; min = Min[Length[m1], Length[m3]]; Riffle[Take[m3, min], Take[m1, min]]]; a[1] = 2; a[n_] := A111745[[n-1]]; Table[a[n], {n, 1, terms}] (* Jean-François Alcover, May 18 2017, using Harvey P. Dale's code for A111745 *)
  • PARI
    up_to = 10000;
A108546list(up_to) = { my(v=vector(up_to), p,q); v[1] = 2; v[2] = 3; v[3] = 5; for(n=4,up_to, p = v[n-2]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[n] = q); (v); };
v108546 = A108546list(up_to);
A108546(n) = v108546[n]; \\ Antti Karttunen, Feb 27 2020

Formula

a(n) mod 4 = 3 - 2 * (n mod 2) for n>1.
For n > 1: a(n) = A111745(n-1).
a(2*n+1) - a(2*n) = A102261(n).
From Antti Karttunen, Feb 27 2020: (Start)
a(1) = 2, a(2n) = A002145(n), a(2n+1) = A002144(n).
a(n) = A000040(A332807(n)).
(End)

A354102 a(n) = phi(A267099(n)), where A267099 is fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes, and phi is Euler totient function.

Original entry on oeis.org

1, 1, 4, 2, 2, 4, 12, 4, 20, 2, 16, 8, 6, 12, 8, 8, 10, 20, 28, 4, 48, 16, 36, 16, 6, 6, 100, 24, 18, 8, 40, 16, 64, 10, 24, 40, 22, 28, 24, 8, 30, 48, 52, 32, 40, 36, 60, 32, 156, 6, 40, 12, 42, 100, 32, 48, 112, 18, 72, 16, 46, 40, 240, 32, 12, 64, 88, 20, 144, 24, 96, 80, 58, 22, 24, 56, 192, 24, 100, 16, 500
Offset: 1

Views

Author

Antti Karttunen, May 18 2022

Keywords

Links

Crossrefs

Möbius transform of A267099.
Cf. A000720, A008683, A267101, A354101, A354103, A354104 (Dirichlet inverse), A354105 (sum with it), A354106, A354107 (a(n) mod 4), A354190, A354191.
Coincides with A000010 on A354189.

Programs

Formula

Multiplicative with a(p^e) = (q-1) * q^(e-1), where q = A267101(A000720(p)).
a(n) = A000010(A267099(n)).
a(n) = Sum_{d|n} A008683(n/d) * A267099(d).
a(n) = A354101(n) + A000010(n) = A354190(n) - A354191(n).
For all n >= 0, a(4n+2) = a(2n+1).

A267097 a(n) = number of 4k+1 primes among first n primes; least monotonic left inverse of A080147.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 3, 3, 4, 4, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 11, 12, 12, 12, 13, 14, 14, 14, 15, 15, 16, 16, 17, 17, 17, 18, 18, 19, 19, 20, 21, 21, 21, 21, 21, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 28, 28, 29, 29, 29, 30, 31, 31, 32, 32, 33, 34, 34, 34, 35, 35, 35, 36, 37, 38, 39, 39, 40, 40, 41
Offset: 1

Views

Author

Antti Karttunen, Feb 01 2016

Keywords

Comments

a(n) = number of 4k+1 primes (A002144) among primes in range 2 .. A000040(n).

Links

Crossrefs

Cf. A000040, A002144, A080147, A267098.
Cf. also A267101.

Programs

  • Mathematica
    a[n_]:=Length[Select[Prime[Range[n]],IntegerQ[(#-1)/4] &]]; Array[a,84] (* Stefano Spezia, May 01 2025 *)

Formula

Other identities. For all n >= 1:
a(A080147(n)) = n.
a(n) + A267098(n) = n-1.

A267098 a(n) = number of 4k+3 primes among first n primes; least monotonic left inverse of A080148.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 3, 4, 5, 5, 6, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 13, 13, 13, 13, 14, 15, 15, 15, 16, 17, 17, 18, 18, 19, 19, 20, 21, 21, 22, 22, 23, 23, 23, 24, 25, 26, 27, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 31, 32, 32, 33, 34, 34, 34, 35, 35, 36, 36, 36, 37, 38, 38, 39, 40, 40, 40, 40, 40, 41, 41, 42
Offset: 1

Views

Author

Antti Karttunen, Feb 01 2016

Keywords

Comments

a(n) = number of 4k+3 primes (A002145) among primes in range 2 .. A000040(n).

Links

Crossrefs

Cf. A000040, A002145, A080148, A267097.
Cf. also A267101.

Programs

  • Mathematica
    Accumulate[Table[If[IntegerQ[(n-3)/4],1,0],{n,Prime[Range[90]]}]] (* Harvey P. Dale, Mar 07 2018 *)

Formula

Other identities. For all n >= 1:
a(A080148(n)) = n.
a(n) + A267097(n) = n-1.

A267100 Self-inverse permutation of natural numbers: a(1) = 1, a(A080147(n)) = A080148(n), a(A080148(n)) = A080147(n).

Original entry on oeis.org

1, 3, 2, 6, 7, 4, 5, 10, 12, 8, 13, 9, 11, 16, 18, 14, 21, 15, 24, 25, 17, 26, 29, 19, 20, 22, 30, 33, 23, 27, 35, 37, 28, 40, 31, 42, 32, 44, 45, 34, 50, 36, 51, 38, 39, 53, 55, 57, 59, 41, 43, 60, 46, 62, 47, 65, 48, 66, 49, 52, 68, 54, 70, 71, 56, 58, 74, 61, 77, 63, 64, 78, 79, 67, 80, 82, 69, 72, 73, 75, 84, 76, 87, 81
Offset: 1

Views

Author

Antti Karttunen, Feb 01 2016

Keywords

Links

Crossrefs

Cf. A000040, A000720, A080147, A080148, A267101, A267097, A267098, A267099.
Cf. also A267107 (a more recursed variant).

Formula

a(1) = 1; for n > 1, if prime(n) mod 4 = 1, then a(n) = A080148(A267097(n)), otherwise a(n) = A080147(A267098(n)).
Other identities. For all n >= 1:
a(n) = A000720(A267101(n)).

A354104 Dirichlet inverse of A354102.

Original entry on oeis.org

1, -1, -4, -1, -2, 4, -12, -1, -4, 2, -16, 4, -6, 12, 8, -1, -10, 4, -28, 2, 48, 16, -36, 4, -2, 6, -4, 12, -18, -8, -40, -1, 64, 10, 24, 4, -22, 28, 24, 2, -30, -48, -52, 16, 8, 36, -60, 4, -12, 2, 40, 6, -42, 4, 32, 12, 112, 18, -72, -8, -46, 40, 48, -1, 12, -64, -88, 10, 144, -24, -96, 4, -58, 22, 8, 28, 192, -24
Offset: 1

Views

Author

Antti Karttunen, May 18 2022

Keywords

Links

Crossrefs

Cf. A023900, A267099, A267101, A354102, A354105.

Programs

Formula

Multiplicative with a(p^e) = (1-q), where q = A267101(A000720(p)).
a(n) = A023900(A267099(n)).
a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d < n} A354102(n/d) * a(d).
a(n) = A354105(n) - A354102(n).
Showing 1-7 of 7 results.

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