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 1-5 of 5 results.

A332808 Fully multiplicative with a(p) = A332806(A000720(p)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11, 14, 15, 16, 17, 18, 19, 20, 21, 26, 29, 24, 25, 22, 27, 28, 23, 30, 37, 32, 39, 34, 35, 36, 31, 38, 33, 40, 41, 42, 43, 52, 45, 58, 53, 48, 49, 50, 51, 44, 47, 54, 65, 56, 57, 46, 61, 60, 59, 74, 63, 64, 55, 78, 71, 68, 87, 70, 79, 72, 67, 62, 75, 76, 91, 66, 89, 80, 81, 82, 101
Offset: 1

Views

Author

Antti Karttunen, Feb 27 2020

Keywords

Links

Crossrefs

Cf. A000720, A332806, A108549 (fixed points), A332818, A332819.
Inverse permutation is A108548, from which this differs for the first time at n=67, where a(67) = 71, while A108548(67) = 73.

Programs

  • PARI
    up_to = 10000;
A332806list(up_to) = { my(v=vector(2), xs=Map(), lista=List([]), p,q,u); v[2] = 3; v[1] = 5; mapput(xs,1,1); mapput(xs,2,2); mapput(xs,3,3);  for(n=4,up_to, p = v[2-(n%2)]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[2-(n%2)] = q; mapput(xs,primepi(q),n)); for(i=1, oo, if(!mapisdefined(xs, i, &u), return(Vec(lista)), listput(lista, prime(u)))); };
v332806 = A332806list(up_to);
A332806(n) = v332806[n];
A332808(n) = { my(f=factor(n)); f[,1] = apply(A332806,apply(primepi,f[,1])); factorback(f); };

A332805 a(n) = A000720(A332806(n)).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 8, 10, 9, 12, 11, 13, 14, 16, 15, 18, 17, 20, 22, 19, 24, 26, 21, 23, 25, 28, 30, 27, 29, 32, 34, 31, 36, 33, 38, 35, 40, 42, 37, 44, 39, 46, 41, 43, 48, 50, 52, 54, 45, 47, 56, 49, 58, 51, 60, 53, 62, 55, 57, 64, 59, 66, 68, 61, 63, 70, 65, 72, 67, 69, 74, 76, 71, 78, 80, 73, 75, 77, 79, 82, 81, 84, 83, 86, 88, 85
Offset: 1

Views

Author

Antti Karttunen, Feb 27 2020

Keywords

Links

Crossrefs

Cf. A000040, A108546, A332806, A332807 (inverse permutation).
Fixed points are given by A000720(A108547(n)), n>=1.
Cf. also A267100.

Programs

  • PARI
    up_to = 10000;
A332805list(up_to) = { my(v=vector(2), xs=Map(), lista=List([]), p,q); v[2] = 3; v[1] = 5; mapput(xs,1,1); mapput(xs,2,2); mapput(xs,3,3);  for(n=4,up_to, p = v[2-(n%2)]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[2-(n%2)] = q; mapput(xs,primepi(q),n)); for(i=1, oo, if(!mapisdefined(xs, i, &u), return(Vec(lista)), listput(lista, u))); };
v332805 = A332805list(up_to);
A332805(n) = v332805[n];

Formula

For all n >= 1, A108546(a(n)) = A000040(n).

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)

A332816 a(n) = A156552(A332808(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 7, 6, 9, 32, 11, 16, 17, 10, 15, 64, 13, 128, 19, 18, 65, 512, 23, 12, 33, 14, 35, 256, 21, 2048, 31, 66, 129, 20, 27, 1024, 257, 34, 39, 4096, 37, 8192, 131, 22, 1025, 32768, 47, 24, 25, 130, 67, 16384, 29, 68, 71, 258, 513, 131072, 43, 65536, 4097, 38, 63, 36, 133, 524288, 259, 1026, 41, 2097152, 55, 262144, 2049, 26, 515
Offset: 1

Views

Author

Antti Karttunen, Feb 28 2020

Keywords

Links

Crossrefs

Cf. A332815 (inverse permutation).
Cf. A070939, A156552, A332808, A332894, A332899.

Programs

  • PARI
    up_to = 26927;
A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A332806list(up_to) = { my(v=vector(2), xs=Map(), lista=List([]), p,q,u); v[2] = 3; v[1] = 5; mapput(xs,1,1); mapput(xs,2,2); mapput(xs,3,3);  for(n=4,up_to, p = v[2-(n%2)]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[2-(n%2)] = q; mapput(xs,primepi(q),n)); for(i=1, oo, if(!mapisdefined(xs, i, &u), return(Vec(lista)), listput(lista, prime(u)))); };
v332806 = A332806list(up_to);
A332806(n) = v332806[n];
A332808(n) = { my(f=factor(n)); f[,1] = apply(A332806,apply(primepi,f[,1])); factorback(f); };
A332816(n) = A156552(A332808(n));

Formula

a(n) = A156552(A332808(n)).
For n > 1, A070939(a(n)) = A332894(n).
For n >= 1: (Start)
A080791(a(n)) = A332899(n)-1.
Among many identities given in A156552 that apply here as well we have for example the following ones:
A000120(a(n)) = A001222(n).
A069010(a(n)) = A001221(n).
A106737(a(n)) = A000005(n).
(End)

A332811 a(n) = A243071(A332808(n)).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 15, 4, 5, 14, 63, 12, 31, 30, 13, 8, 127, 10, 255, 28, 29, 126, 1023, 24, 11, 62, 9, 60, 511, 26, 4095, 16, 125, 254, 27, 20, 2047, 510, 61, 56, 8191, 58, 16383, 252, 25, 2046, 65535, 48, 23, 22, 253, 124, 32767, 18, 123, 120, 509, 1022, 262143, 52, 131071, 8190, 57, 32, 59, 250, 1048575, 508, 2045, 54, 4194303, 40, 524287, 4094, 21
Offset: 1

Views

Author

Antti Karttunen, Mar 05 2020

Keywords

Links

Crossrefs

Cf. A332817 (inverse permutation).
Cf. A054429, A243071, A332808, A332816, A332895, A332896, A332899.
Cf. also A332215.

Programs

  • PARI
    up_to = 26927;
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A243071(n) = if(n<=2, n-1, if(!(n%2), 2*A243071(n/2), 1+(2*A243071(A064989(n)))));
A332806list(up_to) = { my(v=vector(2), xs=Map(), lista=List([]), p,q,u); v[2] = 3; v[1] = 5; mapput(xs,1,1); mapput(xs,2,2); mapput(xs,3,3);  for(n=4,up_to, p = v[2-(n%2)]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[2-(n%2)] = q; mapput(xs,primepi(q),n)); for(i=1, oo, if(!mapisdefined(xs, i, &u), return(Vec(lista)), listput(lista, prime(u)))); };
v332806 = A332806list(up_to);
A332806(n) = v332806[n];
A332808(n) = { my(f=factor(n)); f[,1] = apply(A332806,apply(primepi,f[,1])); factorback(f); };
A332811(n) = A243071(A332808(n));

Formula

a(n) = A243071(A332808(n)).
For n > 1, a(n) = A054429(A332816(n)).
a(n) = A332895(n) + A332896(n).
a(n) = A332895(n) OR A332896(n) = A332895(n) XOR A332896(n).
A000120(a(n)) = A332899(n).
Showing 1-5 of 5 results.

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