A108546 -id:A108546 - cp's OEIS frontend

A332894 a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A332819(2n+1)); also binary width of terms of A332816.

0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 6, 4, 5, 5, 4, 4, 7, 4, 8, 5, 5, 7, 10, 5, 4, 6, 4, 6, 9, 5, 12, 5, 7, 8, 5, 5, 11, 9, 6, 6, 13, 6, 14, 8, 5, 11, 16, 6, 5, 5, 8, 7, 15, 5, 7, 7, 9, 10, 18, 6, 17, 13, 6, 6, 6, 8, 20, 9, 11, 6, 22, 6, 19, 12, 5, 10, 7, 7, 24, 7, 5, 14, 26, 7, 8, 15, 10, 9, 21, 6, 6, 12, 13, 17, 9, 7, 23, 6, 8, 6, 25, 9, 28, 8, 6

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

a(n) tells how many iterations of A332893 are needed before 1 is reached, i.e., the distance of n from 1 in binary trees like A332815.

Each n > 0 occurs 2^(n-1) times in total.

Antti Karttunen, Table of n, a(n) for n = 1..65537

A left inverse of A000079 and A108546.

Cf. A070939, A252464, A332808, A332815, A332816, A332819, A332893, A332897, A332898, A332899.

A332894(n) = if(1==n,0,if(!(n%2),1+A332894(n/2),1+A332894(A332819(n))));

A332894(n) = if(1==n,0,1+A332894(A332893(n)));

a(n) = A252464(A332808(n)).
a(1) = 0, and for n > 1, a(n) = 1 + a(A332893(n)).
For n >= 1, a(A108546(n)) = n; for all n >= 0, a(2^n) = n.
For n > 1: (Start)
a(n) = 1 + a(n/2) if n is even, and a(n) = 1 + a(A332819(n)), if n is odd.
a(n) = A070939(A332816(n)).
a(n) >= A332899(n).
(End)

A332895 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A332893(n)) + [n == 1 (mod 4)].

Original entry on oeis.org

0, 1, 2, 2, 5, 4, 10, 4, 5, 10, 42, 8, 21, 20, 8, 8, 85, 10, 170, 20, 21, 84, 682, 16, 11, 42, 8, 40, 341, 16, 2730, 16, 85, 170, 16, 20, 1365, 340, 40, 40, 5461, 42, 10922, 168, 17, 1364, 43690, 32, 23, 22, 168, 84, 21845, 16, 80, 80, 341, 682, 174762, 32, 87381, 5460, 40, 32, 43, 170, 699050, 340, 1365, 32, 2796202

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000975, A108546, A332815, A332893, A332896, A332897.

Cf. also A292385.

A332895(n) = if(n<=2,n-1,2*A332895(A332893(n)) + (1==(n%4)));

a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A332893(n)) + [n == 1 (mod 4)].
For n > 1, a(2n) = 2*a(n).
For n >= 1, a(A108546(n)) = A000975(n); A000120(a(n)) = A332897(n).

A348744 Lexicographically earliest bijection from primes to odd primes where each prime of the form 4k+1 is mapped to the next larger prime that is of the same form.

Original entry on oeis.org

3, 5, 13, 7, 11, 17, 29, 19, 23, 37, 31, 41, 53, 43, 47, 61, 59, 73, 67, 71, 89, 79, 83, 97, 101, 109, 103, 107, 113, 137, 127, 131, 149, 139, 157, 151, 173, 163, 167, 181, 179, 193, 191, 197, 229, 199, 211, 223, 227, 233, 241, 239, 257, 251, 269, 263, 277, 271, 281, 293, 283, 313, 307, 311, 317, 337, 331, 349, 347

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

Terms of A002144 map each to the next term there, as: 5 -> 13 -> 17 -> 29 -> 37 -> 41, etc., and the remaining positions are filled with the terms of A002145: 3, 7, 11, 19, 23, 31, 43, etc., which gives the result that 2 is mapped to 3, 3 is mapped 5, and the rest of 4k+3 primes are fixed.

Cf. A000040, A002144, A002145, A348745, A348746.

Cf. also A108546, A332211.

up_to = 10000;
A348744list(up_to) = { my(v=vector(up_to), xs=Map(), i=2, p, q); mapput(xs,v[1]=3,1); for(n=2,up_to, p = prime(n); if(1==(p%4), for(k=1+n,oo,q=prime(k);if((1==(q%4))&&!mapisdefined(xs,q),v[n]=q;break)), while(mapisdefined(xs,prime(i)), i++); v[n] = prime(i)); mapput(xs,v[n],n)); (v); };
v348744 = A348744list(up_to);
A348744(n) = v348744[n];

a(n) = A348746(A000040(n)).

A111745 a(2k-1) = k-th prime congruent to 3 mod 4, a(2k) = k-th prime congruent to 1 mod 4.

Original entry on oeis.org

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, 263

Offset: 1

Jon Perry, Nov 19 2005

nonn

The graph shows the "race" between the two types of primes. - T. D. Noe, Nov 15 2006

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. bisections A002144, A002145.

Module[{prs=Prime[Range[70]],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]]] (* Harvey P. Dale, Apr 15 2012 *)

a(n)=A108546(n+1). [From R. J. Mathar, Aug 15 2008]

Edited, corrected and extended by Franklin T. Adams-Watters, Jul 19 2006

Corrected by T. D. Noe, Nov 15 2006

A108549 Fixed points for A108548.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 27, 28, 30, 32, 34, 35, 36, 38, 40, 41, 42, 43, 45, 48, 49, 50, 51, 54, 56, 57, 60, 63, 64, 68, 70, 72, 75, 76, 80, 81, 82, 84, 85, 86, 90, 95, 96, 98, 100, 102, 105, 108, 112, 114, 119, 120, 123, 125

Offset: 1

Reinhard Zumkeller, Jun 10 2005

nonn

A108548(a(n)) = a(n); multiplicative closure of A108547.

Cf. A108546.

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

Antti Karttunen, Feb 27 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..50507
Index entries for sequences that are permutations of the natural numbers

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

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];

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

A102261 a(n) = A002144(n) - A002145(n).

Original entry on oeis.org

2, 6, 6, 10, 14, 10, 10, 14, 14, 22, 26, 22, 26, 10, 30, 22, 26, 34, 30, 30, 30, 50, 42, 42, 46, 46, 50, 42, 42, 50, 46, 54, 42, 42, 42, 42, 38, 34, 30, 38, 14, 18, 18, 18, 46, 54, 62, 70, 78, 78, 90, 78, 66, 54, 70, 66, 62, 66, 58, 70, 66, 86, 98, 78, 78, 54, 70, 70, 78, 78

Offset: 1

Paul Curtz, Sep 06 2008

a(n) = A108546(2*n+1) - A108546(2*n).

Ivan Neretin, Table of n, a(n) for n = 1..30000

A002144 := proc(n) option remember ; if n = 1 then RETURN(5) ; fi; for a from procname(n-1)+2 do if isprime(a) and (a mod 4 = 1 ) then RETURN(a) ; fi; od: end; A002145 := proc(n) option remember ; if n = 1 then RETURN(3) ; fi; for a from procname(n-1)+2 do if isprime(a) and (a mod 4 = 3 ) then RETURN(a) ; fi; od: end; A102261 := proc(n) A002144(n)-A002145(n) ; end: seq(A102261(n),n=1..120) ; # R. J. Mathar, Feb 07 2009

nmax = 70; a1 = Select[Range[1, Prime[3*nmax], 4], PrimeQ]; a3 = Select[Range[3, Prime[3*nmax], 4], PrimeQ]; a[n_] := a1[[n]] - a3[[n]]; Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Dec 17 2013 *)

Edited by N. J. A. Sloane, Sep 06 2008

More terms from R. J. Mathar, Feb 07 2009

A339900 Lexicographically earliest permutation of odd primes such that A007814(a(n)-1) = 1+A007814(n), where A007814 gives the 2-adic valuation of n.

Original entry on oeis.org

3, 5, 7, 41, 11, 13, 19, 17, 23, 29, 31, 73, 43, 37, 47, 97, 59, 53, 67, 89, 71, 61, 79, 113, 83, 101, 103, 137, 107, 109, 127, 193, 131, 149, 139, 233, 151, 157, 163, 241, 167, 173, 179, 281, 191, 181, 199, 353, 211, 197, 223, 313, 227, 229, 239, 337, 251, 269, 263, 409, 271, 277, 283, 641, 307, 293, 311, 457, 331

Offset: 1

Antti Karttunen, Dec 25 2020

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001511, A007814, A065091.
Cf. A002145 (odd bisection), A007521 (quadrisection starting from 5), A105126, A105127, A105128, A105129, A105130, A105131, A105132.
Cf. also A108546, A111745.

A339900(n) = { my(lev=1+valuation(n,2), k=(1+(n>>(lev-1)))/2); forprime(p=3,,if(valuation(p-1,2)==lev, k--; if(!k, return(p)))); };

cp's OEIS Frontend

A332894 a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A332819(2n+1)); also binary width of terms of A332816.

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A332895 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A332893(n)) + [n == 1 (mod 4)].

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A348744 Lexicographically earliest bijection from primes to odd primes where each prime of the form 4k+1 is mapped to the next larger prime that is of the same form.

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PARI

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A111745 a(2k-1) = k-th prime congruent to 3 mod 4, a(2k) = k-th prime congruent to 1 mod 4.

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A108549 Fixed points for A108548.

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

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PARI

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A102261 a(n) = A002144(n) - A002145(n).

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Maple

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A339900 Lexicographically earliest permutation of odd primes such that A007814(a(n)-1) = 1+A007814(n), where A007814 gives the 2-adic valuation of n.

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PARI