A348745 -id:A348745 - cp's OEIS frontend

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.

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)).

A348747 Fully multiplicative with a(2) = 1, a(3) = 2, a(5) = 3, a(A002144(1+n)) = A002144(n) and a(A002145(1+n)) = a(A002145(1+n)) for all n >= 1, where A002144 and A002145 give the primes of the form 4k+1 and 4k+3 respectively.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 7, 1, 4, 3, 11, 2, 5, 7, 6, 1, 13, 4, 19, 3, 14, 11, 23, 2, 9, 5, 8, 7, 17, 6, 31, 1, 22, 13, 21, 4, 29, 19, 10, 3, 37, 14, 43, 11, 12, 23, 47, 2, 49, 9, 26, 5, 41, 8, 33, 7, 38, 17, 59, 6, 53, 31, 28, 1, 15, 22, 67, 13, 46, 21, 71, 4, 61, 29, 18, 19, 77, 10, 79, 3, 16, 37, 83, 14, 39, 43, 34, 11, 73

Offset: 1

Antti Karttunen, Nov 02 2021

Left inverse of A348746.
Cf. A000265, A000720, A002144, A002145, A348745.
Cf. also A064989, A332819 for similar maps.

A348747(n) = { my(f=factor(n)); for(k=1,#f~, if(f[k,1]<=3, f[k,1]--, if(5==f[k,1], f[k,1]=3, if(1==(f[k,1]%4), forstep(i=primepi(f[k,1])-1,0,-1,if(1==(prime(i)%4), f[k,1]=prime(i); break)))))); factorback(f); };

Fully multiplicative with a(p) = A348745(A000720(p)).
a(A348746(n)) = n.
a(2n) = a(A000265(n)) = a(n).

cp's OEIS Frontend

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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