A355833 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342671(i) = A342671(j) and A348717(i) = A348717(j) for all i, j >= 1.

1, 2, 3, 4, 3, 5, 3, 6, 4, 7, 3, 8, 3, 9, 10, 11, 3, 12, 3, 13, 14, 15, 3, 16, 4, 17, 18, 19, 3, 20, 3, 21, 22, 23, 10, 24, 3, 25, 26, 27, 3, 28, 3, 29, 8, 30, 3, 31, 4, 32, 33, 34, 3, 35, 14, 36, 37, 38, 3, 39, 3, 40, 41, 42, 43, 44, 3, 45, 46, 47, 3, 48, 3, 49, 50, 51, 10, 52, 3, 53, 11, 54, 3, 55, 26, 56, 57, 58, 3, 59, 14, 60, 61, 62, 33, 63, 3, 64, 65, 66

Offset: 1

Author

Antti Karttunen, Jul 20 2022

Keywords

• nonn

Comments

Restricted growth sequence transform of the ordered pair [A342671(n), A348717(n)].

Terms that occur in positions given by A349166 may occur only a finite number of times in this sequence. See also the array A355924.

Examples

			a(100) = a(3025) [= 66 as allotted by the rgs-transform] because 3025 = A003961(A003961(100)), therefore it is in the same column of the prime shift array A246278 as 100 is], and as A342671(100) = A342671(3025) = 7.
a(300) = a(21175) [= 200 as allotted by the rgs-transform], as 21175 = A003961(A003961(300)) and as A342671(300) = A342671(21175) = 7.
a(1215) = a(21875) [= 831 as allotted by the rgs-transform] because 21875 = A003961(1215), therefore it is in the same column of the prime shift array A246278 as 1215 is, and as A342671(1215) = A342671(21875) = 7.
a(2835) = a(48125) [= 1953 as allotted by the rgs-transform] because 48125 = A003961(2835) and as A342671(2835) = A342671(48125) = 11.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A003961, A246278, A342671, A348717, A349165, A349166, A355924.

Cf. also A305801, A305897, A355834, A355835.

Programs

PARI

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));
A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
Aux355833(n) = [A342671(n), A348717(n)];
v355833 = rgs_transform(vector(up_to,n,Aux355833(n)));
A355833(n) = v355833[n];