A322023 -id:A322023 - cp's OEIS frontend

A322025 Ordinal transform of A322023.

1, 1, 2, 2, 3, 1, 4, 1, 3, 1, 5, 1, 6, 2, 1, 2, 7, 1, 8, 1, 1, 4, 9, 1, 10, 2, 5, 2, 11, 1, 12, 1, 1, 3, 2, 1, 13, 3, 3, 1, 14, 1, 15, 1, 2, 6, 16, 1, 7, 1, 3, 3, 17, 2, 2, 1, 2, 8, 18, 1, 19, 9, 1, 4, 1, 1, 20, 3, 4, 2, 21, 1, 22, 2, 2, 2, 3, 1, 23, 1, 24, 3, 25, 1, 1, 4, 5, 3, 26, 1, 1, 5, 3, 10, 5, 1, 27, 3, 2, 1, 28, 1, 29, 1, 1

Offset: 1

Antti Karttunen, Dec 01 2018

nonn

Positions where 1, 2, 3, 4, 5, ... occur for the first time are 1, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 187, 191, 193, ... Note that this is not a subsequence of A000961; for example, 187 = 11*17 is a semiprime.

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

Cf. A000010, A002322, A081373, A303753, A303755, A303756, A303777, A322023.

\\ Needs also code from A322023.
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
v322025 = ordinal_transform(v322023);
A322025(n) = v322025[n];

A322024 Lexicographically earliest such sequence a that a(i) = a(j) => A014197(i) = A014197(j) and A081373(i) = A081373(j), for all i, j. Here A081373(n) gives the number of k, 1 <= k <= n, with phi(k) = phi(n), while A014197(n) gives the number of integers m with phi(m) = n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 29 2018

nonn

Restricted growth sequence transform of the ordered pair [A014197(n), A081373(n)].

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

Cf. A014197, A081373, A305896, A319339, A322023.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
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; };
A014197(n, m=1) = { n==1 && return(1+(m<2)); my(p, q); sumdiv(n, d, if( d>=m && isprime(d+1), sum( i=0, valuation(q=n\d, p=d+1), A014197(q\p^i, p))))}; \\ From A014197
v081373 = ordinal_transform(vector(up_to,n,eulerphi(n)));
A081373(n) = v081373[n];
v322024 = rgs_transform(vector(up_to, n, [A014197(n), A081373(n)]));
A322024(n) = v322024[n];

cp's OEIS Frontend

A322025 Ordinal transform of A322023.

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