A286378 - OEIS

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A286378

Restricted growth sequence computed for Stern-polynomial related filter-sequence A278243.

1, 2, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 8, 5, 9, 2, 10, 6, 11, 4, 12, 7, 13, 3, 13, 8, 14, 5, 15, 9, 16, 2, 17, 10, 18, 6, 19, 11, 20, 4, 21, 12, 22, 7, 23, 13, 24, 3, 24, 13, 25, 8, 26, 14, 27, 5, 28, 15, 29, 9, 30, 16, 31, 2, 32, 17, 33, 10, 34, 18, 35, 6, 36, 19, 37, 11, 38, 20, 39, 4, 40, 21, 41, 12, 42, 22, 43, 7, 44, 23, 45, 13, 46, 24, 47, 3, 47, 24, 48

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OFFSET

0,2

COMMENTS

Construction: we start with a(0)=1 for A278243(0)=1, and then after, for n > 0, we use the least unused natural number k for a(n) if A278243(n) has not been encountered before, otherwise [whenever A278243(n) = A278243(m), for some m < n], we set a(n) = a(m).

When filtering sequences (by equivalence class partitioning), this sequence (with its modestly sized terms) can be used instead of A278243, because for all i, j it holds that: a(i) = a(j) <=> A278243(i) = A278243(j).

For example, for all i, j: a(i) = a(j) => A002487(i) = A002487(j).

For pairs of distinct primes p, q for which a(p) = a(q) see comments in A317945. - Antti Karttunen, Aug 12 2018

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for sequences related to Stern's sequences

EXAMPLE

For n=1, A278243(1) = 2, which has not been encountered before, thus we allot for a(1) the least so far unused number, which is 2, thus a(1) = 2.

For n=2, A278243(2) = 2, which was already encountered as A278243(1), thus we set a(2) = a(1) = 2.

For n=3, A278243(3) = 6, which has not been encountered before, thus we allot for a(3) the least so far unused number, which is 3, thus a(3) = 3.

For n=23, A278243(23) = 2520, which has not been encountered before, thus we allot for a(23) the least so far unused number, which is 13, thus a(23) = 3.

For n=25, A278243(25) = 2520, which was already encountered at n=23, thus we set a(25) = a(23) = 13.

MATHEMATICA

a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; With[{nn = 100}, Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Table[Times @@ MapIndexed[Prime[First@#2]^#1 &, Sort[FactorInteger[#][[All, -1]], Greater]] - Boole[# == 1] &@ a@ n, {n, 0, nn}]] (* Michael De Vlieger, May 12 2017 *)

PROG

(PARI)

up_to = 65537;

rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences, invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences, invec[i], i); outvec[i] = u; u++ )); outvec; };

write_to_bfile(start_offset, vec, bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961

A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2))));

A278243(n) = A046523(A260443(n));

v286378 = rgs_transform(vector(up_to+1, n, A278243(n-1)));

A286378(n) = v286378[1+n];

CROSSREFS

Cf. A002487, A125184, A260443, A278243, A286377.

Cf. also A101296, A286603, A286605, A286610, A286619, A286621, A286622, A286626 for similarly constructed sequences.

Cf. also A317942, A317943, A317944, A317945.

Differs from A103391(1+n) for the first time at n=25, where a(25)=13, while A103391(26) = 14.

Sequence in context: A214126 A205378 A323889 * A331745 A103391 A331743

Adjacent sequences: A286375 A286376 A286377 * A286379 A286380 A286381

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 09 2017

STATUS

approved