cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A286603 Restricted growth sequence computed for sigma, A000203.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, May 11 2017

Keywords

Comments

When filtering sequences (by equivalence class partitioning), this sequence can be used instead of A000203, because for all i, j it holds that: a(i) = a(j) <=> A000203(i) = A000203(j) <=> A286358(i) = A286358(j).
Note that the latter equivalence indicates that this is also the restricted growth sequence of A286358.

Examples

			Construction: we start with a(1)=1 for sigma(1)=1 (where sigma = A000203), and then after, for all n > 1, whenever the value of sigma(n) has not been encountered before, we set a(n) to the least natural number k not already in sequence among a(1) .. a(n-1), otherwise [whenever sigma(n) = sigma(m), for some m < n], we set a(n) = a(m), i.e., to the same value that was assigned to a(m).
For n=2, sigma(2) = 3, not encountered before, thus we allot for a(2) the least so far unused number, which is 2, thus a(2) = 2.
For n=3, sigma(3) = 4, not encountered before, thus we allot for a(3) the least so far unused number, which is 3, thus a(3) = 3.
For n=4, sigma(4) = 7, not encountered before, thus we allot for a(4) the least so far unused number, which is 4, thus a(4) = 4.
For n=5, sigma(5) = 6, not encountered before, thus we allot for a(5) the least so far unused number, which is 5, thus a(5) = 5.
For n=6, sigma(6) = 12, not encountered before, thus we allot for a(6) the least so far unused number, which is 6, thus a(6) = 6.
And this continues for n=7..10 because also for those n sigma obtains fresh new values, so here a(n) = n up to n = 10.
But then comes n=11, where sigma(11) = 12, a value which was already encountered at n=6 for the first time, thus we set a(11) = a(6) = 6.

Links

Crossrefs

Cf. A000203, A206036, A211656, A286358.
Cf. also A101296, A286605, A286610, A286619, A286621, A286622, A286626, A286378.

Programs

  • Mathematica
    With[{nn = 93}, Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Array[DivisorSigma[1, #] &, nn]] (* Michael De Vlieger, May 12 2017, Version 10 *)
  • PARI
    A000203(n) = sigma(n);
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])); }
write_to_bfile(1,rgs_transform(vector(10000,n,A000203(n))),"b286603.txt");

A286619 Restricted growth sequence computed for filter-sequence A278219, related to run-lengths in the binary representation of n.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 5, 2, 3, 6, 7, 4, 5, 6, 5, 2, 3, 6, 8, 6, 7, 9, 10, 4, 5, 11, 10, 6, 5, 6, 5, 2, 3, 6, 8, 6, 8, 12, 13, 6, 7, 14, 15, 9, 10, 12, 10, 4, 5, 11, 13, 11, 10, 14, 13, 6, 5, 11, 10, 6, 5, 6, 5, 2, 3, 6, 8, 6, 8, 12, 13, 6, 8, 16, 17, 12, 13, 16, 13, 6, 7, 14, 17, 14, 15, 18, 19, 9, 10, 20, 21, 12, 10, 12, 10, 4, 5, 11, 13, 11, 13, 20, 22, 11, 10
Offset: 0

Views

Author

Antti Karttunen, May 11 2017

Keywords

Comments

When filtering sequences (by equivalence class partitioning), this sequence can be used instead of A278219, because for all i, j it holds that: a(i) = a(j) <=> A278219(i) = A278219(j).
For example, for all i, j: a(i) = a(j) => A005811(i) = A005811(j). (The same is true for A073334, as it is a sequence computed from A005811).

Links

Crossrefs

Cf. A278219, A005811, A073334.
Cf. also A101296, A286603, A286605, A286610, A286621, A286622, A286626, A286378 for similarly constructed sequences.

Programs

  • Mathematica
    f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; g[n_] := If[n == 1, 1, Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]]; With[{nn = 99}, Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Table[g@ f[BitXor[n, Floor[n/2]], 1, 1], {n, 0, nn}]] (* Michael De Vlieger, May 12 2017, Version 10 *)
  • PARI
    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])); }
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A278222(n) = A046523(A005940(1+n));
A003188(n) = bitxor(n, n>>1);
A278219(n) = A278222(A003188(n));
write_to_bfile(0,rgs_transform(vector(65538,n,A278219(n-1))),"b286619.txt");

A286626 Restricted growth sequence computed for primorial base related filter-sequence A278226.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 2, 3, 3, 6, 5, 7, 4, 5, 5, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 16, 17, 18, 19, 2, 3, 3, 6, 5, 7, 3, 6, 6, 20, 7, 21, 5, 7, 7, 21, 9, 22, 11, 12, 12, 23, 14, 24, 16, 17, 17, 25, 19, 26, 4, 5, 5, 7, 8, 9, 5, 7, 7, 21, 9, 22, 8, 9, 9, 22, 27, 28, 13, 14, 14, 24, 29, 30, 18, 19, 19, 26, 31, 32, 10, 11, 11, 12, 13, 14, 11, 12, 12, 23, 14, 24
Offset: 0

Views

Author

Antti Karttunen, May 11 2017

Keywords

Comments

When filtering sequences (by equivalence class partitioning), this sequence (with its modestly sized terms) can be used instead of A278226, because for all i, j it holds that: a(i) = a(j) <=> A278226(i) = A278226(j).
For example, for all i, j: a(i) = a(j) => A276150(i) = A276150(j).

Links

Crossrefs

Cf. A278226, A276150.
Cf. also A101296, A286603, A286605, A286610, A286619, A286621, A286622, A286378 for similarly constructed sequences.

Programs

  • Mathematica
    b = MixedRadix[Reverse@ Prime@ Range@ 12]; f[n_] := Times @@ MapIndexed[Prime[#2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]] - Boole[n == 1]; With[{nn = 102}, Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Table[Function[k, f[Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]]@ IntegerDigits[n, b], {n, 0, nn}]] (* Michael De Vlieger, May 12 2017, Version 10.2 *)
  • PARI
    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]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A278226(n) = A046523(A276086(n));
write_to_bfile(0,rgs_transform(vector(30031,n,A278226(n-1))),"b286626.txt");

A286378 Restricted growth sequence computed for Stern-polynomial related filter-sequence A278243.

Original entry on oeis.org

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
Offset: 0

Views

Author

Antti Karttunen, May 09 2017

Keywords

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

Examples

			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.

Links

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.

Programs

  • 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 *)
  • 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];

A304101 Restricted growth sequence transform of A278222(A048679(n)).

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 4, 3, 2, 4, 4, 3, 5, 2, 4, 4, 4, 6, 3, 6, 5, 2, 4, 4, 4, 6, 4, 7, 6, 3, 6, 6, 5, 8, 2, 4, 4, 4, 6, 4, 7, 6, 4, 7, 7, 6, 9, 3, 6, 6, 6, 10, 5, 9, 8, 2, 4, 4, 4, 6, 4, 7, 6, 4, 7, 7, 6, 9, 4, 7, 7, 7, 11, 6, 11, 9, 3, 6, 6, 6, 10, 6, 11, 10, 5, 9, 9, 8, 12, 2, 4, 4, 4, 6, 4, 7, 6, 4, 7, 7, 6, 9, 4, 7, 7, 7
Offset: 0

Views

Author

Antti Karttunen, May 13 2018

Keywords

Comments

Positions of 2's is given by the positive Fibonacci numbers: 1, 2, 3, 5, 8, 13, 21, ..., that is, A000045(n) from n >= 2 onward.
Positions of 3's is given by Lucas numbers larger than 3: 4, 7, 11, 18, ..., that is, A000032(n) from n >= 3 onward.
Sequence allots a distinct value for each distinct multiset formed from the lengths of 1-runs in the binary representation of A048679(n). Compare to the scatter plot of A286622.

Links

Crossrefs

Cf. A000032, A000045, A003603, A007895, A035513, A048679, A278222, A304100, A304102, A304103, A304104, A304105.
Cf. also A286622 (compare the scatter-plots).

Programs

  • PARI
    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; };
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A106151(n) = if(n<=1, n, if(n%2, 1+(2*A106151((n-1)/2)), A106151(n>>valuation(n, 2))<<(valuation(n, 2)-1)));
A048679(n) = if(!n,n,A106151(2*A003714(n)));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
v304101 = rgs_transform(vector(1+up_to, n, A278222(A048679(n-1))));
A304101(n) = v304101[1+n];

A286602 Restricted growth sequence transform of A286601.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 3, 5, 6, 5, 3, 7, 4, 7, 4, 2, 4, 2, 4, 7, 4, 8, 9, 7, 6, 10, 9, 5, 11, 7, 3, 7, 11, 7, 3, 7, 11, 12, 13, 7, 6, 14, 15, 10, 9, 5, 9, 12, 4, 7, 4, 2, 9, 7, 4, 8, 13, 7, 9, 14, 4, 8, 13, 8, 16, 8, 13, 17, 4, 8, 13, 8, 13, 18, 13, 7, 19, 14, 9, 17, 4, 8, 9, 7, 4, 2, 4, 7, 13, 8, 4, 8, 9, 14, 13, 7, 11, 12, 13, 7, 11, 7, 3, 7
Offset: 0

Views

Author

Antti Karttunen, Jun 04 2017

Keywords

Comments

The scatter plot looks complex.

Links

Crossrefs

Cf. A193231, A234022.
Cf. A286581, A286589, A286597, A286599, A286600, A286601, A286617, A286619, A286622 for similarly formed sequences.

A286610 Restricted growth sequence computed for Euler totient function phi, A000010.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, May 11 2017

Keywords

Examples

			Construction: we start with a(1)=1 for phi(1)=1 (where phi = A000010), and then after, for all n > 1, whenever the value of phi(n) has not been encountered before, we set a(n) to the least natural number k not already in sequence among a(1) .. a(n-1), otherwise [whenever phi(n) = phi(m), for some m < n], we set a(n) = a(m), i.e., to the same value that was assigned to a(m).
For n=2, phi(2) = 1, which value was already encountered as phi(1), thus we set also a(2) = 1.
For n=3, phi(3) = 2, which has not been encountered before, thus we allot for a(3) the least so far unused number, which is 2, thus a(3) = 2.
For n=4, phi(4) = 2, which was already encountered as at n=3 for the first time, thus we set a(4) = a(3) = 2.
For n=5, phi(5) = 4, which has not been encountered before, thus we allot for a(5) the least so far unused number, which is now 3, thus a(5) = 3.

Links

Crossrefs

Cf. A000010, A210719 (positions of records, and also the first occurrence of each n).
Cf. also A101296, A286603, A286605, A286619, A286621, A286622, A286626, A286378 for similarly constructed sequences.

Programs

  • Mathematica
    With[{nn = 99}, Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Array[EulerPhi, nn]] (* Michael De Vlieger, May 12 2017, Version 10 *)
  • PARI
    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])); }
A000010(n) = eulerphi(n);
write_to_bfile(1,rgs_transform(vector(10000,n,A000010(n))),"b286610.txt");

A286531 Restricted growth sequence of A278531 (prime-signature of A163511).

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 7, 5, 5, 2, 8, 6, 9, 4, 10, 7, 7, 3, 9, 7, 11, 5, 7, 5, 5, 2, 12, 8, 13, 6, 14, 9, 9, 4, 14, 10, 15, 7, 10, 7, 7, 3, 13, 9, 15, 7, 15, 11, 11, 5, 9, 7, 11, 5, 7, 5, 5, 2, 16, 12, 17, 8, 18, 13, 13, 6, 19, 14, 20, 9, 14, 9, 9, 4, 18, 14, 21, 10, 21, 15, 15, 7, 14, 10, 15, 7, 10, 7, 7, 3, 17, 13, 20, 9, 21, 15, 15, 7, 20, 15
Offset: 0

Views

Author

Antti Karttunen, May 17 2017

Keywords

Links

Crossrefs

Cf. A046523, A163511, A286533, A286535, A286219, A286622, A305433, A305434.

Programs

  • PARI
    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])); }
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A278222(n) = A046523(A005940(1+n));
A054429(n) = ((3<<#binary(n\2))-n-1); \\ After M. F. Hasler, Aug 18 2014
A278531(n) = if(!n,1,A278222(A054429(n)));
write_to_bfile(0,rgs_transform(vector(65538,n,A278531(n-1))),"b286531.txt");

A286605 Restricted growth sequence computed for number of divisors, d(n) (A000005).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 6, 2, 5, 2, 5, 4, 4, 2, 7, 3, 4, 4, 5, 2, 7, 2, 5, 4, 4, 4, 8, 2, 4, 4, 7, 2, 7, 2, 5, 5, 4, 2, 9, 3, 5, 4, 5, 2, 7, 4, 7, 4, 4, 2, 10, 2, 4, 5, 11, 4, 7, 2, 5, 4, 7, 2, 10, 2, 4, 5, 5, 4, 7, 2, 9, 6, 4, 2, 10, 4, 4, 4, 7, 2, 10, 4, 5, 4, 4, 4, 10, 2, 5, 5, 8, 2, 7, 2, 7, 7, 4, 2, 10, 2, 7, 4, 9, 2, 7, 4, 5, 5, 4, 4
Offset: 1

Views

Author

Antti Karttunen, May 11 2017

Keywords

Comments

For all i, j: A101296(i) = A101296(j) => a(i) = a(j).
For all i, j: a(i) = a(j) <=> A000005(i) = A000005(j).

Links

Crossrefs

Cf. A000005, A007416 (positions of records, and also the first occurrence of each n).
Cf. also A101296, A286603, A286610, A286619, A286621, A286622, A286626, A286378.

Programs

  • Mathematica
    With[{nn = 119}, Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Array[DivisorSigma[0, #] &, nn]] (* Michael De Vlieger, May 12 2017, Version 10 *)
  • PARI
    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])); }
A000005(n) = numdiv(n);
write_to_bfile(1,rgs_transform(vector(10000,n,A000005(n))),"b286605.txt");

A305793 Restricted growth sequence transform of A305792, a filter sequence constructed from binary expansions of the proper divisors of n.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 11 2018

Keywords

Links

Crossrefs

Cf. A278222, A286622, A305792, A305795.
Cf. also A293215, A300833, A304103, A305813.

Programs

  • PARI
    \\ Needs also code from A286622:
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; };
A305792(n) = { my(m=1); fordiv(n,d,if(dA286622(d)-1))); (m); };
v305793 = rgs_transform(vector(up_to, n, A305792(n)));
A305793(n) = v305793[n];

Formula

For all i, j:
a(i) = a(j) => A000005(i) = A000005(j).
a(i) = a(j) => A292257(i) = A292257(j).
a(i) = a(j) => A305426(i) = A305426(j).
a(i) = a(j) => A305435(i) = A305435(j).
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