A305793 -id:A305793 - cp's OEIS frontend

A305982 a(n) = Product_{d|n, dA305793(1+d)-1), where A305973(k) records the prime signature of 2k-1.

1, 2, 2, 4, 2, 8, 2, 12, 4, 8, 2, 48, 2, 20, 8, 24, 2, 32, 2, 120, 20, 8, 2, 288, 4, 28, 8, 120, 2, 320, 2, 120, 8, 20, 20, 576, 2, 20, 28, 480, 2, 320, 2, 264, 32, 8, 2, 4320, 10, 200, 20, 168, 2, 320, 8, 1200, 20, 8, 2, 11520, 2, 44, 80, 600, 28, 704, 2, 300, 8, 800, 2, 6912, 2, 44, 80, 300, 20, 448, 2, 31200, 40, 8, 2, 72000, 20, 20, 8

Offset: 1

Antti Karttunen, Jun 15 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A305973, A305983 (rgs-transform), A305984.

A305982(n) = { my(m=1); fordiv(n, d, if((dA305973(1+d)-1))); (m); }; \\ Needs also code from A305973.

a(n) = Product_{d|n, dA008578(A305793(1+d)).

A305984 a(n) = Product_{d|n, d>1} prime(A305793(1+d)-1), where A305973(k) records the prime signature of 2k-1.

Original entry on oeis.org

1, 2, 2, 6, 2, 8, 5, 12, 4, 20, 2, 72, 7, 20, 8, 60, 5, 32, 5, 120, 20, 44, 2, 432, 10, 28, 20, 300, 2, 320, 11, 300, 8, 50, 20, 576, 11, 50, 28, 3120, 2, 800, 5, 264, 80, 20, 5, 4320, 55, 200, 20, 1428, 2, 320, 20, 1200, 50, 44, 5, 17280, 5, 154, 80, 1500, 28, 1760, 19, 300, 8, 2000, 5, 17280, 11, 44, 80, 1650, 50, 448, 5, 78000, 40, 68, 2

Offset: 1

Antti Karttunen, Jun 15 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A305973, A305982, A305985 (rgs-transform).

A305984(n) = { my(m=1); fordiv(n, d, if((d>1), m *= prime(A305973(1+d)-1))); (m); }; \\ Needs also code from A305973.

a(n) = Product_{d|n, d>1} A008578(A305793(1+d)).

A286622 Restricted growth sequence computed for filter-sequence A278222, related to 1-runs in the binary representation of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 11 2017

When filtering sequences (by equivalence class partitioning), this sequence can be used instead of A278222, because for all i, j it holds that: a(i) = a(j) <=> A278222(i) = A278222(j).
For example, for all i, j: a(i) = a(j) => A000120(i) = A000120(j), and for all i, j: a(i) = a(j) => A001316(i) = A001316(j).
The sequence allots a distinct value for each distinct multiset formed from the lengths of 1-runs in the binary representation of n. See the examples. - Antti Karttunen, Jun 04 2017

			For n = 0, there are no 1-runs, thus the multiset is empty [], and it is allotted the number 1, thus a(0) = 1.
For n = 1, in binary also "1", there is one 1-run of length 1, thus the multiset is [1], which has not been encountered before, and a new number is allotted for that, thus a(1) = 2.
For n = 2, in binary "10", there is one 1-run of length 1, thus the multiset is [1], which was already encountered at n=1, thus a(2) = a(1) = 2.
For n = 3, in binary "11", there is one 1-run of length 2, thus the multiset is [2], which has not been encountered before, and a new number is allotted for that, thus a(3) = 3.
For n = 4, in binary "100", there is one 1-run of length 1, thus the multiset is [1], which was already encountered at n=1 for the first time, thus a(4) = a(1) = 2.
For n = 5, in binary "101", there are two 1-runs, both of length 1, thus the multiset is [1,1], which has not been encountered before, and a new number is allotted for that, thus a(5) = 4.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A278222, A000120, A001316.
Cf. A286552 (ordinal transform).
Cf. also A101296, A286581, A286589, A286597, A286599, A286600, A286602, A286603, A286605, A286610, A286619, A286621, A286626, A286378, A304101 for similarly constructed or related sequences.
Cf. also A305793, A305795.

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));
v286622 = rgs_transform(vector(1+65537, n, A278222(n-1)));
A286622(n) = v286622[1+n];

Example section added by Antti Karttunen, Jun 04 2017

A305795 Restricted growth sequence transform of A305794, a filter sequence constructed from the binary expansions of the divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 11, 13, 14, 15, 5, 16, 11, 17, 18, 19, 20, 21, 22, 19, 23, 24, 20, 25, 26, 27, 28, 10, 29, 30, 31, 19, 32, 33, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 23, 36, 35, 43, 44, 45, 32, 38, 46, 47, 39, 48, 49, 50, 51, 52, 11, 17, 53, 54, 20, 55, 31, 56, 57, 36, 58, 59, 39, 60, 61, 56, 35, 62, 63, 64, 65, 66, 35, 67

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A278222, A286622, A305794.

Cf. also A304105, A305793, A305815.

\\ 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; };
A305794(n) = { my(m=1); fordiv(n, d, if(d>1, m *= prime(A286622(d)-1))); (m); };
v305795 = rgs_transform(vector(up_to, n, A305794(n)));
A305795(n) = v305795[n];

For all i, j:
  a(i) = a(j) => A000005(i) = A000005(j).
  a(i) = a(j) => A007814(i) = A007814(j).
  a(i) = a(j) => A093653(i) = A093653(j).
  a(i) = a(j) => A154402(i) = A154402(j).
  a(i) = a(j) => A305436(i) = A305436(j).

A304103 Restricted growth sequence transform of A304102, a filter sequence related to the proper divisors of n expressed in Fibonacci number system.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 13 2018

nonn

For all i, j: a(i) = a(j) => b(i) = b(j), where b can be any of {A000005, A293435, A304095 or A300836} for example.

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

Cf. A293435, A304095, A300836, A304102.
Cf. also A300835, A304105, A305800.
Cf. A305793 (analogous filter for base 2).

\\ Needs also code from A304101.
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A304102(n) = { my(m=1); fordiv(n,d,if(dA304101(d)-1))); (m); };
write_to_bfile(1,rgs_transform(vector(up_to,n,A304102(n))),"b304103.txt");

A305792 a(n) = Product_{d|n, dA286622(d)-1).

Original entry on oeis.org

1, 2, 2, 4, 2, 12, 2, 8, 6, 20, 2, 72, 2, 28, 30, 16, 2, 180, 2, 200, 42, 44, 2, 432, 10, 44, 30, 392, 2, 11700, 2, 32, 66, 20, 70, 5400, 2, 44, 66, 2000, 2, 29988, 2, 968, 1950, 76, 2, 2592, 14, 1100, 30, 968, 2, 20700, 110, 5488, 66, 76, 2, 4563000, 2, 116, 3570, 64, 110, 21780, 2, 200, 114, 53900, 2, 162000, 2, 68, 4290, 968, 154, 82764, 2, 20000

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A008578, A278222, A286622, A305793 (rgs-transform), A305794.

Cf. also A293214, A304102.

A305792(n) = { my(m=1); fordiv(n,d,if(dA286622(d)-1))); (m); }; \\ Needs also code from A286622.

a(n) = Product_{d|n, dA008578(A286622(d)).

For all k >= 0, a(2^k) = 2^k.

A317943 Filter sequence constructed from the coefficients of the Stern polynomials B(d,t) collected for each proper divisor d of n; Restricted growth sequence transform of A317942.

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, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 40, 41, 42, 43, 2, 44, 2, 45, 46, 47, 48, 49, 2, 50, 51, 52, 2, 53, 2, 54, 55, 56, 57, 58, 2, 59, 60, 61, 2, 62, 63, 64, 65, 66, 2, 67, 68, 69, 70, 71, 72, 73, 2, 74, 75, 76, 2, 77, 2, 78, 79, 80, 2, 81, 2, 82, 83, 84, 2, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 86

Offset: 1

Antti Karttunen, Aug 12 2018

nonn

For all i, j: a(i) = a(j) => A317837(i) = A317837(j).

			Proper divisors of 115 are 1, 5 and 23 and proper divisors of 125 are 1, 5 and 25. The divisors 1 and 5 occur in both, while for the Stern polynomials B(23,t) and B(25,t) (see A125184) the nonzero coefficients are {1, 2, 3, 1} and {1, 3, 2, 1}, that is, they are equal as multisets, thus A286378(23) = A286378(25). From this follows that a(115) = a(125).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to Stern's sequences

Cf. A125184, A286378, A317837, A317942, A317945.
Cf. also A293217, A305793.
Differs from A305800 and A296073 for the first time at n=125, where a(125) = 86.

\\ Needs also code from A286378:
up_to = 65537;
A317942(n) = { my(m=1); fordiv(n,d,if(dA286378(d)-1))); (m); };
v317943 = rgs_transform(vector(up_to, n, A317942(n)));
A317943(n) = v317943[n];

A305813 Restricted growth sequence transform of A305812, a filter sequence constructed from the GF(2)[X]-factorization signatures of the proper divisors of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A278233, A305788, A305812, A305815.

Cf. also A305793.

\\ Needs also code from A305788:
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; };
A305812(n) = if(1==n,0, my(m=1); fordiv(n,d,if((d>1)&&(dA305788(d)-1))); (m));
v305813 = rgs_transform(vector(up_to, n, A305812(n)));
A305813(n) = v305813[n];

For all i, j:
  a(i) = a(j) => A000005(i) = A000005(j).
  a(i) = a(j) => A294881(i) = A294881(j).
  a(i) = a(j) => A294882(i) = A294882(j).

cp's OEIS Frontend

A305982 a(n) = Product_{d|n, dA305793(1+d)-1), where A305973(k) records the prime signature of 2k-1.

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A305984 a(n) = Product_{d|n, d>1} prime(A305793(1+d)-1), where A305973(k) records the prime signature of 2k-1.

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A286622 Restricted growth sequence computed for filter-sequence A278222, related to 1-runs in the binary representation of n.

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A305795 Restricted growth sequence transform of A305794, a filter sequence constructed from the binary expansions of the divisors of n.

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A304103 Restricted growth sequence transform of A304102, a filter sequence related to the proper divisors of n expressed in Fibonacci number system.

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A305792 a(n) = Product_{d|n, dA286622(d)-1).

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A317943 Filter sequence constructed from the coefficients of the Stern polynomials B(d,t) collected for each proper divisor d of n; Restricted growth sequence transform of A317942.

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A305813 Restricted growth sequence transform of A305812, a filter sequence constructed from the GF(2)[X]-factorization signatures of the proper divisors of n.

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