A366275 -id:A366275 - cp's OEIS frontend

A366283 a(n) = gcd(n, A366275(n)), where A366275 is the Cat's tongue permutation.

1, 1, 2, 3, 4, 1, 6, 1, 8, 9, 2, 1, 12, 1, 2, 1, 16, 1, 18, 1, 4, 3, 2, 1, 24, 25, 2, 1, 4, 1, 2, 1, 32, 3, 2, 5, 36, 1, 2, 3, 8, 1, 6, 1, 4, 3, 2, 1, 48, 1, 50, 1, 4, 1, 2, 55, 8, 1, 2, 1, 4, 1, 2, 1, 64, 1, 6, 1, 4, 3, 10, 1, 72, 1, 2, 15, 4, 7, 6, 1, 16, 3, 2, 1, 12, 5, 2, 3, 8, 1, 6, 7, 4, 3, 2, 1, 96, 1, 2, 1, 100

Offset: 0

Antti Karttunen, Oct 07 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A057889, A163511, A366275, A366282, A366284, A366285, A366286.

Differs from related A364255 for the first time at n=25, where a(25) = 25, while A364255(25) = 5.

A366283(n) = gcd(n,A366275(n)); \\ Uses the program given in A366275.

a(n) = gcd(n,A366282(n)) = gcd(A366275(n),A366282(n)).

a(n) = n / A366284(n) = A366275(n) / A366285(n).

A366282 a(n) = A366275(n) - n, where A366275 is the Cat's tongue permutation.

Original entry on oeis.org

1, 1, 2, 0, 4, 4, 0, -2, 8, 18, 8, 4, 0, 12, -4, -8, 16, 64, 36, 26, 16, 54, 8, -2, 0, 100, 24, 8, -8, 20, -16, -20, 32, 210, 128, 100, 72, 188, 52, 24, 32, 334, 108, 62, 16, 102, -4, -14, 0, 576, 200, 124, 48, 192, 16, 0, -16, 286, 40, 18, -32, 60, -40, -50, 64, 664, 420, 338, 256, 606, 200, 118, 144, 1052, 376

Offset: 0

Antti Karttunen, Oct 07 2023

sign

Antti Karttunen, Table of n, a(n) for n = 0..12288
Index entries for sequences related to binary expansion of n

Cf. A057889, A163511, A366275, A366277 (positions of 0's), A366283.

Cf. also A364258.

A366282(n) = (A366275(n)-n); \\ Uses the program given in A366275.

A366285 a(n) = A366275(n) / gcd(n, A366275(n)), where A366275 is the Cat's tongue permutation.

Original entry on oeis.org

1, 2, 2, 1, 2, 9, 1, 5, 2, 3, 9, 15, 1, 25, 5, 7, 2, 81, 3, 45, 9, 25, 15, 21, 1, 5, 25, 35, 5, 49, 7, 11, 2, 81, 81, 27, 3, 225, 45, 21, 9, 375, 25, 105, 15, 49, 21, 33, 1, 625, 5, 175, 25, 245, 35, 1, 5, 343, 49, 77, 7, 121, 11, 13, 2, 729, 81, 405, 81, 225, 27, 189, 3, 1125, 225, 21, 45, 63, 21, 99, 9, 625, 375, 525

Offset: 0

Antti Karttunen, Oct 07 2023

Denominator of n / A366275(n).

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A057889, A163511, A366275, A366282, A366283, A366284 (numerators), A366286 (rgs-transform).

Cf. also A364492.

A366285(n) = { my(u=A366275(n)); (u/gcd(n,u)); }; \\ Uses the program given in A366275.

a(n) = A366275(n) / A366283(n) = A366275(n) / gcd(n, A366275(n))

A366286 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366285(i) = A366285(j) for all i, j >= 0, where A366285(n) is the denominator of n / A366275(n).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 07 2023

Restricted growth sequence transform of A366285.

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

Cf. A057889, A163511, A366275, A366283, A366285.

Cf. also A365393, A365431 (compare the scatter plots).

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; };
A366285(n) = { my(u=A366275(n)); (u/gcd(n,u)); }; \\ Uses also the program given in A366275.
v366286 = rgs_transform(vector(1+up_to,n,A366285(n-1)));
A366286(n) = v366286[1+n];

A366284 a(n) = n / gcd(n, A366275(n)), where A366275 is the Cat's tongue permutation.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 7, 1, 1, 5, 11, 1, 13, 7, 15, 1, 17, 1, 19, 5, 7, 11, 23, 1, 1, 13, 27, 7, 29, 15, 31, 1, 11, 17, 7, 1, 37, 19, 13, 5, 41, 7, 43, 11, 15, 23, 47, 1, 49, 1, 51, 13, 53, 27, 1, 7, 57, 29, 59, 15, 61, 31, 63, 1, 65, 11, 67, 17, 23, 7, 71, 1, 73, 37, 5, 19, 11, 13, 79, 5, 27, 41, 83, 7, 17, 43, 29, 11

Offset: 0

Antti Karttunen, Oct 07 2023

Numerator of n / A366275(n).

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A057889, A163511, A366275, A366282, A366283, A366285 (denominators), A366286.

Cf. also A364491.

A366284(n) = (n/gcd(n,A366275(n))); \\ Uses the program given in A366275.

a(n) = n / A366283(n) = n / gcd(n, A366275(n))

A366277 Fixed points of map n -> A366275(n).

Original entry on oeis.org

3, 6, 12, 24, 48, 55, 96, 110, 192, 220, 384, 440, 768, 880, 1536, 1760, 3072, 3520, 6144, 7040, 12288, 14080, 24576, 28160, 49152, 56320, 98304, 112640, 196608, 225280, 393216, 450560, 786432, 901120, 1572864, 1802240, 3145728, 3604480, 6291456, 7208960, 12582912, 14417920, 25165824, 28835840, 50331648, 57671680

Offset: 1

Antti Karttunen, Oct 06 2023

nonn

Equally, fixed points of map n -> A366276(n).

If n is a term, then 2*n is also a term, and vice versa, thus the sequence is wholly determined by its odd terms: 3, 55. Are there any others?

Cf. A007283 (subsequence), A057889, A163511, A366275, A366276.

isA366277(n) = (n==A366275(n)); \\ Uses also the program given in A366275.

A366278 Perfect powers k such that A052409(k) is equal to A052409(A366275(k)).

Original entry on oeis.org

196, 2116, 2500, 3136, 3844, 9409, 15376, 33856, 37636, 40000, 49729, 50176, 58564, 64516, 148225, 150544, 198916, 229441, 231361, 237169, 246016, 255025, 528529, 541696, 543169, 555025, 592900, 602176, 628849, 640000, 654481, 790321, 795664, 801025, 802816, 917764, 925444, 937024, 948676, 984064, 1020100, 1032256

Offset: 1

Antti Karttunen, Oct 06 2023

nonn

Numbers k such that A052409(k) > 1 and A052409(k) = A366281(k).

Conjecture: all terms are squares, and no higher powers occur.

Antti Karttunen, Table of n, a(n) for n = 1..408

Subsequence of A001597, and also by conjecture, of A153158 (thus also of A000290).
Cf. A052409, A366275, A366281.
Cf. also A365805, A365808.

A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2)));
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A366275(n) = A163511(A057889(n));
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); };
isA366278(n) = (A052409(n)>1 && A052409(n)==A052409(A366275(n)));

A366279 The least number with same prime signature as A366275, where A366275(n) = A163511(A057889(n)).

Original entry on oeis.org

1, 2, 4, 2, 8, 4, 6, 2, 16, 8, 12, 6, 12, 4, 6, 2, 32, 16, 24, 12, 36, 12, 30, 6, 24, 8, 12, 6, 12, 4, 6, 2, 64, 32, 48, 24, 72, 36, 60, 12, 72, 24, 60, 30, 60, 12, 30, 6, 48, 16, 24, 12, 36, 12, 30, 6, 24, 8, 12, 6, 12, 4, 6, 2, 128, 64, 96, 48, 144, 72, 120, 24, 216, 72, 180, 60, 180, 36, 60, 12, 144, 48, 120, 60

Offset: 0

Antti Karttunen, Oct 06 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383

Cf. A046523, A057889, A163511, A278531, A366275, A366280 (rgs-transform).

Cf. also A286601, A366261.

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2)));
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A366275(n) = A163511(A057889(n));
A366279(n) = A046523(A366275(n));

a(n) = A046523(A366275(n)) = A046523(A163511(A057889(n))).

a(n) = A278531(A057889(n)).

A366281 a(n) = largest exponent m for which a representation of the form A366275(n) = k^m exists (for some k). a(0) = 0 by convention.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 1, 1, 4, 3, 1, 1, 1, 2, 1, 1, 5, 4, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 6, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 7, 6, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 0

Antti Karttunen, Oct 06 2023

nonn

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

Cf. A052409, A057889, A365805, A366275, A366278 [where a(n) = A052409(n)].

A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2)));
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A366275(n) = A163511(A057889(n));
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); };
A366281(n) = A052409(A366275(n));

a(n) = A052409(A366275(n)).

a(n) = A365805(A057889(n)).

A366258 Dirichlet inverse of A366283, where A366283(n) = gcd(n, A366275(n)).

Original entry on oeis.org

1, -2, -3, 0, -1, 6, -1, 0, 0, 2, -1, 0, -1, 2, 5, 0, -1, 0, -1, 0, 3, 2, -1, 0, -24, 2, 26, 0, -1, -10, -1, 0, 3, 2, -3, 0, -1, 2, 3, 0, -1, -6, -1, 0, -6, 2, -1, 0, 0, 48, 5, 0, -1, -52, -53, 0, 5, 2, -1, 0, -1, 2, 8, 0, 1, -6, -1, 0, 3, 6, -1, 0, -1, 2, 128, 0, -5, -6, -1, 0, -78, 2, -1, 0, -3, 2, 3, 0, -1, 12

Offset: 1

Antti Karttunen, Oct 07 2023

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A366275, A366283, A366259 (rgs-transform).

Cf. also A364257.

\\ Needs also the program given in A366275:
A366283(n) = gcd(n,A366275(n));
memoA366258 = Map();
A366258(n) = if(1==n,1,my(v); if(mapisdefined(memoA366258,n,&v), v, v = -sumdiv(n,d,if(dA366283(n/d)*A366258(d),0)); mapput(memoA366258,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA366283(n/d) * a(d).

cp's OEIS Frontend

A366283 a(n) = gcd(n, A366275(n)), where A366275 is the Cat's tongue permutation.

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A366282 a(n) = A366275(n) - n, where A366275 is the Cat's tongue permutation.

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A366285 a(n) = A366275(n) / gcd(n, A366275(n)), where A366275 is the Cat's tongue permutation.

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A366286 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366285(i) = A366285(j) for all i, j >= 0, where A366285(n) is the denominator of n / A366275(n).

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A366284 a(n) = n / gcd(n, A366275(n)), where A366275 is the Cat's tongue permutation.

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A366277 Fixed points of map n -> A366275(n).

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A366278 Perfect powers k such that A052409(k) is equal to A052409(A366275(k)).

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A366279 The least number with same prime signature as A366275, where A366275(n) = A163511(A057889(n)).

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A366281 a(n) = largest exponent m for which a representation of the form A366275(n) = k^m exists (for some k). a(0) = 0 by convention.

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