A289626 -id:A289626 - cp's OEIS frontend

A290087 a(1) = 0; for n > 1, a(n) = A289626(A051953(n)).

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 5, 1, 5, 4, 5, 1, 5, 1, 5, 4, 5, 1, 8, 3, 4, 4, 8, 1, 6, 1, 8, 7, 4, 6, 13, 1, 8, 8, 13, 1, 8, 1, 13, 11, 13, 1, 17, 4, 8, 10, 11, 1, 11, 8, 17, 11, 8, 1, 18, 1, 17, 10, 17, 9, 12, 1, 11, 14, 12, 1, 21, 1, 10, 19, 21, 9, 10, 1, 21, 10, 11, 1, 21, 11, 18, 16, 21, 1, 18, 10, 21, 18, 21, 12, 25, 1, 28, 19, 21, 1, 19, 1, 28, 29

Offset: 1

Antti Karttunen, Aug 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A051953, A054571, A289626, A290086, A290085, A290086, A290088, A290089.

a(1) = 0; for n > 1, a(n) = A289626(A051953(n)).

A290088 a(n) = A289626(A000203(n)).

Original entry on oeis.org

1, 2, 2, 4, 2, 5, 5, 8, 7, 4, 5, 11, 4, 13, 13, 16, 4, 19, 8, 11, 17, 11, 13, 21, 16, 11, 21, 28, 8, 28, 17, 32, 21, 10, 21, 47, 10, 21, 28, 19, 11, 49, 18, 28, 19, 28, 21, 38, 29, 38, 28, 23, 10, 59, 28, 59, 40, 19, 21, 83, 16, 49, 53, 62, 28, 53, 25, 32, 49, 53, 28, 93, 20, 29, 38, 53, 49, 83, 40, 38, 60, 32, 28, 105, 29, 45, 59, 53, 19, 47, 53, 83, 63, 53, 59

Offset: 1

Antti Karttunen, Aug 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000203, A062401, A289626, A290085, A290086, A290087.

a(n) = A289626(A000203(n)).

A290085 a(n) = A289626(A000005(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 5, 2, 2, 2, 2, 1, 5, 1, 2, 2, 2, 2, 4, 1, 2, 2, 5, 1, 5, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 5, 2, 5, 2, 2, 1, 5, 1, 2, 2, 4, 2, 5, 1, 2, 2, 5, 1, 5, 1, 2, 2, 2, 2, 5, 1, 3, 3, 2, 1, 5, 2, 2, 2, 5, 1, 5, 2, 2, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 5, 5, 2, 1, 5, 1, 5, 2, 3, 1, 5, 2, 2, 2, 2, 2, 8

Offset: 1

Antti Karttunen, Aug 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A163109, A289626, A290086, A290087, A290088.

a(n) = A289626(A000005(n)).

A290086 a(n) = A289626(A000010(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 5, 2, 5, 5, 8, 2, 4, 5, 5, 3, 6, 5, 8, 5, 4, 5, 11, 5, 8, 8, 8, 8, 13, 5, 11, 4, 13, 8, 21, 5, 11, 8, 13, 6, 12, 8, 11, 8, 17, 13, 19, 4, 21, 13, 11, 11, 15, 8, 21, 8, 11, 17, 21, 8, 18, 17, 18, 13, 19, 13, 28, 11, 21, 11, 21, 13, 19, 17, 10, 21, 22, 13, 25, 11, 28, 21, 45, 13, 28, 18, 21, 12, 28, 17, 49, 11, 21, 21, 27, 17

Offset: 1

Antti Karttunen, Aug 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A010554, A077197, A277906, A289626, A290085, A290087, A290088.

a(n) = A289626(A000010(n)).

A289628 Compound filter (for the structure of the multiplicative group of integers modulo n & prime signature of n): a(n) = P(A289626(n), A101296(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 8, 9, 12, 14, 41, 19, 18, 27, 50, 35, 25, 63, 99, 54, 40, 65, 86, 102, 42, 90, 203, 134, 52, 101, 131, 135, 128, 152, 342, 228, 75, 250, 221, 230, 88, 250, 399, 275, 182, 299, 271, 295, 117, 324, 517, 323, 185, 403, 295, 377, 146, 462, 623, 525, 168, 495, 549, 527, 187, 698, 728, 663, 343, 629, 460, 738, 370, 702, 889, 740, 273, 523, 590, 858, 370

Offset: 1

Antti Karttunen, Jul 19 2017

nonn

Here, instead of A046523 and A289625 we use as the components of a(n) their rgs-versions A101296 and A289626 because of the latter sequence's more moderate growth rate.
For all i, j: a(i) = a(j) => A286160(i) = A286160(j).
For all i, j: a(i) = a(j) => A289622(i) = A289622(j).

Antti Karttunen, Table of n, a(n) for n = 1..65536

Cf. A000027, A046523, A101296, A286160, A289622, A289626, A286454.

(define (A289628 n) (* (/ 1 2) (+ (expt (+ (A289626 n) (A101296 n)) 2) (- (A289626 n)) (- (* 3 (A101296 n))) 2)))

a(n) = (1/2)*(2 + ((A289626(n)+A101296(n))^2) - A289626(n) - 3*A101296(n)).

A290083 Odd bisection of A289626.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 7, 8, 9, 10, 11, 12, 14, 10, 15, 16, 18, 19, 20, 19, 22, 23, 19, 24, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 27, 38, 39, 41, 42, 43, 44, 46, 47, 38, 48, 50, 38, 51, 52, 53, 54, 55, 48, 56, 57, 47, 58, 60, 61, 51, 62, 64, 65, 66, 48, 68, 69, 70, 71, 72, 64, 73, 74, 58, 71, 76, 77, 79, 80, 81, 82, 76, 66, 84, 71, 85, 86, 87, 71

Offset: 1

Antti Karttunen, Jul 19 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A037225, A278223, A289626.

(define (A290083 n) (A289626 (+ n n -1)))

a(n) = A289626(2n-1).

A305800 Filter sequence for a(prime) = constant sequences.

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, 96

Offset: 1

Antti Karttunen, Jun 14 2018

nonn

Restricted growth sequence transform of A239968.
In the following, A stands for this sequence, A305800, and S -> T (where S and T are sequence A-numbers) indicates that for all i, j: S(i) = S(i) => T(i) = T(j).
For example, the following implications hold:
  A -> A300247 -> A305897 -> A077462 -> A101296,
  A -> A290110 -> A300250 -> A101296.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000720, A239968.
Differs from A296073 for the first time at n=125, as a(125) = 96, while A296073(125) = 33.
Cf. also A305900, A305801, A295300, A289626 for other "upper level" filters.

Join[{1},Table[If[PrimeQ[n],2,1+n-PrimePi[n]],{n,2,150}]] (* Harvey P. Dale, Jul 12 2019 *)

A305800(n) = if(1==n,n,if(isprime(n),2,1+n-primepi(n)));

a(1) = 1; for n > 1, a(n) = 2 for prime n, and a(n) = 1+n-A000720(n) for composite n.

A289625 a(n) = prime factorization encoding of the structure of the multiplicative group of integers modulo n.

Original entry on oeis.org

1, 1, 4, 4, 16, 4, 64, 36, 64, 16, 1024, 36, 4096, 64, 144, 144, 65536, 64, 262144, 144, 576, 1024, 4194304, 900, 1048576, 4096, 262144, 576, 268435456, 144, 1073741824, 2304, 9216, 65536, 36864, 576, 68719476736, 262144, 36864, 3600, 1099511627776, 576, 4398046511104, 9216, 36864, 4194304, 70368744177664, 3600, 4398046511104, 1048576, 589824, 36864

Offset: 1

Antti Karttunen, Jul 17 2017

nonn

Here multiplicative group of integers modulo n is decomposed as a product of cyclic groups C_{k_1} x C_{k_2} x ... x C_{k_m}, where k_i divides k_j for i > j, like PARI-function znstar does. a(n) is then 2^{k_1} * 3^{k_2} * 5^{k_3} * ... * prime(m)^{k_m}.

			For n=5, the multiplicative group modulo 5 is isomorphic to C_4, which does not factorize to smaller subgroups, thus a(5) = 2^4 = 16.
For n=8, the multiplicative group modulo 8 is isomorphic to C_2 x C_2, thus a(8) = 2^2 * 3^2 = 36.
For n=15, the multiplicative group modulo 15 is isomorphic to C_4 x C_2, thus a(15) = 2^4 * 3^2 = 144.

Antti Karttunen, Table of n, a(n) for n = 1..1024
The PARI group, Catalogue of GP/PARI Functions: Arithmetic functions, function znstar
Eric Weisstein's World of Mathematics, Modulo Multiplication Group.
Wikipedia, Multiplicative group of integers modulo n

Cf. A000010, A002322, A046072.
Cf. A033948 (positions of terms that are powers of 2).
Cf. A289626 (rgs-transform of this sequence).

A289625(n) = { my(m=1,p=2,v=znstar(n)[2]); for(i=1,length(v),m *= p^v[i]; p = nextprime(p+1)); (m); };

A005361(a(n)) = A000010(n).
A072411(a(n)) = A002322(n).
A007814(a(n)) = A002322(n) for n > 2.
A001221(a(n)) = A046072(n) for n > 2.

A322592 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 for odd primes, and f(n) = A289625(n) for any other number.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 18 2018

nonn

For all i, j:
  a(i) = a(j) => A034380(i) = A034380(j),
  a(i) = a(j) => A104194(i) = A104194(j),
  a(i) = a(j) => A290084(i) = A290084(j).

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

Cf. A289625, A289626, A322587, A322588, A322589, A322591.

default(parisizemax,2^31);
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; };
A289625(n) = { my(m=1,p=2,v=znstar(n)[2]); for(i=1,length(v),m *= p^v[i]; p = nextprime(p+1)); (m); };
Aux322592(n) = if((n>2)&&isprime(n),0,A289625(n));
v322592 = rgs_transform(vector(up_to, n, Aux322592(n)));
A322592(n) = v322592[n];
for(n=1,up_to,write("b322592.txt", n, " ", A322592(n)));

A296233 Numbers k such that U(i) is not isomorphic to U(k) for all i < k, where U(k) is the multiplicative group of integers modulo k.

Original entry on oeis.org

1, 3, 5, 7, 8, 11, 13, 15, 17, 19, 21, 23, 24, 25, 29, 31, 32, 33, 35, 37, 40, 41, 43, 47, 51, 53, 55, 56, 57, 59, 61, 63, 65, 67, 69, 71, 73, 77, 79, 80, 81, 83, 85, 87, 88, 89, 91, 95, 96, 97, 101, 103, 104, 107, 109, 113, 115, 119, 120, 121, 123, 127, 128

Offset: 1

Jianing Song, Apr 29 2018

nonn

Numbers k such that A289626(i) < A289626(k) for all i < k.
All odd primes are in this sequence. This sequence contains almost all odd numbers.
Numbers k divisible by 2 but not by 4 are not members since U(k) is isomorphic to U(k/2) (i.e., 2, 6, 10, 14, ... are not terms).
Numbers k divisible by 4 but not by 3 or 8 are not members since U(k) is isomorphic to U(3/4*k) (i.e., 4, 20, 28, 44, ... are not terms).
Numbers k divisible by 12 but not by 24 or 36 are not members since U(k) is isomorphic to U(2/3*k) (i.e., 12, 60, 84, 132, ... are not terms).
Numbers k divisible by 9 but not by 7 or 27 are not members since U(k) is isomorphic to U(7/9*k) (i.e., 9, 18, 36, 45, 72, ... are not terms).
Numbers k divisible by 27 but not by 19 or 81 are not members since U(k) is isomorphic to U(19/27*k) (i.e., 27, 54, 108, 135, ... are not terms).
First term == 4 (mod 8) is 252.

			75 is not a term because U(55) and U(75) are both isomorphic to C_2 x C_20.
93 is not a term because U(77) and U(93) are both isomorphic to C_2 x C_30.
96 is a term because U(96) is isomorphic to C_2 x C_2 x C_8 and U(k) is not isomorphic to C_2 x C_2 x C_8 for all k < 96.

Jianing Song, Table of n, a(n) for n = 1..6527 (All terms <= 16384.)
Wikipedia, Multiplicative group of integers modulo n

Cf. A289625, A289626. A319928 is a subsequence.

isA296233(n) = !(sum(i=1,n-1,znstar(i)[2]==znstar(n)[2])) \\ Jianing Song, Oct 04 2018

a(n) = min{k : A289626(k) = n}. - Jianing Song, Jun 30 2018

cp's OEIS Frontend

A290087 a(1) = 0; for n > 1, a(n) = A289626(A051953(n)).

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A290088 a(n) = A289626(A000203(n)).

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A290085 a(n) = A289626(A000005(n)).

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A290086 a(n) = A289626(A000010(n)).

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A289628 Compound filter (for the structure of the multiplicative group of integers modulo n & prime signature of n): a(n) = P(A289626(n), A101296(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A290083 Odd bisection of A289626.

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A305800 Filter sequence for a(prime) = constant sequences.

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PARI

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A289625 a(n) = prime factorization encoding of the structure of the multiplicative group of integers modulo n.

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A322592 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 for odd primes, and f(n) = A289625(n) for any other number.

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PARI

A296233 Numbers k such that U(i) is not isomorphic to U(k) for all i < k, where U(k) is the multiplicative group of integers modulo k.