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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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

Views

Author

Jianing Song, Apr 29 2018

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A289625, A289626. A319928 is a subsequence.

Programs

Formula

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

A270492 a(n) = gcd(r) where r ranges over the orders of all subgroups whose direct product gives the multiplicative group modulo n.

Original entry on oeis.org

0, 0, 2, 2, 4, 2, 6, 2, 6, 4, 10, 2, 12, 6, 2, 2, 16, 6, 18, 2, 2, 10, 22, 2, 20, 12, 18, 2, 28, 2, 30, 2, 2, 16, 2, 2, 36, 18, 2, 2, 40, 2, 42, 2, 2, 22, 46, 2, 42, 20, 2, 2, 52, 18, 2, 2, 2, 28, 58, 2, 60, 30, 6, 2, 4, 2, 66, 2, 2, 2, 70, 2, 72, 36, 2, 2, 2, 2, 78, 2, 54, 40, 82, 2, 4, 42, 2, 2, 88, 2, 6, 2
Offset: 1

Views

Author

Joerg Arndt, Mar 18 2016

Keywords

Links

Crossrefs

Cf. A002322 (LCM over the orders of all subgroups), A052409, A289625, A290084.

Programs

  • PARI
    a(n)=gcd(znstar(n)[2]);

Formula

a(p) = p - 1 for odd primes p.
a(p^k) = phi(p^k) = (p-1)*p^(k-1) for odd primes p and k >= 1.
a(n) = A052409(A289625(n)). - Antti Karttunen, Aug 07 2017

Extensions

Terms a(1) and a(2) changed from 1 to 0 by Antti Karttunen, Aug 07 2017

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

Views

Author

Antti Karttunen, Jul 19 2017

Keywords

Comments

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).

Links

Crossrefs

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

Programs

Formula

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

A329896 Lexicographically earliest infinite sequence such that a(i) = a(j) => A219175(i) = A219175(j) and A322592(i) = A322592(j) for all i, j.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 2, 4, 5, 6, 2, 4, 2, 7, 8, 9, 2, 10, 2, 9, 11, 12, 2, 13, 14, 15, 16, 17, 2, 18, 2, 19, 20, 21, 22, 23, 2, 24, 25, 26, 2, 23, 2, 27, 28, 29, 2, 26, 30, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 26, 2, 39, 40, 41, 42, 43, 2, 44, 45, 46, 2, 47, 2, 48, 35, 49, 50, 51, 2, 52, 53, 54, 2, 47, 55, 56, 57, 58, 2, 51, 59, 60, 61, 62, 63, 64, 2, 65, 66, 67, 2, 68, 2, 69, 70
Offset: 1

Views

Author

Antti Karttunen, Dec 07 2019

Keywords

Links

Crossrefs

Cf. A002322, A219175, A289625, A322592, A327979, A329895.

Programs

  • PARI
    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; };
A219175(n) = (n%lcm(znstar(n)[2]));
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); };
Aux329896(n) = if((n>2)&&isprime(n),0,[A219175(n),A289625(n)]);
v329896 = rgs_transform(vector(up_to, n, Aux329896(n)));
A329896(n) = v329896[n];
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