A296233 -id:A296233 - cp's OEIS frontend

A319928 Numbers k such that there is no other m such that (Z/mZ)* is isomorphic to (Z/kZ), where (Z/kZ) is the multiplicative group of integers modulo k.

24, 32, 80, 96, 120, 128, 160, 168, 240, 252, 256, 264, 324, 384, 400, 408, 416, 456, 480, 504, 512, 544, 552, 640, 648, 672, 696, 768, 840, 928, 1040, 1088, 1128, 1272, 1280, 1312, 1320, 1360, 1408, 1416, 1504, 1536, 1632, 1696, 1704, 1840, 1848, 1896, 1920, 1992

Offset: 1

Jianing Song, Oct 03 2018

nonn

Numbers such that A317993(k) = 1.
To find such k, it's sufficient to check for A015126(k) <= m <= A028476(k).
This is a subsequence of A296233. As a result, all members in this sequence should not satisfy any congruence mentioned there. Specially, all terms here are divisible by 4.
There are only 218 terms <= 10000 and 396 terms <= 20000.

			(Z/24Z)* = C_2 X C_2 X C_2, and there is no other m such that (Z/mZ)* = C_2 X C_2 X C_2, so 24 is a term.
(Z/96Z)* = C_2 X C_2 X C_8, and there is no other m such that (Z/mZ)* = C_2 X C_2 X C_8, so 24 is a term.

Jianing Song, Table of n, a(n) for n = 1..396 (all terms <= 20000)
Wikipedia, Multiplicative group of integers modulo n

Cf. A015126, A028476, A057635, A296233, A317993.

b(n) = my(i=0, search_max = A057635(eulerphi(n))); for(j=eulerphi(n)+1, search_max, if(znstar(j)[2]==znstar(n)[2], i++)); i \\ search_max is the largest k such that phi(k) = phi(n). See A057635 for its program
isA319928(n) = if(n>2, b(n)==1, 0)

A320045 Smallest k such that (Z/kZ)* is isomorphic to (Z/nZ)*.

Original entry on oeis.org

1, 1, 3, 3, 5, 3, 7, 8, 7, 5, 11, 8, 13, 7, 15, 15, 17, 7, 19, 15, 21, 11, 23, 24, 25, 13, 19, 21, 29, 15, 31, 32, 33, 17, 35, 21, 37, 19, 35, 40, 41, 21, 43, 33, 35, 23, 47, 40, 43, 25, 51, 35, 53, 19, 55, 56, 57, 29, 59, 40, 61, 31, 63, 51, 65, 33, 67, 51, 69, 35

Offset: 1

Jianing Song, Oct 04 2018

nonn

a(n) = n iff n is a term in A296233.
Most terms are odd. Among the first 10000 terms there are 8837 odd ones. The even terms are divisible by 4 because (Z/kZ)* is isomorphic to (Z/(2k)Z)* for odd k.
A015126(n) <= a(n) <= n.

			The solutions to (Z/kZ)* = C_6 are k = 7, 9, 14 and 18, so a(7) = a(9) = a(14) = a(18) = 7.
The solutions to (Z/kZ)* = C_2 X C_20 are k = 55, 75, 100, 110 and 150, so a(55) = a(75) = a(100) = a(110) = a(150) = 55.
The solutions to (Z/kZ)* = C_2 X C_12 are k = 35, 39, 45, 52, 70, 78 and 90, so a(35) = a(39) = a(45) = a(52) = a(70) = a(78) = a(90) = 35.

Jianing Song, Table of n, a(n) for n = 1..10000
Wikipedia, Multiplicative group of integers modulo n

Cf. A015126, A296233, A320046.

a(n) = my(i=eulerphi(n)); while(znstar(i)[2]!=znstar(n)[2], i++); i

A372755 Terms of A319928 that are congruent to 4 modulo 8: Numbers k == 4 (mod 8) such that there is no other m such that (Z/mZ)* is isomorphic to (Z/kZ), where (Z/kZ) is the multiplicative group of integers modulo k.

Original entry on oeis.org

252, 324, 2052, 2268, 3276, 4788, 6156, 7452, 7812, 10836, 12348, 14364, 14868, 15228, 16884, 17172, 18396, 19908, 20916, 22572, 23652, 24444, 25596, 25956, 26244, 26892, 26964, 31428, 34668, 35028, 35316, 38052, 38988, 41076, 43092, 43596, 45108, 48636, 48924, 52812, 56052, 56196, 57204

Offset: 1

Jianing Song, May 12 2024

nonn

This is a subsequence of A296233. As a result, all members in this sequence should not satisfy any congruence mentioned there, so terms of A319928 that are congruent to 4 modulo 8 are rare. In particular, all terms are divisible by 252 = 4 * 3^2 * 7, 324 = 4 * 3^4 or 2052 = 4 * 3^3 * 19.

			252 is a term because there is no other k such that (Z/kZ)* = (Z/252Z)* = C_2 X C_6 X C_6.
324 is a term because there is no other k such that (Z/kZ)* = (Z/324Z)* = C_2 X C_54.
2052 is a term because there is no other k such that (Z/kZ)* = (Z/2052Z)* = C_2 X C_18 X C_18.

Jianing Song, Table of n, a(n) for n = 1..145
Wikipedia, Multiplicative group of integers modulo n

Cf. A296233, A319928.

isA372755(n) = (n % 8 == 4) && isA319928(n)

cp's OEIS Frontend

A319928 Numbers k such that there is no other m such that (Z/mZ)* is isomorphic to (Z/kZ), where (Z/kZ) is the multiplicative group of integers modulo k.

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A320045 Smallest k such that (Z/kZ)* is isomorphic to (Z/nZ)*.

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A372755 Terms of A319928 that are congruent to 4 modulo 8: Numbers k == 4 (mod 8) such that there is no other m such that (Z/mZ)* is isomorphic to (Z/kZ), where (Z/kZ) is the multiplicative group of integers modulo k.

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cp's OEIS Frontend

A319928 Numbers k such that there is no other m such that (Z/mZ)* is isomorphic to (Z/kZ)*, where (Z/kZ)* is the multiplicative group of integers modulo k.

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Author

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Comments

Examples

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Crossrefs

Programs

PARI

A320045 Smallest k such that (Z/kZ)* is isomorphic to (Z/nZ)*.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A372755 Terms of A319928 that are congruent to 4 modulo 8: Numbers k == 4 (mod 8) such that there is no other m such that (Z/mZ)* is isomorphic to (Z/kZ)*, where (Z/kZ)* is the multiplicative group of integers modulo k.

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Author

Keywords

Comments

Examples

Links

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Programs

PARI

A319928 Numbers k such that there is no other m such that (Z/mZ)* is isomorphic to (Z/kZ), where (Z/kZ) is the multiplicative group of integers modulo k.

A372755 Terms of A319928 that are congruent to 4 modulo 8: Numbers k == 4 (mod 8) such that there is no other m such that (Z/mZ)* is isomorphic to (Z/kZ), where (Z/kZ) is the multiplicative group of integers modulo k.