A114874 -id:A114874 - cp's OEIS frontend

A305236 Numbers n such that the multiplicative group of integers modulo n is isomorphic to C_m X C_m, m > 1.

8, 12, 63, 126, 513, 1026, 2107, 4214, 12625, 25250, 26533, 39609, 53066, 79218, 355023, 710046, 3190833, 4457713, 6381666, 8915426, 19854847, 38463283, 39709694, 76926566, 242138449, 370634743, 484276898, 516465451, 574336561, 701607583, 741269486, 1032930902, 1148673122, 1380336193, 1403215166, 2324581983, 2760672386, 4649163966, 4882890625, 6174434113, 9765781250

Offset: 1

Jianing Song, Jun 19 2018

nonn

Note that 24 is only number k such that the multiplicative group of integers modulo k is isomorphic to C_m X C_m X C_m, m > 1.
The number of elements in the multiplicative group of integers modulo a(n) of order d is A007434(d), whenever d is divisible by A002322(a(n)).
The corresponding m (=A002322(a(n))) are 2, 2, 6, 6, 18, 18, 42, 42, 100, 100, 156, 162, 156, 162, 486, 486, 1458, 2028, 1458, 2028, ... Each term in A114874, except for those of the form 2^k, k >= 2, occurs exactly twice in this list.
Numbers k such that A046072(k) = 2 and A316089(k) = 1. - Jianing Song, Sep 15 2018
Except for 8 and 12, these are numbers of the form p^e*((p-1)*p^(e-1) + 1) or 2*p^e*((p-1)*p^(e-1) + 1) where p is an odd prime and (p-1)*p^(e-1) + 1 is prime. - Jianing Song, Apr 13 2019

			The multiplicative group of integers modulo 63 is isomorphic to C_6 X C_6. There are A007434(1) = 1 element of order 1, A007434(2) = 3 elements of order 2, A007434(3) = 8 elements of order 3, A007434(6) = 24 elements of order 6 modulo 63.
The multiplicative group of integers modulo 513 is isomorphic to C_18 X C_18. There are A007434(1) = 1 element of order 1, A007434(2) = 3 elements of order 2, A007434(3) = 8 elements of order 3, A007434(6) = 24 elements of order 6, A007434(9) = 72 elements of order 9, A007434(18) = 216 elements of order 18 modulo 513.

Jianing Song, Table of n, a(n) for n = 1..287 (all terms below 10^16)
Wikipedia, Multiplicative group of integers modulo n.

Cf. A114874.

Odd terms are given by A307527.

for(n=1,10^7,if(#znstar(n)[2]==2 && znstar(n)[2][1]==znstar(n)[2][2], print1(n, ", "))) \\ Jianing Song, Sep 15 2018

the_first_entries(nn) = my(u=[]); for(n=2, sqrt(nn), my(v=factor(n), d=#v[, 1], p=v[d, 1], e=v[d, 2]); if(isprime(n+1) && p!=2 && n==(p-1)*p^e, u=concat(u, [(n+1)*p^(e+1)]))); t=concat([8, 12], concat(u, 2*u)); t=vecsort(select(i->(iJianing Song, Apr 13 2019

A302257(a(n)) = A258615(a(n))/2.

Missing a(40) inserted by Jianing Song, Apr 20 2019

A114873 Numbers representable in exactly one way as (p-1)p^k (where p is a prime and k>=0), in ascending order.

Original entry on oeis.org

1, 8, 10, 12, 20, 22, 28, 30, 32, 36, 40, 46, 52, 54, 58, 60, 64, 66, 70, 72, 78, 82, 88, 96, 102, 106, 108, 110, 112, 126, 128, 130, 136, 138, 148, 150, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 262, 268, 270, 272, 276, 280

Offset: 1

Franz Vrabec, Jan 03 2006

nonn

			(2-1)*2^3 is the only representation of 8 in the required form.

Cf. A114871, A114874.

s = Split@ Sort@ Flatten@ Table[(Prime[n] - 1)Prime[n]^k, {n, 60}, {k, 0, 6}]; Take[Union@ Flatten@ Select[s, Length@# == 1 &], 80] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jan 05 2006

A134269 Number of solutions to the equation p^k - p^(k-1) = n, where k is a positive integer and p is prime.

Original entry on oeis.org

1, 2, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0

Offset: 1

Anthony C Robin, Jan 15 2008

nonn

The Euler phi function A000010 (number of integers less than n which are coprime with n) involves calculating the expression p^(k-1)*(p-1), where p is prime. For example phi(120) = phi(2^3*3*5) = (2^3-2^2)*(3-1)*(5-1) = 4*2*4 = 32.

			Notice that it is not possible to have more than 2 solutions, but say when n=4 there are two solutions, namely 5^1 - 5^0 and 2^3 - 2^2.
a(2) = 2 refers to 2^2 - 2^1 = 2 and 3^1 - 3^0 = 2.
a(6) = 2 as 6 = 3^2 - 3^1 = 7^1 - 7^0.

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

Cf. A000010, A014197, A114871, A114873, A114874.

A134269 := proc(n)
    local a,p,r ;
    a := 0 ;
    p :=2 ;
    while p <= n+1 do
        r := n/(p-1) ;
        if type(r,'integer') then
            if r = 1 then
                a := a+1 ;
            else
                r := ifactors(r)[2] ;
                if nops(r) = 1 then
                    if op(1,op(1,r)) = p then
                        a := a+1 ;
                    end if;
                end if;
            end if;
        end if;
        p := nextprime(p) ;
    end do:
    return a;
end proc: # R. J. Mathar, Aug 06 2013

lista(N=100) = {tab = vector(N); for (i=1, N, p = prime(i); for (j=1, N, v = p^j-p^(j-1); if (v <= #tab, tab[v]++););); for (i=1, #tab, print1(tab[i], ", "));} \\ Michel Marcus, Aug 06 2013

A134269list(up_to) = { my(v=vector(up_to)); forprime(p=2,1+up_to, for(j=1,oo,my(d = (p^j)-(p^(j-1))); if(d>up_to,break,v[d]++))); (v); };
v134269 = A134269list(up_to);
A134269(n) = v134269[n]; \\ Antti Karttunen, Nov 09 2018

a(2) corrected by Michel Marcus, Aug 06 2013

More terms from Antti Karttunen, Nov 09 2018

A307527 Odd terms in A305236.

Original entry on oeis.org

63, 513, 2107, 12625, 26533, 39609, 355023, 3190833, 4457713, 19854847, 38463283, 242138449, 370634743, 516465451, 574336561, 701607583, 1380336193, 2324581983, 4882890625, 6174434113, 12859758577, 14793096853, 20578440583, 43522669657, 85504120021

Offset: 1

Jianing Song, Apr 12 2019

nonn

These are numbers of the form p^e*((p-1)*p^(e-1) + 1) where p is an odd prime and (p-1)*p^(e-1) + 1 is prime.
{A002322(a(n))} = {6, 18, 42, 100, 156, 162, ...} is a permutation of A114874 without the terms that are powers of 2 (but they don't have the same order: 6563187324027001 and 6575415997816513 are both terms but A002322(6563187324027001) = 81009000 while A002322(6575415997816513) = 80995248).
A305236 = {a(n)} U {2*a(n)} U {8, 12}.

			See A305236 for examples.

Jianing Song, Table of n, a(n) for n = 1..156 (all terms below 10^16)

Cf. A305236, A114874, A002322.

the_first_entries(nn) = my(u=[]); for(n=2, sqrt(nn), my(v=factor(n), d=#v[, 1], p=v[d, 1], e=v[d, 2]); if(isprime(n+1) && p!=2 && n==(p-1)*p^e, u=concat(u, [(n+1)*p^(e+1)]))); u=vecsort(select(i->(i

A309502 Totients congruent to 2 mod 4.

Original entry on oeis.org

2, 6, 10, 18, 22, 30, 42, 46, 54, 58, 66, 70, 78, 82, 102, 106, 110, 126, 130, 138, 150, 162, 166, 178, 190, 198, 210, 222, 226, 238, 250, 262, 270, 282, 294, 306, 310, 330, 342, 346, 358, 366, 378, 382, 418, 430, 438, 442, 462, 466, 478, 486, 490, 498, 502

Offset: 1

Franz Vrabec, Aug 05 2019

Intersection of A002202 and A016825.
Let the multiplicity of a(n) be the number of m such that phi(m)=a(n), a(1)=2 has multiplicity 3 (phi(3)=phi(4)=phi(6)=2) and all other terms have multiplicity 2 or 4.
From Jianing Song, Aug 23 2021: (Start)
Numbers of the form (p-1)*p^e for primes p == 3 (mod 4), e >= 0.
The terms with multiplicity 4 are the numbers in A114874 that are congruent to 2 modulo 4 and greater than 2, that is, the numbers of the form k = (p-1)*p^e for primes p == 3 (mod 4), e >= 1, where k+1 is prime. In this case, the numbers m such that phi(m) = k are m = k+1, 2*(k+1), p^(e+1) and 2*p^(e+1). (End)

			10 = phi(11) = phi(22) and 10 == 2 (mod 4), so 10 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andre Contiero and Davi Lima, On the distribution of totients 2 mod. 4, arXiv:1803.01396 [math.NT], 4 Mar 2018.
André Contiero and Davi Lima, 2-Adic Stratification of Totients, arXiv:2005.05475 [math.NT], 2020.
V. L. Klee, Jr., On the equation phi(x)=2m, Amer. Math. Monthly, 53 (1946), 327.

Cf. A002202, A016825, A114874.

Supersequence of A063668.

isok(t) = istotient(t) && ((t % 4) == 2); \\ Michel Marcus, Aug 05 2019

New name using existing comment from Michel Marcus, May 14 2020

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A305236 Numbers n such that the multiplicative group of integers modulo n is isomorphic to C_m X C_m, m > 1.

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A114873 Numbers representable in exactly one way as (p-1)p^k (where p is a prime and k>=0), in ascending order.

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A134269 Number of solutions to the equation p^k - p^(k-1) = n, where k is a positive integer and p is prime.

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A307527 Odd terms in A305236.

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A309502 Totients congruent to 2 mod 4.

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