A229289 -id:A229289 - cp's OEIS frontend

A341858 Numbers k such that psi(k^2) = k, psi = A002322; indices of 1 in A341857.

1, 2, 4, 6, 12, 20, 42, 60, 84, 156, 220, 420, 660, 780, 1092, 1806, 1860, 2436, 3612, 3660, 4620, 5060, 5460, 8268, 8580, 12180, 12324, 13020, 15180, 18060, 20460, 24180, 24492, 25620, 29820, 31668, 40260, 41340, 44220, 46956, 47580, 57876, 60060, 61620, 86268, 88620

Offset: 1

Jianing Song, Feb 21 2021

nonn

For all k we have k divides psi(k^2). This sequence gives those k such that the quotient is 1.
Apart from 5 exceptional terms, every term is the product of 4 and distinct odd primes. The exceptional terms are precisely the 5 terms in A014117.
Except for k = 1, 2, 6, 42, 1806, k is a term if and only if k = 4*(p_1)*(p_2)*...*(p_m), where p_1 < p_2 < ... < p_m are odd primes, (p_i)-1 | 4*(p_1)*(p_2)*...*(p_(i-1)) for all 1 <= i <= m.
The LCM of two terms is again in this sequence.
Is this sequence infinite? If this sequence is finite, it means that there exists a term of the form k = 4*(p_1)*(p_2)*...*(p_s), where p_1 < p_2 < ... < p_s are odd primes such that: for every (e_0, e_1, ..., e_s) in {0, 1}^(s+1), 2^((e_0)+1)*(p_1)^(e_1)*(p_2)^(e_2)*...*(p_s)^(e_s)+1 is either composite or equal to some p_i. That term must be divisible by all other terms, since there are no more odd primes q other than p_1, p_2, ..., p_s such that q-1 | k.
Numbers k such that b^k == 1 (mod k^2) for every b coprime to k. Proof: these are numbers k such that psi(k^2) divides k, which holds if and only if psi(k^2) = k. Subsequence of A124240 (see my comment there). If k is a term of the sequence and k+1 is prime, then k*(k+1) is also a term. - Thomas Ordowski, Jul 26 2024

			1092 = 4 * 3 * 7 * 13 is a term since 3-1 | 4, 7-1 | 4*3 and 13-1 | 4*3*7. Indeed, we have psi(1092^2) = 1092.
5060 = 4 * 5 * 11 * 23 is a term since 5-1 | 4, 11-1 | 4*5 and 23-1 | 4*5*11.

Amiram Eldar, Table of n, a(n) for n = 1..6034 (terms below 10^11; terms 1..249 from Jianing Song)
Jianing Song, Factorization of terms <= 15*10^6 other than 1, 2, 6, 42, 1806

Cf. A002322, A341857, A014117.
A229289 gives the set of prime factors of the terms.
Subsequence of A124240.

Select[Range[10^5], CarmichaelLambda[#^2] == # &] (* Paolo Xausa, Mar 11 2024 *)

isA341858(n) = (A002322(n^2)==n) \\ See A002322 for its program

A229290 n is in the sequence if n is prime, (n-1)/3^A007949(n-1) is a squarefree number, A007949(n-1) < 3 and every prime divisor of n-1 is in the sequence.

Original entry on oeis.org

2, 3, 7, 19, 43, 127, 2287, 4903, 5419, 13723, 14479, 82339, 98299, 101347, 304039, 617767, 688087, 1676827, 3735583, 3736087, 4130323, 4324363, 4693267, 4951819, 10621603, 11032999, 11208259, 11554243, 11737783, 12198859, 26152603, 26563939, 28159603

Offset: 1

José María Grau Ribas, Oct 05 2013

nonn

If n is in A226961 then n is some product of elements of this sequence.

Cf. A007949, A226961, A229289, A229291.

fa = FactorInteger; free[n_] := n == Product[fa[n][[i, 1]], {i,
  Length[fa[ n]]}]; Os[b_, 1] = True; Os[b_, 2] = True; Os[b_, b_] = True; Os[b_, n_] := Os[b, n] = PrimeQ[n] && free[(n-1)/ b^IntegerExponent[n - 1,  b]] && IntegerExponent[n - 1, b] < 3 && Union@Table[Os[b, fa[n - 1][[i,1]]], {i, Length[fa[n - 1]]}] == {True}; G[b_] := Select[Prime[Range[2000]], Os[b, #] &]; G[3]

A229291 n is in the sequence if n is prime, (n-1)/5^A112765(n-1) is a squarefree number, A112765(n-1) < 3 and every prime divisor of n-1 is in the sequence.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 31, 43, 47, 67, 71, 139, 151, 211, 283, 311, 331, 431, 463, 659, 683, 691, 863, 907, 947, 967, 1051, 1151, 1291, 1303, 1319, 1367, 1427, 1511, 1699, 1867, 1979, 1987, 2011, 2111, 2131, 2311, 2351, 2531, 3011, 3023, 3083, 3323, 3851, 4099

Offset: 1

José María Grau Ribas, Oct 06 2013

nonn

If n is in A226963 then n is some product of elements of this sequence.

Cf. A112765, A226963, A229289, A229290.

fa = FactorInteger; free[n_] := n == Product[fa[n][[i, 1]], {i,
  Length[fa[n]]}]; Os[b_, 1] = True; Os[b_, 2] = True; Os[ b_, b_] = True; Os[b_, n_] := Os[b, n] = PrimeQ[n] && free[(n - 1)/b^IntegerExponent[n - 1,b]] && IntegerExponent[n - 1, b] < 3 && Union@Table[Os[b, fa[n - 1][[i, 1]]], {i, Length[fa[n - 1]]}] == {True}; G[b_] := Select[Prime [Range[2000]], Os[b, #] &]; G[5]

A129563 Primes not in a certain recursively defined set of primes.

Original entry on oeis.org

101, 151, 197, 251, 401, 491, 503, 601, 607, 677, 701, 727, 751, 809, 883, 907, 983, 1051, 1151, 1201, 1213, 1301, 1373, 1451, 1453, 1471, 1511, 1601, 1619, 1667, 1801, 1901, 1951, 2029, 2179, 2251, 2351, 2417, 2549, 2551, 2647, 2663, 2719, 2801, 2843, 2851, 2903, 2909

Offset: 1

Jonathan Vos Post, Apr 21 2007

The sequence is the complement of the M-sequence constructed in Section 4 of Smarandache (2007). M is defined as follows: (a) 2, 3 are in M; and (b) if 2, 3, q_1, ..., q_n are distinct primes in M and b_m = 1 + 2^a*3^b*q_1*...*q_n is prime, where 0 <= a <= 41 and 0 <= b <= 46, then b_m is in M. - R. J. Mathar, Jul 03 2017
The restriction of the two exponents to 41 and 46 seems to be based on Smarandache's sentence "and Klee to a multiple of 2^42*3^47". This statement however is hard to locate in Klee's publications. In any case, 42 and 46 should be regarded as temporary lower bounds on the exponents, which may increase as the theory and numerical experiments continue. - R. J. Mathar, Jul 04 2017
The M-sequence in Section 3 of the arXiv paper is A229289, and its complement is A289355. - R. J. Mathar, Ray Chandler, Jul 03 2017

Ray Chandler, Table of n, a(n) for n = 1..1000
H. Donnelly, On a problem concerning Euler's phi-function, Am. Math. Monthly 80 (9) (1973) 1029-1031.
V. Klee, On a conjecture of Carmichael, Bull. Am. Math. Soc. 53 (1947) 1183-1186.
V. Klee, Is there an n for which phi(x)=n has a unique solution?, Am. Math. Monthly 76 (3) (1969) 288-289.
Florentin Smarandache, On Carmichael's Conjecture, arXiv preprint arXiv:0704.2453 [math.GM], Apr 19 2007.

Cf. A229289, A289355.

isM := proc(n)
    option remember;
    local p1,pe,p,e ;
    if not isprime(n) then
        return false;
    elif n in {2,3} then
        return true;
    else
        for pe in ifactors(n-1)[2] do
            p := pe[1] ;
            e := pe[2] ;
            if p = 2 and e > 41 then
                return false;
            elif p = 3 and e > 46 then
                return false;
            elif e > 1 and p> 3 then
                return false;
            elif not procname(p) then
                return false;
            end if;
        end do:
        return true;
    end if;
end proc:
isA129563 := proc(n)
    isprime(n) and not isM(n) ;
end proc:
for n from 2 to 3000 do
    if isA129563(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Jul 03 2017

isM[n_] := isM[n] = Module[{p, e}, Which[!PrimeQ[n], Return[False], 2 <= n <= 3, Return[True], True, Do[{p, e} = pe; Which[p == 2 && e > 41, Return[False], p == 3 && e > 46, Return[False], e > 1 && p > 3, Return[False], !isM[p], Return[False]], {pe, FactorInteger[n-1]}], True, Return[True]]]
Select[Range[2, 3000], PrimeQ[#] && !isM[#]&] (* Jean-François Alcover, Dec 02 2017, after R. J. Mathar *)

Definition of M clarified by R. J. Mathar, Jul 03 2017

cp's OEIS Frontend

A289355 Complement of A229289.

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A341858 Numbers k such that psi(k^2) = k, psi = A002322; indices of 1 in A341857.

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A229290 n is in the sequence if n is prime, (n-1)/3^A007949(n-1) is a squarefree number, A007949(n-1) < 3 and every prime divisor of n-1 is in the sequence.

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A229291 n is in the sequence if n is prime, (n-1)/5^A112765(n-1) is a squarefree number, A112765(n-1) < 3 and every prime divisor of n-1 is in the sequence.

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A129563 Primes not in a certain recursively defined set of primes.

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