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.

Showing 1-5 of 5 results.

A350320 Totient numbers k such that 10*k is a nontotient.

Original entry on oeis.org

110, 13310, 18260, 78980, 130460, 143660, 163460, 164780, 167420, 284900, 325160, 329780, 332420, 370700, 381260, 403700, 418220, 431420, 453860, 514580, 526460, 535700, 554180, 560780, 603020, 628100, 646580, 665060, 675620, 732380, 745580, 765380, 801020
Offset: 1

Views

Author

Jianing Song, Dec 24 2021

Keywords

Comments

10 is the smallest totient number that is not in A301587.
If 10*phi(m) is a nontotient, then m is divisible by 121 but not by 5, so every term is divisible by 110.
Proof. In the following cases, 10*phi(m) is a totient number:
(a) If m is not divisible by 11, then phi(11*m) = phi(11)*phi(m) = 10*phi(m).
(b) If m is divisible by 11 but not by 121 or 5, then phi((m/11)*125) = phi(m/11)*phi(125) = (phi(m)/10)*100 = 10*phi(m).
(c) If m is divisible by 5 but not by 2, then phi(4*5*m) = phi(4)*phi(5*m) = 2*(5*phi(m)) = 10*phi(m).
(d) If m is divisible by 5 and 2, then phi(10*m) = 10*phi(m).
So the only left case is that m is divisible by 121 but not by 5.

Examples

			110 is a term since 110 = phi(121) = phi(242), but phi(n) = 10*110 = 1100 has no solution.
13310 is a term since 13310 = phi(14641) = phi(29282), but phi(n) = 10*13310 = 133100 has no solution.

Links

Crossrefs

Totient numbers k such that m*k is a nontotient: A350316 (m=3), A350317 (m=5), A350318 (m=7), A350319 (m=9), this sequence (m=10), A350321 (m=14).
Cf. A002202, A005277, A301587.

Programs

  • PARI
    isA350320(n) = istotient(n) && !istotient(10*n)

A361058 Least totient number k > 1 such that n*k is a nontotient number, or 0 if no such number exists.

Original entry on oeis.org

0, 0, 30, 0, 10, 0, 2, 0, 10, 110, 22, 0, 2, 22, 6, 0, 2, 0, 2, 0, 54, 22, 10, 0, 2, 22, 22, 212983792, 6
Offset: 1

Views

Author

Jinyuan Wang, Mar 01 2023

Keywords

Comments

After a(30) which is unknown, the sequence continues: 2, 0, 18, 2, 10, 0, 2, 2, 6, 0, 6, 0, 2, 22, 2, 46, 2, 0, 2, 22, 10, 146068, 6, 0, 10, and a(56) is unknown. - Michel Marcus, Mar 11 2023
When n is in A002202, then n*a(n) is a term of A329872; in other words a(n) is the value k, such that k*a(n) is the least term of A329872 that is divisible by n. - Michel Marcus, Mar 26 2023
a(30) > 2.5*10^10, if it is not 0. - Amiram Eldar, May 07 2023
a(568) <= 2^17*71^13 where 568 = 2^3*71 (so similar to a(652) = 2^4*163^3 where 652 = 2^2*163). - Michel Marcus, May 14 2023
From Michel Marcus, Jun 08 2023: (Start)
Experimentally there are 2 cases: n is a totient value or is a nontotient.
If n is a nontotient, then it is relatively easy to find the titular k.
If n is a totient value, then we see that there are 4 cases:
there are no such k and a(n)=0,
k is known, and by definition k is a totient value.
k is not known but we know a large totient value K for which n*K is nontotient,
k is currently unknown.
For several k or K, n*k are squares of terms of A281187. (End)

Examples

			a(3) = 30 because 30 is in A002202 and 3*30 = 90 is in A007617.

Links

Crossrefs

Cf. A002202 (totient numbers), A007617 (nontotient numbers).
Cf. A281187, A301587, A316665, A329872, A350085, A350086.

Programs

  • PARI
    a(n) = if (vecsearch([1, 2, 4, 6, 8, 12, 16, 18, 20, 24], n), return(0)); my(k=2); while (istotient(n*k), k++; while (!istotient(k), k++)); k; \\ Michel Marcus, Mar 08 2023
  • PARI
    check(n, k) = istotient(k) && !istotient(n*k); \\ Michel Marcus, Apr 05 2023; just for checking

Formula

a(n) = 0 if n is in A301587.
a(A007617(n)) = A350085(n). - Michel Marcus, Apr 08 2023

A329872 Nontotients (A005277) that are the product of two totients (A002202).

Original entry on oeis.org

484, 968, 1100, 2116, 3364, 4232, 6084, 6724, 6728, 8464, 10404, 11132, 11236, 13448, 16928, 19044, 22472, 26896, 27556, 29584, 31684, 36100, 44944, 51076, 53792, 55112, 56644, 59168, 63368, 65824, 67416, 68644, 72200, 79524, 80344, 89888, 96100, 99856, 102152, 107584
Offset: 1

Views

Author

Jianing Song, Nov 23 2019

Keywords

Comments

We can have a list of nontotients and their factorizations into two totients. A totient m is in A301587 if and only if m never occurs in this list as a divisor of the nontotients. Using the list, many totients (10, 22, 44, 46, ...) are ruled out of A301587. But generally it's hard to prove that a number is in A301587.

Examples

			484 is here, because 484 = 22*22, and 22 is a totient while 484 isn't. Similarly, if p == 3 (mod 4) is a prime such that (p-1)^2+1 is composite, then (p-1)^2 is here.

Links

Crossrefs

Cf. A002202, A005277, A301587.
Squares of terms of A281187 are terms of this sequence.

Programs

  • PARI
    isA329872(n) = if(!istotient(n), my(v=divisors(n)); for(i=1, (1+#v)\2, if(istotient(v[i])&&istotient(n/v[i]), return(1))); 0); \\ improved by Jinyuan Wang, Mar 25 2023

A350085 a(n) is the smallest totient number k > 1 such that A007617(n)*k is a nontotient number, or 0 if no such number exists.

Original entry on oeis.org

30, 10, 2, 10, 22, 2, 22, 6, 2, 2, 54, 10, 2, 22, 22, 6, 2, 18, 2, 10, 2, 2, 6, 6, 2, 2, 2, 2, 22, 10, 6, 10, 2, 2, 2, 2, 18, 6, 2, 10, 6, 2, 2, 10, 6, 2, 2, 2, 30, 10, 2, 6, 2, 6, 106, 2, 2, 2, 10, 2, 22, 6, 2, 2, 18, 2, 2, 6, 6, 46, 2, 2, 2, 6, 2, 2, 2, 2, 10, 2
Offset: 1

Views

Author

Jianing Song, Dec 12 2021

Keywords

Comments

Conjecture: a(n) != 0 for all n.
Records: 30 (A007617(n) = 3), 54 (A007617(n) = 21), 106 (A007617(n) = 90), 2010 (A007617(n) = 450), ...
By definition, a totient number N > 1 is a term if and only if there exists a nontotient r such that: (i) k*r is a totient for totient numbers 2 <= k < N; (ii) N*r is a nontotient. No term can be of the form m*m', where m > 1 is a totient and m' > 1 is in A301587 (otherwise m*r is a totient implies m*m'*r is a totient).
Conjecture: every totient number > 1 which is not of the form m*m', where m > 1 is a totient and m' > 1 is in A301587, appears in this sequence. For example, the numbers 2, 6, 10, 18, 22, 28, 30 first appears when A005277(n) = 34, 86, 68, 186, 14, 902, 318.
Most of the terms are of the form prime-1, but there are some exceptions. Here is a list of distinct such instances [not(prime-1), the corresponding least nontotient]: [54, 21], [110, 10450], [342, 29214], [506, 63250], [3422, 100050], [294, 118062], [2162, 235824], [1210, 308660], [930, 395070], ... That is, instead of k being invphi(prime), in these cases it appears that k is invphi(prime^q). - Michel Marcus, Jun 25 2023

Examples

			A007617(55) = 90. N = 106 is a totient number > 1 such that 90*k is a totient for totient numbers 2 <= k < N, and 90*N is a nontotient, so a(55) = 106.
A007617(307) = 450. N = 2010 is a totient number > 1 such that 450*k is a totient for totient numbers 2 <= k < N, and 450*N is a nontotient, so a(307) = 2010.
A007617(637) = 902. N = 28 is a totient number > 1 such that 902*k is a totient for totient numbers 2 <= k < N, and 902*N is a nontotient, so a(637) = 28.
A007617(194495) = 241010. N = 100 is a totient number > 1 such that 241010*k is a totient for totient numbers 2 <= k < N, and 241010*N is a nontotient, so a(194495) = 100. Note that although 100 = 10*10 is a product of 2 totient number > 1, neither factor is in A301587, so nothing prevents that 100 is a term of this sequence.

Links

Crossrefs

Cf. A007617, A350086, A301587.

Programs

  • PARI
    b(n) = if(!istotient(n), for(k=2, oo, if(istotient(k) && !istotient(n*k), return(k))))
list(lim) = my(v=[]); for(n=1, lim, if(!istotient(n), v=concat(v,b(n)))); v \\ gives a(n) for A007617(n) <= lim

A350086 a(n) is the smallest totient number k > 1 such that A005277(n)*k is a nontotient number, or 0 if no such number exists.

Original entry on oeis.org

22, 22, 2, 2, 22, 2, 10, 10, 2, 6, 106, 2, 22, 46, 2, 2, 2, 6, 2, 10, 2, 2, 6, 2, 78, 2, 18, 2, 6, 2, 2, 2, 2, 46, 58, 2, 2, 2, 58, 2, 6, 2, 2, 2, 10, 10, 2, 46, 2, 2, 2, 82, 2, 30, 2, 6, 2, 10, 2, 10, 46, 2, 2, 2, 2, 2, 6, 78, 2, 10, 2, 10, 46, 10, 2, 46, 2
Offset: 1

Views

Author

Jianing Song, Dec 12 2021

Keywords

Comments

Subsequence of A350085.
Conjecture: a(n) != 0 for all n.
Records: 22 (A005277(n) = 14), 106 (A005277(n) = 90), 2010 (A005277(n) = 450), ...
By definition, a totient number N > 1 is a term if and only if there exists an even nontotient r such that: (i) k*r is a totient for totient numbers 2 <= k < N; (ii) N*r is a nontotient. No term can be of the form m*m', where m > 1 is a totient and m' > 1 is in A301587 (otherwise m*r is a totient implies m*m'*r is a totient).
Conjecture: every totient number > 1 which is not of the form m*m', where m > 1 is a totient and m' > 1 is in A301587, appears in this sequence. For example, the numbers 2, 6, 10, 18, 22, 28, 30 first appears when A007617(n) = 7, 15, 5, 33, 11, 902, 3.

Examples

			A005277(11) = 90. N = 106 is a totient number > 1 such that 90*k is a totient for totient numbers 2 <= k < N, and 90*N is a nontotient, so a(11) = 106.
A005277(83) = 450. N = 2010 is a totient number > 1 such that 450*k is a totient for totient numbers 2 <= k < N, and 450*N is a nontotient, so a(83) = 2010.
A005277(187) = 902. N = 28 is a totient number > 1 such that 902*k is a totient for totient numbers 2 <= k < N, and 902*N is a nontotient, so a(187) = 28.
A005277(73991) = 241010. N = 100 is a totient number > 1 such that 241010*k is a totient for totient numbers 2 <= k < N, and 241010*N is a nontotient, so a(73991) = 100. Note that although 100 = 10*10 is a product of 2 totient number > 1, neither factor is in A301587, so nothing prevents that 100 is a term of this sequence.

Links

Crossrefs

Cf. A005277, A350085, A301587.

Programs

  • PARI
    b(n) = if(!istotient(n), for(k=2, oo, if(istotient(k) && !istotient(n*k), return(k))))
list(lim) = my(v=[]); forstep(n=2, lim, 2, if(!istotient(n), v=concat(v,b(n)))); v \\ gives a(n) for A005277(n) <= lim
Showing 1-5 of 5 results.

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