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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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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

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
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