Franz Vrabec - cp's OEIS frontend

A379851 Numbers k such that phi(k) does not divide k. Complement of A007694.

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89

Offset: 1

Franz Vrabec, Jan 04 2025

nonn

Let PHI(n) the set of all numbers x such there is a k-fold iteration of Euler's totient function phi = A000010 on x resulting in n. The numbers a(n) are exactly the numbers for which PHI(a(n)) is a finite set (possibly empty).

Contains A007617.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A007617. Complement of A007694.

filter:= n -> n mod numtheory:-phi(n) <> 0:
select(filter, [$1..100]); # Robert Israel, Feb 04 2025

Select[Range[100], ! Divisible[#, EulerPhi[#]] &] (* Amiram Eldar, Jan 08 2025 *)

A377191 Smallest number of digits that must be changed in n to obtain a palindrome (without changing the first digit to 0).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0

Offset: 0

Franz Vrabec, Oct 19 2024

The sequence is unbounded, with a(n) = k first occurring at n = A377192(k).

			a(12) = 1 because 12 is not a palindrome, but changing 1 digit appropriately (either the first or second in this case) yields a palindrome.

Cf. A002113, A029742, A136522, A377192.

a(A002113(n)) = 0, a(A029742(n)) > 0.

a(n) = 0 iff A136522(n) = 1, a(n) > 0 iff A136522(n) = 0.

A377192 Smallest number with the property that you have to change at least n digits to get a palindrome.

Original entry on oeis.org

0, 10, 1010, 100110, 10001110, 1000011110, 100000111110, 10000001111110, 1000000011111110, 100000000111111110, 10000000001111111110, 1000000000011111111110, 100000000000111111111110, 10000000000001111111111110, 1000000000000011111111111110, 100000000000000111111111111110

Offset: 0

Franz Vrabec, Oct 19 2024

Positions of records in A377191.

			a(2) = 1010 because 1010 is the smallest number with the property that you have to change at least 2 digits to get a palindrome.

Paolo Xausa, Table of n, a(n) for n = 0..450
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002113, A007088, A143960, A377191.

A377192[n_] := Ceiling[10^(2*n-1) + (10^n-1)/9 - 1]; Array[A377192, 20, 0] (* or *)
LinearRecurrence[{111, -1110, 1000}, {0, 10, 1010, 100110}, 20] (* Paolo Xausa, Nov 06 2024 *)

a(n) = 10^(2*n-1) + (10^n-1)/9 - 1 for n > 0.
From Stefano Spezia, Oct 20 2024: (Start)
G.f.: 10*x*(1 - 10*x - 90*x^2)/((1 - x)*(1 - 10*x)*(1 - 100*x)).
E.g.f.: (81 - 100*exp(x) + 10*exp(10*x) + 9*exp(100*x))/90. (End)
a(n) = A007088(A143960(n)) for n > 0. - Rémy Sigrist, Nov 05 2024

A365228 Denominator of Sum_{1<=j<=k<=n, gcd(j,k)=1} 1/(j*k).

Original entry on oeis.org

1, 2, 1, 3, 4, 20, 10, 210, 168, 504, 105, 1155, 792, 1560, 60060, 180180, 16016, 123760, 510510, 29099070, 21162960, 335920, 3233230, 74364290, 45762640, 187210800, 13385572200, 40156716600, 97349616, 31054527504, 166363540200, 12033629407800, 2831442213600, 1698865328160

Offset: 1

Franz Vrabec, Aug 28 2023

Cf. A365227 (numerator of this sum).

A365228 := proc(n)
   local j,k,s; s := 0;
   for j from 1 to n do
      for k from j to n do
         if gcd(j,k) = 1 then s := s + 1/(j*k);
         end if;
      end do;
   end do;
   denom(s);
end proc;
seq(A365228(n), n = 1..30);
# second Maple program:
a:= n-> denom(add(add(`if`(igcd(j, k)=1, 1/j, 0), j=1..k)/k, k=1..n)):
seq(a(n), n=1..45);  # Alois P. Heinz, Aug 28 2023

a[n_Integer]:=Module[{sum,j,k},sum=Sum[If[GCD[j,k]==1,1/(j*k),0],{j,1,n},{k,j,n}]; Denominator[sum]]; Table[a[n],{n,1,34}] (* Robert P. P. McKone, Aug 29 2023 *)

a(n) = denominator(sum(j=1, n, sum(k=j, n, if (gcd(j,k)==1, 1/(j*k))))); \\ Michel Marcus, Aug 28 2023

from fractions import Fraction
from math import gcd
def A365228(n): return sum(sum(Fraction(1,j) for j in range(1,k+1) if gcd(j,k)==1)/k for k in range(1,n+1)).denominator # Chai Wah Wu, Aug 29 2023

A365227 Numerator of Sum_{1<=j<=k<=n, gcd(j,k)=1} 1/(j*k).

Original entry on oeis.org

1, 3, 2, 7, 11, 59, 33, 737, 631, 1973, 439, 4967, 3595, 7283, 289433, 891067, 82391, 647449, 2764637, 160300109, 119168603, 1923477, 19032303, 442903921, 278705461, 1155909107, 84109239017, 255355122859, 632225777, 203232858383, 1110186816983, 81194050820693

Offset: 1

Franz Vrabec, Aug 27 2023

Cf. A365228 (denominator of this sum).

A365227 := proc(n)
   local j,k,s; s := 0;
   for j from 1 to n do
      for k from j to n do
         if gcd(j,k) = 1 then s := s + 1/(j*k);
         end if;
      end do;
   end do;
   numer(s);
end proc;
seq(A365227(n), n = 1..20);
# second Maple program:
a:= n-> numer(add(add(`if`(igcd(j, k)=1, 1/j, 0), j=1..k)/k, k=1..n)):
seq(a(n), n=1..45);  # Alois P. Heinz, Aug 28 2023

a(n) = numerator(sum(j=1, n, sum(k=j, n, if (gcd(j,k)==1, 1/(j*k))))); \\ Michel Marcus, Aug 28 2023

from math import gcd
from fractions import Fraction
def A365227(n): return sum(sum(Fraction(1,j) for j in range(1,k+1) if gcd(j,k)==1)/k for k in range(1,n+1)).numerator # Chai Wah Wu, Aug 29 2023

A347277 Table T(n,k) read by downward antidiagonals: A quotient belonging to a generalization of Euler's theorem.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 2, 0, 4, 6, 8, 3, 0, 5, 10, 20, 18, 6, 0, 6, 15, 40, 60, 48, 8, 0, 7, 21, 70, 150, 204, 108, 18, 0, 8, 28, 112, 315, 624, 640, 312, 30, 0, 9, 36, 168, 588, 1554, 2500, 2340, 810, 56, 0, 10, 45, 240, 1008, 3360, 7560, 11160, 8160, 2184, 96, 0

Offset: 1

Franz Vrabec, Aug 26 2021

The quotient T(n,k) = (k^n - k^(n-phi(n)))/n results from the generalization k^n == k^(n-phi(n)) (mod n) of Euler's theorem (see Sierpiński, p. 243).

The n-th row of the table is equal to the n-th row of A074650 iff n = p^j (p prime, j>=1).

			T(4,3) = (3^4 - 3^2)/4 = 18.
Square array starts:
  0, 1,   2,   3,    4,    5, ...
  0, 1,   3,   6,   10,   15, ...
  0, 2,   8,  20,   40,   70, ...
  0, 3,  18,  60,  150,  315, ...
  0, 6,  48, 204,  624, 1554, ...
  0, 8, 108, 640, 2500, 7560, ...

W. Sierpiński, Elementary Theory of Numbers, Warszawa, 1964.

Cf. A074650.

with(numtheory):
T:= proc(n, k) (k^n-k^(n-phi(n)))/n end:
seq(seq(T(i, 1+d-i), i=1..d), d=1..11);

T(n,k) = (k^n - k^(n - eulerphi(n)))/n; \\ Jinyuan Wang, Aug 28 2021

T(n,k) = (k^n - k^(n - phi(n)))/n.

More terms from Jinyuan Wang, Aug 28 2021

A345289 Indices of low records in A346064.

Original entry on oeis.org

10, 11, 13, 846400, 5845240, 67621810, 67647250, 436771450

Offset: 1

Franz Vrabec, Jul 16 2021

The corresponding record values are 21, 17, 15, 14, 13, 12, 11, 10.

Cf. A346064.

A346064 Number of primes that may be generated by changing any two digits of n simultaneously.

Original entry on oeis.org

21, 17, 20, 15, 21, 20, 21, 16, 21, 17, 23, 18, 22, 17, 23, 22, 23, 17, 23, 19, 23, 19, 22, 16, 23, 22, 23, 18, 23, 18, 22, 18, 21, 16, 22, 21, 22, 17, 22, 17, 23, 18, 22, 17, 23, 22, 23, 17, 23, 19, 23, 19, 22, 16, 23, 22, 23, 18, 23, 18, 22, 18, 21, 16, 22

Offset: 10

Franz Vrabec, Jul 03 2021

Indices of low records are given by A345289. By heuristic considerations it is conjectured that a(n) > 0 for all n >= 10.

			Changing two digits of the number 17 simultaneously yields the primes 02,03,05,23,29,31,41,43,53,59,61,71,73,79,83,89, so a(17) = 16.

M. Filaseta, M. Kozek, Ch. Nicol and J. Selfridge, Composites that Remain Composite After Changing a Digit, J. Comb. Number Theory, Vol. 2, No. 1 (2010), pp. 25-36.

Cf. A345289 (indices of record lows).

Cf. A209252 (changing one digit).

A346064 := proc(n)
local a, d, e, r, s, l, N, NN, nn, i;
a := 0;
N := convert(n, base, 10);
l := nops(N);
for d to l - 1 do
   for e from d + 1 to l do
      for r from 0 to 9 do
         for s from 0 to 9 do
        if r <> op(d, N) and s <> op(e, N) then
           NN := subsop(d = r, e = s, N);
           nn := add(op(i, NN)*10^(i - 1), i = 1 .. l);
           if isprime(nn) then a := a + 1; end if;
        end if;
        end do;
      end do;
   end do;
end do;
a;
end proc:

Table[Count[Flatten[FromDigits/@Tuples[ReplacePart[t=List/@IntegerDigits[n],{#->Complement[Range[0,9],t[[#]]],#2->Complement[Range[0,9],t[[#2]]]}]&@@#]&/@Subsets[Range@IntegerLength@n,{2}]],?PrimeQ],{n,10,100}] (* _Giorgos Kalogeropoulos, Jul 23 2021 *)

from sympy import isprime
from itertools import combinations, product
def change2(s):
    for i, j in combinations(range(len(s)), 2):
        for c, d in product("0123456789", repeat=2):
            if c != s[i] and d != s[j]:
                yield s[:i] + c + s[i+1:j] + d + s[j+1:]
def a(n): return sum(isprime(int(t)) for t in change2(str(n)))
print([a(n) for n in range(10, 101)]) # Michael S. Branicky, Jul 23 2021

A342446 Square table read by antidiagonals downwards: T(n,k) = floor((4/(4^(1/2^n)-1))^(1/2^k)).

Original entry on oeis.org

1, 4, 1, 9, 2, 1, 21, 3, 1, 1, 44, 4, 1, 1, 1, 90, 6, 2, 1, 1, 1, 182, 9, 2, 1, 1, 1, 1, 367, 13, 3, 1, 1, 1, 1, 1, 736, 19, 3, 1, 1, 1, 1, 1, 1, 1475, 27, 4, 1, 1, 1, 1, 1, 1, 1, 2952, 38, 5, 2, 1, 1, 1, 1, 1, 1, 1, 5907, 54, 6, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Franz Vrabec, Mar 12 2021

Every positive integer occurs infinitely often.

			T(2,1) = floor((4/(4^(1/4)-1))^(1/2)) = floor(sqrt(4/(sqrt(2)-1))) = floor(3.1075...) = 3.

N. J. Fine and Marcin E. Kuczma, Writing Integers with Exactly Three Fours, Problem E 3363 [1990, 63], Amer. Math. Monthly, Vol. 99 (1992), No. 2, pp. 163-164.

T(n,k) = floor((4/(4^(1/2^n)-1))^(1/2^k)).

A340765 Numbers k such that iterations of phi(k), phi(phi(k)), ... end in ... 6, 2, 1.

Original entry on oeis.org

6, 7, 9, 14, 18, 19, 27, 38, 54, 81, 162, 163, 243, 326, 486, 487, 729, 974, 1458, 1459, 2187, 2918, 4374, 6561, 13122, 19683, 39366, 39367, 59049, 78734, 118098, 177147, 354294, 531441, 1062882, 1594323, 3188646, 4782969, 9565938, 14348907, 28697814, 43046721, 86093442, 86093443, 129140163, 172186886

Offset: 1

Franz Vrabec, Jan 20 2021

nonn

Infinite set (see reference).

Contains 3^k for k >= 2 and 2*3^k for k >= 1, and all members of A111974 except 3. - Robert Israel, Dec 23 2021

			19 is in the list because phi(phi(19)) = phi(18) = 6.

Robert Israel, Table of n, a(n) for n = 1..1000
Eliot T. Jacobson and Alan E. Parks, Infinite branches of the phi-tree, Amer. Math. Monthly, Vol. 93, No. 7 (August-September 1986), pp. 552-554.
Keith Matthews, Solving phi(x)=n, where phi(x) is Euler's totient function.

Cf. A000010, A340762 (complement relative to {n>=4}).

Cf. A003434, A032358, A049108, A111974.

R:= {6}: Agenda:= {6}: count:= 1:
while count - nops(Agenda) < 99 do
  v:= min(Agenda);
  W:= convert(numtheory:-invphi(v),set);
  count:= count + nops(W);
  Agenda:= Agenda minus {v} union W;
  R:= R union W;
od:
sort(select(`<=`, convert(R,list),min(Agenda))); # Robert Israel, Dec 23 2021

Select[Range[4, 10000], FixedPointList[EulerPhi, #][[-4]] == 6 &] (* Amiram Eldar, Jan 27 2021 *)

isok(k) = if (k>=6, while((k!=6) && (k!=4), k=eulerphi(k))); k == 6; \\ Michel Marcus, Feb 01 2021

cp's OEIS Frontend

User: Franz Vrabec

A379851 Numbers k such that phi(k) does not divide k. Complement of A007694.

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A377191 Smallest number of digits that must be changed in n to obtain a palindrome (without changing the first digit to 0).

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A377192 Smallest number with the property that you have to change at least n digits to get a palindrome.

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A365228 Denominator of Sum_{1<=j<=k<=n, gcd(j,k)=1} 1/(j*k).

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A365227 Numerator of Sum_{1<=j<=k<=n, gcd(j,k)=1} 1/(j*k).

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A347277 Table T(n,k) read by downward antidiagonals: A quotient belonging to a generalization of Euler's theorem.

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A345289 Indices of low records in A346064.

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A346064 Number of primes that may be generated by changing any two digits of n simultaneously.

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