A083535 -id:A083535 - cp's OEIS frontend

A083533 First difference sequence of A002202. Difference between consecutive possible values of phi(n), the Euler totient function A000010.

1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 4, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 4, 2, 6, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 4, 4, 6, 2, 2, 2, 4, 2, 2, 4, 4, 2, 6, 4, 2, 2, 2, 2, 4, 4, 2, 2, 4, 6, 2, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 2, 2, 4, 6, 2, 10, 2, 4, 4, 2, 2, 4, 2, 2, 4, 4, 2, 6, 4, 2, 2, 4, 6, 4, 2, 4

Offset: 1

Labos Elemer, May 20 2003

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000010, A002202, A005277, A083531, A083532, A083534, A083535, A083536.

a083533 n = a083533_list !! (n-1)
a083533_list = zipWith (-) (tail a002202_list) a002202_list
-- Reinhard Zumkeller, Nov 26 2015

t=Table[EulerPhi[w], {w, 1, 25000}]; u=Union[%]; Delete[u-RotateRight[u], 1]

lista(lim) = {my(k1 = 1, k2 = 1); while(k1 < lim, until(istotient(k2), k2++); print1(k2 - k1, ", "); k1 = k2);} \\ Amiram Eldar, Nov 16 2024

a(n) = A002202(n+1) - A002202(n).

A083531 First difference sequence of A002191. Differences between possible values for sum of divisors of n.

Original entry on oeis.org

2, 1, 2, 1, 1, 4, 1, 1, 1, 3, 2, 4, 4, 2, 1, 1, 4, 2, 1, 1, 2, 2, 4, 6, 2, 1, 3, 2, 1, 5, 4, 2, 4, 2, 4, 6, 1, 2, 3, 2, 4, 2, 4, 2, 2, 2, 6, 1, 3, 2, 1, 1, 4, 1, 5, 2, 4, 6, 2, 4, 2, 2, 2, 2, 4, 3, 3, 2, 4, 2, 1, 3, 6, 2, 1, 3, 2, 4, 6, 2, 4, 1, 5, 2, 4, 2, 4, 6, 2, 6, 4, 3, 1, 2, 2, 4, 2, 4, 2, 6, 2, 2, 2, 4, 6

Offset: 1

Labos Elemer, May 20 2003

nonn

			8 and 12 are the 6th and 7th possible values for sigma(x), since they are sum of divisors of x = 7 and x = 11 respectively, while 9, 10, 11 are impossible ones so 12 - 8 = 4 = a(6) = A002191(7) - A002191(6).
From _Michael De Vlieger_, Jul 22 2017: (Start)
First position of values:
Value   First position
    1         2
    2         1
    3        10
    4         6
    5        30
    6        24
    7       277
    8       165
    9       509
   10       150
   11       824
   12       400
   13     10970
   14      1400
   15     10448
   16      1182
   17     18731
   18      2218
   19    209237
   20      3420
   21    127385
   22      6910
   23     28899
   24      5377
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002191, A007609, A007369, A083532, A083533, A083534, A083535, A083536, A109323 (start of record gaps in A002191).

t=Table[DivisorSigma[1, w], {w, 1, 25000}]; u=Union[%]; Delete[u-RotateRight[u], 1]
(* Second program: *)
With[{nn = 300}, Differences@ TakeWhile[Union@ DivisorSigma[1, Range@ nn], # < nn &]] (* Michael De Vlieger, Jul 22 2017 *)

A035158 Floor of the Chebyshev function theta(n): a(n) = floor(Sum_{primes p <= n } log(p)).

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 5, 5, 5, 5, 7, 7, 10, 10, 10, 10, 13, 13, 16, 16, 16, 16, 19, 19, 19, 19, 19, 19, 22, 22, 26, 26, 26, 26, 26, 26, 29, 29, 29, 29, 33, 33, 37, 37, 37, 37, 40, 40, 40, 40, 40, 40, 44, 44, 44, 44, 44, 44, 49, 49, 53, 53, 53, 53, 53, 53, 57, 57, 57, 57, 61, 61, 65, 65, 65, 65

Offset: 1

N. J. A. Sloane, Oct 02 2008

nonn

The old entry with this sequence number was a duplicate of A002325.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, see Chap. 22.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.35. (For inequalities, etc.)

J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. Journ. Math. 6 (1962) 64-94.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers (scan of some key pages from an ancient annotated photocopy)
J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions theta(x) and psi(x), Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. Math. Comp. 29 (1975), 243-269.
J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions theta (x) and psi (x). II. Math. Comp. 30 (1976), number 134, 337-360.
J. Barkley Rosser and Lowell Schoenfeld, Corrigendum: "Sharper bounds for the Chebyshev functions theta (x) and psi (x). II" (Math. Comput. 30 (1976), number 134, 337-360), Math. Comp. 30 (1976), number 136, 900.

Cf. A057872, A083535, A016040 (records), A000040 (places of records)

(Maple for A035158, A057872, A083535:)
Digits:=2000;
tf:=[]; tr:=[]; tc:=[];
for n from 1 to 120 do
t2:=0;
j:=pi(n);
for i from 1 to j do t2:=t2+log(ithprime(i)); od;
tf:=[op(tf),floor(evalf(t2))];
tr:=[op(tr),round(evalf(t2))];
tc:=[op(tc),ceil(evalf(t2))];
od:

a(n) ~ n by the prime number theorem. - Charles R Greathouse IV, Aug 02 2012

A083534 First difference sequence of A007617. Difference between consecutive values not being in the range of phi (A000010).

Original entry on oeis.org

2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2

Offset: 1

Labos Elemer, May 20 2003

nonn

a(n) is either 2 or 1 since odd numbers are in A007619.

If a(n) = 1 then A007619(n+1) is an even number not in the range of phi.

			{11,13,14,15,17} are not in the range of phi and the corresponding differences are {2,1,1,2}.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000010, A002202, A005277, A007617, A007619, A083531, A083532, A083533, A083535, A083536.

a083534 n = a083534_list !! (n-1)
a083534_list = zipWith (-) (tail a007617_list) a007617_list
-- Reinhard Zumkeller, Nov 26 2015

t0[x_] := Table[j, {j, 1, x}]; t=Table[EulerPhi[w], {w, 1, 10000}]; u=Union[%]; c=Complement[t0[10000], u]; Delete[c-RotateRight[c], 1]

list(lim) = {my(k1 = 3, k2 = 3); while(k1 < lim, until(!istotient(k2), k2++); print1(k2 - k1, ", "); k1 = k2); } \\ Amiram Eldar, Feb 22 2025

a(n) = A007617(n+1) - A007617(n).

A252398 Successive n with minimal relative distance |1-theta(n)/n|, where theta(n) = log(A034386(n)) is Chebyshev's theta function.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 43, 47, 73, 103, 107, 109, 113, 199, 283, 467, 661, 887, 1063, 1069, 1097, 1103, 1109, 1123, 1129, 1303, 1307, 1321, 1327, 1621, 1627, 2803, 3931, 3947, 4273, 4289, 4297, 5867, 5869, 5881, 6373, 6379, 9439, 9473, 9479, 9497, 9551, 9859, 9931, 9949

Offset: 1

Jean-François Alcover, Dec 17 2014

nonn

The first 10000 terms are the same as A108310 (see that sequence for comments). - Charles R Greathouse IV, Dec 18 2014

This sequence, unlike A108310, is presumably infinite; it is finite if and only if theta(n) = n for some number n.

			Given that 1 - theta(3)/3 = 1 - log(6)/3 = 0.40..., 1 - theta(4)/4 = 1 - log(6)/4 = 0.55... and 1 - theta(5)/5 = 1 - log(30)/5 = 0.31..., the next term after 3 is 5.

Jean-François Alcover and Charles R Greathouse IV, Table of n, a(n) for n = 1..5000 (first 88 terms from Alcover)
Eric Weisstein's MathWorld, Chebyshev functions
Wikipedia, Chebyshev function

Cf. A016040, A035158, A057872, A083535, A108310, A215013.

(* Adapted from PARI *) Reap[For[record = 2; theta = 0; p = 2, p < 2 * 10^8, p = NextPrime[p], theta = theta + Log[p] //N; d = Abs[1 - theta/p]; If[d < record, record = d; Print[p]; Sow[p]]]][[2, 1]]

/* Note: This program may fail if you replace 1e6 with a number larger than 1e17, and will certainly fail at some point below 1e316. These large numbers are not remotely feasible at the moment. */
r=th=0; forprime(p=2,1e6, th+=log(p); t=th/p; if(t>r, r=t; print1(p", "); if(t>1, warning("theta(n) > n, possible missed terms")))) \\ Charles R Greathouse IV, Dec 17 2014

cp's OEIS Frontend

A083533 First difference sequence of A002202. Difference between consecutive possible values of phi(n), the Euler totient function A000010.

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A083531 First difference sequence of A002191. Differences between possible values for sum of divisors of n.

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A035158 Floor of the Chebyshev function theta(n): a(n) = floor(Sum_{primes p <= n } log(p)).

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A083534 First difference sequence of A007617. Difference between consecutive values not being in the range of phi (A000010).

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A252398 Successive n with minimal relative distance |1-theta(n)/n|, where theta(n) = log(A034386(n)) is Chebyshev's theta function.

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