Farideh Firoozbakht - cp's OEIS frontend

A262061 Least prime(i) such that prime(i)^(1+1/i) - prime(i) > n.

2, 3, 5, 7, 11, 11, 17, 17, 23, 29, 29, 37, 41, 53, 59, 67, 79, 89, 97, 127, 127, 137, 163, 179, 211, 223, 251, 293, 307, 337, 373, 419, 479, 521, 541, 587, 691, 727, 797, 853, 929, 1009, 1151, 1201, 1277, 1399, 1523, 1693, 1777, 1931, 2053, 2203, 2333, 2521, 2647, 2953, 3119, 3299, 3527, 3847, 4127

Offset: 1

Farideh Firoozbakht and Robert G. Wilson v, Sep 11 2015

nonn

Where A246778(i) first exceeds n, stated by p_i.
Similar to A245396.
Number of terms < 10^n: 4, 19, 41, 75, 120, 176, 242, 319, 407, 506, ..., .
Concerning Firoozbakht's Conjecture (1982): (prime(n+1))^(1/(n+1)) < prime(n)^(1/n), for all n = 1 or prime(n+1) < prime(n)^(1+1/n), which can be rewritten as: (log(prime(n+1))/log(prime(n)))^n < (1+1/n)^n. This suggests a weaker conjecture: (log(prime(n+1))/log(prime(n)))^n < e.
Prime index of a(n): 1, 1, 3, 4, 5, 5, 7, 7, 9, 10, 10, 12, 13, 16, 17, 19, 22, 24, 25, 31, 31, ..., .
All terms are unique for n > 21. Indices not unique: 1 & 2, 5 & 6, 7 & 8, 10 & 11 and 20 & 21.
The distribution of initial digits, 1...9, for a(n), n<508: 140, 91, 60, 50, 44, 36, 32, 27 and 26.

			a(20) = 127 since for all primes less than the 31st prime, 127, p_k^(32/31) - p_k are less than 20.
a(100) = 38113,
a(200) = 2400407,
a(300) = 57189007,
a(400) = 828882731,
a(500) = 8748565643,
a(1000) = 91215796479037,
a(1064) = 246842748060263, limit of Mathematica by direct computation, i.e., the first Mathematica line.

Paulo Ribenboim, The little book Of bigger primes, second edition, Springer, 2004, p. 185.

Farideh Firoozbakht and Robert G. Wilson v, Table of n, a(n) for n = 1..507
Alexei Kourbatov, Upper Bounds for Prime Gaps Related to Firoozbakht's Conjecture, arXiv:1506.03042 [math.NT], 2015.
Alexei Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015

Cf. A144104, A182134, A182514, A245396, A246777, A246778, A246781, A246810, A248855, A249566.

f[n_] := Block[{p = 2, k = 1}, While[n > p^(1 + 1/k) - p, p = NextPrime@ p; k++]; p]; Array[f, 60] (* or  quicker *)
(* or quicker *) p = 2; i = 1; lst = {}; Do[ While[ p^(1 + 1/i) < n + p, p = NextPrime@ p; i++]; AppendTo[lst, p]; Print[{n, p}], {n, 100}]; lst

a(n) = {i = 0; forprime(p=2,, i++; if (p^(1+1/i) - p > n, return (p)););} \\ Michel Marcus, Oct 04 2015

Log(y) ~= g + x^(1/2) where g = Euler's Gamma.

a(2) corrected in b-file by Andrew Howroyd, Feb 22 2018

A248858 a(n) is number of digits of the smallest term of the sequence A248857 which is of the form 4^n(5^(2n-1)10^m-1).

Original entry on oeis.org

4, 5, 7, 10, 14, 14, 16, 17, 42, 24, 47, 25, 29, 39, 31, 40, 41, 45, 40, 69, 46, 65, 807, 128, 67, 89, 55, 217, 87, 76, 74, 72, 99, 70, 75, 144, 78, 213, 96, 233, 114, 103, 457, 108, 156, 163, 140, 97, 127, 270, 109, 127, 167, 135, 128, 131, 196, 133, 309, 138, 148, 140, 813, 169, 135, 148, 235, 7235

Offset: 1

Farideh Firoozbakht, Jan 06 2015

			a(1) = number of digits of 1996 = 4.
a(2) = number of digits of 19984 = 5.
a(3) = number of digits of 1999936 = 7 and
a(68)=7235, because smallest prime of the form 5^(2*68-1)*10^m -1 is 5^135*10^7099-1 and 4^68(5^135*10^7099-1) has 7235 digits.

Cf. A248857.

A248857 Composite numbers n such that n - phi(n) is a power of 10.

Original entry on oeis.org

1320, 1640, 1768, 1996, 13200, 16400, 19984, 19996, 132000, 164000, 199996, 1320000, 1640000, 1999936, 13200000, 16400000, 16666240, 17999488, 18515584, 19999984, 19999996, 132000000, 164000000, 164296960, 166662400, 199999936, 199999984, 1320000000

Offset: 1

Farideh Firoozbakht, Dec 31 2014

nonn

Numbers n such that n - phi(n) is a positive power of 10.
Numbers n such that phi(n) = n - 10^floor(log(10,n)).
The most significant digit of all terms is equal to 1, since all terms are even and for even numbers n, phi(n) <= n/2.
If p = 5^(2n-1)*10^m-1 is prime then s = 4^n*p is in the sequence, because s - phi(s) = 10^(2n+m-1).
For n=1,2, ..., 6, ... smallest such term of the sequence respectively are 1996, 19984, 1999936, 1999999744, 19999999998976,19999999995904, ... .
Sequence A248858 gives number of digits of these terms.

			1320 is in the sequence because 1320 - phi(1320) = 10^3.

Giovanni Resta, Table of n, a(n) for n = 1..47 (terms < 10^12, first 41 terms from Robert G. Wilson v)

Cf. A000010, A067206, A066663, A244440, A248854, A248858.

a248857[n_] := Select[Select[Range@n, CompositeQ[#] &], IntegerQ[Log10[# - EulerPhi[#]]] &]; a248857[10^7] (* Michael De Vlieger, Jan 07 2015 *)

lista(nn) = forcomposite(n=2, nn, if (ispower(n-eulerphi(n),,&d) && (d==10), print1(n, ", "))); \\ Michel Marcus, Jan 06 2015

a(22)-a(28) from Giovanni Resta, Apr 17 2017

A248856 Numbers n such that n + pi(n) is a power of 10.

Original entry on oeis.org

1, 853, 91182, 926756, 9374193, 94535668, 951496285, 9563906973, 963706466000, 9665127969899, 96891533076641, 970995550452370, 9728143518403637, 97441817594570206, 975843062833251485, 9771174122943813068

Offset: 1

Farideh Firoozbakht, Dec 31 2014

Numbers n such that pi(n) equals 10^ceiling(log(10,n)) - n.
853 is the only known prime term of the sequence. If n is a prime term of the sequence and m = pi(n) then prime(m) + m is a power of 10. So 147 = pi(853) is the only known number m such that prime(m) + m is a power of 10. What is the next such number?
For each number n there exists at most one n-digit term.
a(11) = 96891533076641 is also prime. - Chai Wah Wu, May 25 2018

			pi(96891533076641) + 96891533076641 = 10^14 so 96891533076641 is in the sequence.

Cf. A000720, A244440, A248854.

Select[Range[1000], IntegerQ[Log[10, # + PrimePi[#]]] &] (* Alonso del Arte, Dec 31 2014 *)

for(n=1,10^3,s=digits(n+primepi(n)-1);if(s==[]||vecmin(s)==9,print1(n,", "))) \\ Derek Orr, Jan 02 2015

a(12)-a(16) from Chai Wah Wu, May 25 2018

A248854 Numbers n such that n - pi(n) is a power of 10.

Original entry on oeis.org

1, 2, 3, 16, 17, 132, 1196, 11373, 110486, 1084604, 10708554, 106091744, 1053422338, 10475688326, 104287176418, 1039019056245, 10358018863852, 103307491450819, 1030734020030317, 10287026204717357, 102692313540015923, 1025351434864118025, 10239531292310798955, 102270102190290407385

Offset: 1

Farideh Firoozbakht, Dec 31 2014

For n > 1 there exist at most two n-digit terms. If n is a term of the sequence and n + 1 is prime then n + 1 is also in the sequence.

Numbers n such that pi(n) is equal to n - 10^floor(log(10, n)). - Farideh Firoozbakht, Jan 01 2015

Cf. A000720, A248856, A248857.

A[1]:= 1: A[2]:= 2: A[3]:= 3:
count:= 3:
for k from 1 to 8 do
  x:= 10^k; y:= x + numtheory:-pi(x);
  while x < y and y < 10^(k+1) do
    x:= y; y:= 10^k + numtheory:-pi(x);
  od;
  count:= count+1; A[count]:= x;
  if isprime(x+1) then
      count:= count+1; A[count]:= x+1
  fi;
od:
seq(A[i],i=1..count); # Robert Israel, Dec 31 2014

Select[Range[1000], IntegerQ[Log[10, # - PrimePi[#]]] &] (* Alonso del Arte, Jan 01 2015 *)

for(n=1,10^3,s=digits(n-primepi(n)-1);if(s==[]||vecmin(s)==9,print1(n,", "))) \\ Derek Orr, Jan 02 2015

a(19)-a(24) from Giovanni Resta, Jun 07 2020

A248862 Primes p such that 900*p^2 is in the sequence A248861.

Original entry on oeis.org

2, 47, 59, 89, 173, 55439, 561599, 19824479

Offset: 1

Farideh Firoozbakht, Dec 12 2014

From Jason Yuen, Jul 01 2024: (Start)
For p>5, an equivalent condition is (240*p*(p-1))^(240*p*(p-1)) == 1 (mod 2821*(1+p+p^2)).
a(9) > 10^12 if it exists. (End)

Cf. A000040, A248861.

lastP=2;lst={2};While[lastP<200,If[
Mod[EulerPhi[900*NextPrime[lastP]^2]^EulerPhi[900*NextPrime[lastP]^2],DivisorSigma[1,900*NextPrime[lastP]^2]]==1,
AppendTo[lst,NextPrime[lastP]]];lastP=NextPrime[lastP]];lst (* Ivan N. Ianakiev, Dec 15 2014 *)

A248861 Numbers k such that phi(k)^phi(k) == 1 (mod sigma(k)).

Original entry on oeis.org

1, 2, 8, 36, 128, 225, 289, 578, 900, 2025, 2601, 3600, 10404, 32768, 41616, 45369, 57600, 242064, 665856, 725904, 783225, 1134225, 1140624, 1782225, 1988100, 2903616, 3132900, 4862025, 6155361, 6275025, 7128900, 7868025, 8625969, 10208025, 13505625

Offset: 1

Farideh Firoozbakht, Dec 12 2014

nonn

2^m is a term of the sequence if and only if m=2^j-1 where j is a nonnegative integer. Hence the sequence is infinite.
289 is a term of the sequence which is of the form p^2 where p is prime. What is the next such term?
578 is a term of the sequence which is not of the form 2^m or m^2. What is the next such term?
A248862 gives primes p such that 900*p^2 is a term of the sequence.
Subsequence of A055008. - Jason Yuen, Jul 01 2024

Jason Yuen, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A177006, A248862, A055008.

Prepend[Select[Range[30000], Mod[EulerPhi[#]^EulerPhi[#], DivisorSigma[1, #]] == 1 &], 1] (* Michael De Vlieger, Dec 13 2014 *)

isok(n) = my(in = eulerphi(n)); lift(Mod(in, sigma(n))^in - 1) == 0; \\ Michel Marcus, Dec 13 2014

A248902 Numbers k such that A248891(k) = 2.

Original entry on oeis.org

6, 7, 8, 10, 19, 21, 23, 24, 28, 33, 72, 80, 270, 271, 323, 404, 2058, 4118, 5986, 7118, 13387, 16041, 20164, 30024, 30025, 76955, 151818, 622213, 1012549, 2482211, 2482212, 6330859, 9988608, 14984623, 105181797, 180589455

Offset: 1

Jahangeer Kholdi and Farideh Firoozbakht, Nov 22 2014

If k is in {6, 7, 23, 270, 30024, 2482211, ...} then both numbers k and k+1 are in the sequence. It seems that this set has more members.

Cf. A001359, A248891, A248901.

a(27)-a(36) from Jinyuan Wang, Nov 28 2020

A248894 Numbers k such that both numbers k and k+1 are in the sequence A248903.

Original entry on oeis.org

17, 69, 70, 241, 405, 2606, 3339, 4116, 7116, 15148, 15149, 20162, 137633, 324410, 332504, 439298, 1012547, 1121608, 2482209, 5028662, 6330857, 7180864, 7180865, 9569168, 14452770, 17021632, 110229972, 110229973, 193329301

Offset: 1

Jahangeer Kholdi and Farideh Firoozbakht, Nov 22 2014

If k is in the set {69, 15148, 7180864, 110229972, ...} then both numbers k and k+1 are in the sequence. This means that k, k+1 and k+2 are in the sequence A248903. It seems that there exist more such numbers k.

Cf. A001359, A248891, A248903.

a(13)-a(29) from Jinyuan Wang, Nov 28 2020

A248855 a(n) is the smallest positive integer m such that if k >= m then a(k+1,n)^(1/(k+1)) <= a(k,n)^(1/k), where a(k,n) is the k-th term of the sequence {p | p and p+2n are primes}.

Original entry on oeis.org

1, 1, 1, 1, 3556, 1, 34, 3, 4, 1, 2, 1, 11285, 5, 2, 124, 569, 1, 290, 3, 1, 165, 2, 1, 1, 2, 1, 316, 1, 2, 58957, 1, 3, 58617, 522, 2, 1, 1, 4, 1, 2, 1, 1, 2, 1, 7932, 4, 1, 5875, 1679, 4, 4, 3, 3, 1, 2, 307, 1, 1, 1, 1, 1, 4, 3206, 2, 1, 1, 3, 2, 1, 1, 1, 1, 5, 2, 11170, 1, 2, 4245, 1, 1, 81, 2, 1, 1, 2, 58, 1, 3, 4, 7303, 1, 1, 5, 1, 3, 3, 3, 383, 111408, 1

Offset: 0

Farideh Firoozbakht and Jahangeer Kholdi, Nov 25 2014

nonn

All terms conjecturally are found. Note that according to the definition a(k,0) is the k-th term of the sequence {p | p is prime} namely for every positive integer k, a(k,0) = prime(k). Hence if Firoozbakht's conjecture is true then a(0)=1.

			a(0)=a(1)=a(2)=a(3)=1 conjecturally states that the four sequences A000040, A001359, A023200 and A023201 have this property: For every positive integer n, b(n) exists and b(n+1) < b(n)^(1+1/n). Namely b(n)^(1/n) is a strictly decreasing function of n.
If in the definition instead of the sequence {p | p and p+2n are primes} we set {p | p is prime and nextprime(p)=p+2n} then it seems that except for n=3 all terms of the new sequence {c(n)} are equal to 1 and for n=3, c(3)=7746. Note that c(3)=7746 means that the sequence {p | p is prime and nextprime(p)=p+6} = A031924 has this property: For all k >= 7746, A031924(k+1)^(1/(k+1)) < A031924(k)^(1/k).

Wikipedia, Firoozbakht's conjecture

Cf. A000040, A001359, A023200, A023201, A023202, A023203, A031924.

cp's OEIS Frontend

User: Farideh Firoozbakht

A262061 Least prime(i) such that prime(i)^(1+1/i) - prime(i) > n.

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A248858 a(n) is number of digits of the smallest term of the sequence A248857 which is of the form 4^n*(5^(2n-1)*10^m-1).

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A248857 Composite numbers n such that n - phi(n) is a power of 10.

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A248856 Numbers n such that n + pi(n) is a power of 10.

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A248854 Numbers n such that n - pi(n) is a power of 10.

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A248862 Primes p such that 900*p^2 is in the sequence A248861.

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A248861 Numbers k such that phi(k)^phi(k) == 1 (mod sigma(k)).

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A248902 Numbers k such that A248891(k) = 2.

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A248858 a(n) is number of digits of the smallest term of the sequence A248857 which is of the form 4^n(5^(2n-1)10^m-1).