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A045707 Primes with first digit 1. 27
11, 13, 17, 19, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Also primes with all divisors starting with digit 1. Complement of A206288 (nonprime numbers with all divisors starting with digit 1) with respect to A206287 (numbers with all divisors starting with digit 1). - Jaroslav Krizek, Mar 04 2012
Cohen and Katz show that the set of primes with first digit 1 has no natural density, but has supernatural/Dirichlet density log_{10} (2) ~= 0.3, the primes with first digit 2 have (supernatural) density log_{10} (3/2) ~= 0.176, ... and the primes with first digit 9 have density log_{10} (10/9) ~= 0.046. This would seem to explain the first digit phenomenon. Note that sum_{k = 1}^9 log_{10} (k+1)/k = 1. - Gary McGuire, Dec 22 2004
Lower density is 1/9, upper density is 5/9. The Dirichlet density, if it exists, is always between the lower and upper density (as it does and is in this case). - Charles R Greathouse IV, Sep 26 2022
LINKS
Daniel I. A. Cohen and Talbot M. Katz, Prime numbers and the first digit phenomenon, J. Number Theory 18 (1984), 261-268.
MATHEMATICA
Select[Table[Prime[n], {n, 500}], First[IntegerDigits[#]] == 1 &]
Flatten[Table[Prime[Range[PrimePi[10^n] + 1, PrimePi[2 * 10^n]]], {n, 3}]] (* Alonso del Arte, Jul 18 2014 *)
PROG
(Magma) [p: p in PrimesUpTo(10^4) | IsOne(Intseq(p)[#Intseq(p)])]; // Bruno Berselli, Jul 19 2014
(PARI) list(lim)=my(v=[]); for(d=1, #digits(lim\=1)-1, v=concat(v, primes([10^d, min(lim, 2*10^d-1)]))); v \\ Charles R Greathouse IV, Sep 26 2022
CROSSREFS
For primes with initial digit d (1 <= d <= 9) see A045707 (1, this sequence), A045708 (2), A045709 (3), A045710 (4), A045711 (5), A045712 (6), A045713 (7), A045714 (8), A045715 (9).
Column k=1 of A262369.
Sequence in context: A068492 A206287 A306661 * A032591 A088265 A356987
KEYWORD
nonn,base,easy
AUTHOR
EXTENSIONS
More terms from Erich Friedman.
Cohen-Katz reference from Victor S. Miller, Dec 21 2004
STATUS
approved

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Last modified March 28 10:31 EDT 2024. Contains 371240 sequences. (Running on oeis4.)