cp's OEIS Frontend

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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A073507 Number of primes == 7 (mod 10) less than 10^n.

Original entry on oeis.org

1, 6, 46, 308, 2411, 19621, 166211, 1440495, 12712314, 113764039, 1029518337, 9401997000, 86516367790
Offset: 1

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Author

Shyam Sunder Gupta, Aug 14 2002

Keywords

Comments

This and the related sequences A073505-A073517 and A006880, A073548-A073565 are included because there is interest in the distribution of primes by their initial or final digits.

Examples

			a(2)=6 because there are 6 primes == 7 (mod 10) less than 10^2. They are 7, 17, 37, 47, 67 and 97.
		

Crossrefs

Programs

  • Mathematica
    c = 0; k = 7; Do[While[k < 10^n, If[PrimeQ[k], c++ ]; k += 10]; Print[c], {n, 1, 10}]

Formula

A073505(n) + A073506(n) + a(n) + A073508(n) + 2 = A006880(n).

Extensions

Edited by Robert G. Wilson v, Oct 03 2002
a(10) from Robert G. Wilson v, Dec 22 2003
a(11)-a(13) from Giovanni Resta, Aug 07 2018

A073508 Number of primes == 9 (mod 10) less than 10^n.

Original entry on oeis.org

0, 5, 38, 303, 2390, 19593, 166032, 1440186, 12711333, 113761326, 1029509896, 9401974132, 86516371101
Offset: 1

Views

Author

Shyam Sunder Gupta, Aug 14 2002

Keywords

Comments

This and the related sequences A073505-A073517 and A006880, A073548-A073565 are included because there is interest in the distribution of primes by their initial or final digits.

Examples

			a(2) = 5 because there are 5 primes == 9 (mod 10) less than 10^2. They are 19, 29, 59, 79 and 89.
		

Crossrefs

Programs

  • Mathematica
    c = 0; k = 9; Do[While[k < 10^n, If[PrimeQ[k], c++ ]; k += 10]; Print[c], {n, 1, 10}]

Formula

A073505(n) + A073506(n) + A073507(n) + a(n) + 2 = A006880(n).

Extensions

Edited by Robert G. Wilson v, Oct 03 2002
a(10) from Robert G. Wilson v, Dec 22 2003
a(11)-a(13) from Giovanni Resta, Aug 07 2018

A308945 Number of totient numbers, phi(k), k <= 10^n, whose initial digit is 1.

Original entry on oeis.org

2, 20, 213, 2152, 21594, 216009, 2159776, 21595522, 215951111, 2159507603, 21595061256, 215950604593
Offset: 1

Views

Author

Frank M Jackson, Jul 02 2019

Keywords

Comments

The probability that a totient number starts with an initial 1 does not obey Benford's law however it does appear to tend to a constant value. In a sample of 10^9 totient numbers the distribution of initial digits 1 - 9 is approx. 21.595%, 20.774%, 16.457%, 12.682%, 7.904%, 6.633%, 5.505%, 4.634%, 3.816%.

Examples

			a(1)=2 as the first 10 totient numbers are {1, 1, 2, 2, 4, 2, 6, 4, 6, 4} and the occurrence of numbers with an initial 1 is 2.
		

Crossrefs

Programs

  • Mathematica
    lst1={}; Do[lst=Table[0, {n, 1, 9}]; Do[++lst[[First@IntegerDigits@EulerPhi[n]]], {n, 1, 10^m}]; AppendTo[lst1, lst[[1]]], {m, 1, 7}]; lst1
  • PARI
    a(n) = {k=0; for(j=1, 10^n, if(digits(eulerphi(j))[1]==1, k++)); k} \\ Jinyuan Wang, Jul 04 2019

Extensions

a(10)-a(12) from Giovanni Resta, Jul 04 2019
Previous Showing 21-23 of 23 results.