A045708 -id:A045708 - cp's OEIS frontend

A065680 Number of primes <= prime(n) which begin with a 1.

0, 0, 0, 0, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25

Offset: 1

Reinhard Zumkeller, Nov 13 2001

Considering the frequency of all decimal digits in leading position of prime numbers (A065681 - A065687), we cannot apply Benford's Law. But we observe at 10^e - levels that the frequency for 0 to 9 decreases monotonically, at least in the small range until 10^7.
The "begins with 9" sequence is too dull to include. - N. J. A. Sloane
Note that the primes do not satisfy Benford's law (see A000040). - N. J. A. Sloane, Feb 08 2017

			13 is the second prime beginning with 1: A000040(6) = 13, therefore a(6) = 2. a(664579) = 80020 (A000040(664579) = 9999991 is the largest prime < 10^7).

Harry J. Smith, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Benford's Law
Index entries for sequences related to Benford's law

Cf. A065681, A065682, A065683, A065684, A065685, A065686, A065687, A000040.

For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509.

Accumulate[If[First[IntegerDigits[#]]==1,1,0]&/@Prime[Range[80]]] (* Harvey P. Dale, Jan 22 2013 *)

lista(n) = { my(a=[p\10^logint(p,10)==1 | p<-primes(n)]); for(i=2, #a, a[i]+=a[i-1]); a} \\ Harry J. Smith, Oct 26 2009

A355430 Primes starting with an even decimal digit.

Original entry on oeis.org

2, 23, 29, 41, 43, 47, 61, 67, 83, 89, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 809, 811, 821

Offset: 1

Bernard Schott, Jul 20 2022

Primes whose reversal is an even integer.

			43 is a term because 43 is prime and 34 is an even number.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Intersection of A000040 and A273892.
Equals disjoint union of A045708, A045710, A045712 and A045714.
Primes whose reversal is a multiple of k: this sequence (k=2), {3} (k=3), A045711 (k=5), A087762 (k=7), {11} (k=11), A087764 (k=13), A087765 (k=17), A087766 (k=19), A087767 (k=23).

imax=142; a={}; For[i=1, i<=imax, i++, If[EvenQ[FromDigits[Reverse[IntegerDigits[Prime[i]]]]], AppendTo[a,Prime[i]]]]; a (* Stefano Spezia, Jul 20 2022 *)
Select[Prime[Range[200]],EvenQ[IntegerDigits[#][[1]]]&] (* Harvey P. Dale, May 18 2025 *)

isok(k) = isprime(k) && !(fromdigits(Vecrev(digits(k))) % 2); \\ Michel Marcus, Jul 20 2022

from sympy import isprime
def ok(n): return str(n)[0] in "2468" and isprime(n)
print([k for k in range(822) if ok(k)]) # Michael S. Branicky, Jul 25 2022

from sympy import isprime
from itertools import chain, count, islice, product
def agen(): yield from chain((2,), (t for t in (b+i for d in count(1) for b in range(2*10**d, 10*10**d, 2*10**d) for i in range(1, 10**d, 2)) if isprime(t)))
print(list(islice(agen(), 62))) # Michael S. Branicky, Jul 25 2022

A036433 Number of divisors is a digit in the base 10 representation of n.

Original entry on oeis.org

1, 2, 14, 23, 29, 34, 46, 63, 68, 74, 76, 78, 88, 94, 116, 127, 128, 134, 138, 141, 142, 143, 145, 146, 164, 182, 184, 186, 189, 194, 196, 211, 214, 223, 227, 229, 233, 236, 238, 239, 241, 247, 248, 249, 251, 254, 257, 258, 261, 263, 268, 269, 271, 274, 277

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Invented by the HR concept formation program.

			14 has 4 divisors and 4 is a digit in the base 10 representation of 14.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics

Cf. A045708, A208272.

Cf. A000005.

a036433 n = a036433_list !! (n-1)
a036433_list = filter f [1..] where
   f x = d < 10 && ("0123456789" !! d) `elem` show x where d = a000005 x
-- Reinhard Zumkeller, Mar 15 2012

Select[Range[300],MemberQ[IntegerDigits[#],DivisorSigma[0,#]]&] (* Harvey P. Dale, Sep 02 2013 *)

from sympy import divisor_count
A036433_list = []
for i in range(1,10**5):
    d = divisor_count(i)
    if d < 10 and str(d) in str(i):
        A036433_list.append(i) # Chai Wah Wu, Jan 07 2015

A065681 Number of primes <= prime(n) which begin with a 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19

Offset: 1

Reinhard Zumkeller, Nov 13 2001

			After 2 and 23, 29 is the third prime beginning with 2: A000040(10) = 29, therefore a(10) = 3. a(664579) = 77025 (A000040(664579) = 9999991 is the largest prime < 10^7).

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A045708, A065680.

Accumulate@ Array[Boole[First@ IntegerDigits@ Prime@ # == 2] &, 87] (* Michael De Vlieger, Jun 14 2018 *)

lista(n) = { my(a=[p\10^logint(p,10)==2 | p<-primes(n)]); for(i=2, #a, a[i]+=a[i-1]); a} \\ Harry J. Smith, Oct 26 2009

cp's OEIS Frontend

A065680 Number of primes <= prime(n) which begin with a 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A355430 Primes starting with an even decimal digit.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Python

A036433 Number of divisors is a digit in the base 10 representation of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Python

A065681 Number of primes <= prime(n) which begin with a 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI