A045711 -id:A045711 - cp's OEIS frontend

A087766 Primes whose reversal is a multiple of 19.

59, 67, 83, 139, 163, 227, 251, 307, 331, 977, 1019, 1051, 1123, 1697, 1777, 1873, 2003, 2011, 2593, 2657, 2729, 2753, 2833, 3449, 3457, 3529, 3617, 4177, 4273, 4337, 4409, 4441, 4513, 4999, 5081, 5153, 5209, 5233, 5879, 6113, 6121

Offset: 1

Zak Seidov, Oct 03 2003

Mohammed Yaseen, Table of n, a(n) for n = 1..10000

Primes whose reversal is a multiple of k: A045711 (k=5), A087762 (k=7), A087764 (k=13), A087765 (k=17), this sequence (k=19), A087767 (k=23).

Select[Prime[Range[800]],Divisible[FromDigits[Reverse [IntegerDigits[#]]],19]&]  (* Harvey P. Dale, Mar 17 2011 *)

forprime(n=2, 7000, if(fromdigits(Vecrev(digits(n)))%19==0, print1(n, ", "))) \\ Mohammed Yaseen, Jul 20 2022

A087767 Primes whose reversal is a multiple of 23.

Original entry on oeis.org

29, 193, 223, 317, 349, 863, 1163, 1289, 1319, 1451, 1483, 1607, 1801, 2579, 2609, 2741, 2803, 3041, 3167, 3229, 3361, 3517, 4363, 4457, 4519, 4651, 5077, 5107, 5333, 5591, 5653, 5779, 6047, 6079, 6367, 6911, 7211, 7243, 7369, 7883

Offset: 1

Zak Seidov, Oct 03 2003

Mohammed Yaseen, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Primes whose reversal is a multiple of k: A045711 (k=5), A087762 (k=7), A087764 (k=13), A087765 (k=17), A087766 (k=19), this sequence (k=23).

Select[Prime[Range[1000]],Divisible[IntegerReverse[#],23]&] (* Harvey P. Dale, Apr 22 2022 *)

forprime(n=2, 8000, if(fromdigits(Vecrev(digits(n)))%23==0, print1(n, ", "))) \\ Mohammed Yaseen, Jul 20 2022

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

Original entry on oeis.org

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

A355983 Primes whose reversal is a multiple of 4.

Original entry on oeis.org

23, 29, 61, 67, 211, 233, 239, 251, 257, 271, 277, 293, 401, 409, 421, 443, 449, 461, 463, 467, 487, 613, 617, 619, 631, 653, 659, 673, 677, 691, 809, 821, 823, 827, 829, 863, 881, 883, 887, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2309, 2311, 2333, 2339, 2341, 2347, 2351

Offset: 1

Bernard Schott, Jul 22 2022

Equivalently, primes starting with 21, 23, 25, 27, 29, 40, 42, 44, 46, 48, 61, 63, 65, 67, 69, 80, 82, 84, 86, 88.

Subsequence of A355430.

			67 is a term since 67 is prime and 76 is divisible by 4.

Primes whose reversal is a multiple of k: A355430 (k=2), {3} (k=3), this sequence (k=4), A045711 (k=5), A087762 (k=7), {11} (k=11), A087764 (k=13), A087765 (k=17), A087766 (k=19), A087767 (k=23).

Select[Prime[Range[350]], Divisible[IntegerReverse[#], 4] &] (* Amiram Eldar, Jul 22 2022 *)

isok(p) = isprime(p) && !(fromdigits(Vecrev(digits(p))) % 4); \\ Michel Marcus, Jul 22 2022

A065684 Number of primes <= prime(n) which begin with a 5.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 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, 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, 3, 3, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11

Offset: 1

Reinhard Zumkeller, Nov 13 2001

			a(i) = 1 for 2 < i < 16 and a(16) = 2 as 53 = A000040(16) is the second prime beginning with 5. a(664579) = 72951 (A000040(664579) = 9999991 is the largest prime < 10^7).

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

Cf. A000040, A045711, A065680.

Accumulate@ Array[Boole[First@ IntegerDigits@ Prime@ # == 5] &, 103] (* Michael De Vlieger, Jun 14 2018 *)

lista(n) = { my(a=[p\10^logint(p,10)==5 | p<-primes(n)]); for(i=2, #a, a[i]+=a[i-1]); a} \\ Harry J. Smith, Oct 26 2009

A355984 Primes whose reversal is a multiple of 8.

Original entry on oeis.org

23, 61, 211, 251, 257, 293, 401, 409, 443, 449, 487, 631, 673, 677, 821, 823, 827, 829, 863, 2111, 2113, 2131, 2137, 2153, 2179, 2309, 2341, 2347, 2381, 2383, 2389, 2531, 2539, 2551, 2557, 2579, 2591, 2593, 2707, 2729, 2741, 2749, 2767, 2789, 2917, 2939, 2953, 2957

Offset: 1

Bernard Schott, Jul 25 2022

Subsequence of A355430 and of A355983.

			251 is a term since 251 is prime and 152 = 8 * 19.

Primes whose reversal is a multiple of k: A355430 (k=2), {3} (k=3), A355983 (k=4), A045711 (k=5), A087762 (k=7), this sequence (k=8), {11} (k=11), A087764 (k=13), A087765 (k=17), A087766 (k=19), A087767 (k=23)

Select[Prime[Range[500]], Divisible[IntegerReverse[#], 8] &] (* Amiram Eldar, Jul 25 2022 *)

is(n) = fromdigits(Vecrev(digits(n)))%8 == 0 \\ David A. Corneth, Jul 25 2022

More terms from David A. Corneth, Jul 25 2022

A355985 Primes whose reversal is a multiple of 16.

Original entry on oeis.org

23, 61, 211, 257, 449, 487, 821, 829, 863, 2131, 2137, 2179, 2551, 2557, 2591, 2593, 2707, 2741, 2749, 2789, 2939, 2971, 4013, 4019, 4051, 4057, 4091, 4093, 4099, 4201, 4241, 4243, 4283, 4289, 4621, 4663, 4813, 4817, 6121, 6163, 6311, 6317, 6353, 6359, 6397

Offset: 1

Bernard Schott, Jul 29 2022

			257 is a term since 257 is prime and 752 = 16 * 47.

Subsequence of A355430, A355983 and A355984.

Primes whose reversal is a multiple of k: A355430 (k=2), {3} (k=3), A355983 (k=4), A045711 (k=5), A087762 (k=7), A355984 (k=8), {11} (k=11), A087764 (k=13), this sequence (k=16), A087765 (k=17), A087766 (k=19), A087767 (k=23).

Select[Prime[Range[1000]], Divisible[IntegerReverse[#], 16] &] (* Amiram Eldar, Jul 29 2022 *)

is(n) = { fromdigits(Vecrev(digits(n)))%16==0 && isprime(n) } \\ Rémy Sigrist, Jul 29 2022

More terms from Rémy Sigrist, Jul 29 2022

A356246 Primes whose reversal is a multiple of 14.

Original entry on oeis.org

41, 89, 211, 223, 281, 293, 463, 487, 499, 691, 827, 839, 2129, 2213, 2237, 2333, 2357, 2441, 2477, 2503, 2539, 2647, 2659, 2693, 2731, 2767, 2851, 2887, 2971, 4021, 4057, 4091, 4153, 4177, 4261, 4273, 4297, 4409, 4517, 4637, 4649, 4721, 4733, 4877, 4889, 4903, 4973

Offset: 1

Bernard Schott, Jul 30 2022

Intersection of A087762 and A355430.

			281 is a term since 281 is prime and 182 = 14 * 13.

Primes whose reversal is a multiple of k: A074895 (k=1), A355430 (k=2), {3} (k=3), A355983 (k=4), A045711 (k=5), A087762 (k=7), A355984 (k=8), {11} (k=11), A087764 (k=13), this sequence (k=14), A355985 (k=16), A087765 (k=17), A087766 (k=19), A087767 (k=23).

Select[Prime[Range[666]], Divisible[IntegerReverse[#], 14] &] (* Amiram Eldar, Jul 30 2022 *)

cp's OEIS Frontend

A087766 Primes whose reversal is a multiple of 19.

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A087767 Primes whose reversal is a multiple of 23.

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

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A355430 Primes starting with an even decimal digit.

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A355983 Primes whose reversal is a multiple of 4.

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A065684 Number of primes <= prime(n) which begin with a 5.

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A355984 Primes whose reversal is a multiple of 8.

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A355985 Primes whose reversal is a multiple of 16.

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