A007652 -id:A007652 - cp's OEIS frontend

A080172 Final digit of n-th Mersenne prime A000668(n).

3, 7, 1, 7, 1, 1, 7, 7, 1, 1, 7, 7, 1, 7, 7, 7, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 7, 7, 1, 7, 7, 1, 7, 1, 1, 1, 1, 1, 7, 7, 7, 1, 1, 7, 1, 1, 1

Offset: 1

Mark Dowdeswell, Feb 04 2003

Distribution of final digit for Mersenne primes appears (naturally) to be different from distribution for regular primes. Unconfirmed 49th, 50th and 51st digits in sequence are 1, 1, 1 (awaiting confirmation of 49th, 50th and 51st Mersenne primes).

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 16.

Great Internet Mersenne Prime Search (GIMPS), Distributed Computing Projects
Mersenne Primes, History, Theorems and Lists

Cf. A000668, A007652, A080173.

(Maple code from N. J. A. Sloane) # let s1 := list of terms in A000043
f:=n->if n mod 4 = 0 then 4 else n mod 4; fi; map(x->2^f(x)-1,s1);

Mod[2^MersennePrimeExponent[Range[48]]-1,10] (* Harvey P. Dale, Aug 09 2023; updated by Mark Dowdeswell, Sep 16 2024 *)

Updated by N. J. A. Sloane, Apr 01 2008
a(40)-a(47) from Ivan Panchenko, Apr 11 2018
a(48) from Mark Dowdeswell, Sep 16 2024

A211890 Triangle read by rows, where row n starts with n-th prime, followed by n primes in arithmetic progression; T(0,0) = 1 by convention.

Original entry on oeis.org

1, 2, 3, 3, 5, 7, 5, 11, 17, 23, 7, 37, 67, 97, 127, 11, 71, 131, 191, 251, 311, 13, 244243, 488473, 732703, 976933, 1221163, 1465393, 17, 6947, 13877, 20807, 27737, 34667, 41597, 48527, 19, 546859, 1093699, 1640539, 2187379, 2734219, 3281059, 3827899

Offset: 0

Reinhard Zumkeller, Jul 13 2012

T(n,0) = A000040(n) and T(n,k+1) - T(n,k) = A211889(n), 0 <= k < n.

			First 9 rows of triangle:
0:  1
1:  2 3
2:  3 5 7
3:  5 11 17 23
4:  7 37 67 97 127
5:  11 71 131 191 251 311
6:  13 244243 488473 732703 976933 1221163 1465393
7:  17 6947 13877 20807 27737 34667 41597 48527
8:  19 546859 1093699 1640539 2187379 2734219 3281059 3827899 4374739

Chai Wah Wu, Rows n=0..10 of triangle, flattened (rows n = 0..9 from Reinhard Zumkeller)
Eric Weisstein's World of Mathematics, Prime Arithmetic Progression.
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions

Cf. A007528, A007652, A132231, A214360, A008578, A002262.

a211890 n k = a211890_tabl !! n !! k
a211890_row n = a211890_tabl !! n
a211890_tabl = zipWith3 (\p k row -> map ((+ p) . (* k)) row)
                        a008578_list (0 : a211889_list) a002262_tabl

A039710 a(n) = n-th prime modulo 12.

Original entry on oeis.org

2, 3, 5, 7, 11, 1, 5, 7, 11, 5, 7, 1, 5, 7, 11, 5, 11, 1, 7, 11, 1, 7, 11, 5, 1, 5, 7, 11, 1, 5, 7, 11, 5, 7, 5, 7, 1, 7, 11, 5, 11, 1, 11, 1, 5, 7, 7, 7, 11, 1, 5, 11, 1, 11, 5, 11, 5, 7, 1, 5, 7, 5, 7, 11, 1, 5, 7, 1, 11, 1, 5, 11, 7, 1, 7, 11, 5, 1, 5, 1, 11

Offset: 1

Clark Kimberling

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A039701-A039706, A038194, A007652, A039709, A039711-A039715.

[p mod(12): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 12, n=1..100); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 12], {n, 100}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 12] (* Vincenzo Librandi, May 06 2014 *)

primes(100)%11 \\ Charles R Greathouse IV, Dec 12 2024

[mod(p, 12) for p in primes(500)] # Bruno Berselli, May 05 2014

Sum_k={1..n} a(k) ~ 6*n. - Amiram Eldar, Dec 11 2024

A039711 a(n) = n-th prime modulo 13.

Original entry on oeis.org

2, 3, 5, 7, 11, 0, 4, 6, 10, 3, 5, 11, 2, 4, 8, 1, 7, 9, 2, 6, 8, 1, 5, 11, 6, 10, 12, 3, 5, 9, 10, 1, 7, 9, 6, 8, 1, 7, 11, 4, 10, 12, 9, 11, 2, 4, 3, 2, 6, 8, 12, 5, 7, 4, 10, 3, 9, 11, 4, 8, 10, 7, 8, 12, 1, 5, 6, 12, 9, 11, 2, 8, 3, 9, 2, 6, 12, 7, 11, 6

Offset: 1

Clark Kimberling

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A039701-A039706, A038194, A007652, A039709, A039710, A039712-A039715.

[p mod(13): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 13, n=1..100); # Nathaniel Johnston, Jun 29 2011

Mod[ Prime@ Range@ 80, 13] (* Robert G. Wilson v, Mar 13 2011 *)

primes(1000)%13 \\ Charles R Greathouse IV, Mar 13 2011

[mod(p, 13) for p in primes(500)] # Bruno Berselli, May 05 2014

Sum_k={1..n} a(k) ~ (13/2)*n. - Amiram Eldar, Dec 11 2024

A039712 a(n) = n-th prime modulo 14.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 3, 5, 9, 1, 3, 9, 13, 1, 5, 11, 3, 5, 11, 1, 3, 9, 13, 5, 13, 3, 5, 9, 11, 1, 1, 5, 11, 13, 9, 11, 3, 9, 13, 5, 11, 13, 9, 11, 1, 3, 1, 13, 3, 5, 9, 1, 3, 13, 5, 11, 3, 5, 11, 1, 3, 13, 13, 3, 5, 9, 9, 1, 11, 13, 3, 9, 3, 9, 1, 5, 11, 5, 9

Offset: 1

Clark Kimberling

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A039701-A039706, A038194, A007652, A039709-A039711, A039713-A039715.

[p mod(14): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 14, n=1..100); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 14], {n, 100}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 14] (* Vincenzo Librandi, May 06 2014 *)

[mod(p, 14) for p in primes(500)] # Bruno Berselli, May 05 2014

Sum_k={1..n} a(k) ~ 7*n. - Amiram Eldar, Dec 12 2024

A110923 Final two digits of prime(n), with leading zero omitted.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 1, 3, 7, 9, 13, 27, 31, 37, 39, 49, 51, 57, 63, 67, 73, 79, 81, 91, 93, 97, 99, 11, 23, 27, 29, 33, 39, 41, 51, 57, 63, 69, 71, 77, 81, 83, 93, 7, 11, 13, 17, 31, 37

Offset: 1

Paolo P. Lava, Sep 23 2005

Primes modulo 100.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Index entries for sequences related to final digits of numbers.

Cf. A007652, A242119.

[p mod(100): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 07 2014

seq(ithprime(n) mod 100, n=1..100); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 100], {n, 100}] (* Ray Chandler, Oct 01 2005 *)
Mod[Prime[Range[100]], 100] (* Vincenzo Librandi, May 07 2014 *)

[mod(p, 100) for p in primes(500)] # Bruno Berselli, May 05 2014

Sum_k={1..n} a(k) ~ 50*n. - Amiram Eldar, Dec 13 2024

Edited, corrected and extended by Ray Chandler, Oct 01 2005

A166499 Concatenation of the rightmost digit of the n-th prime and the leftmost digit of the (n+1)th prime.

Original entry on oeis.org

23, 35, 57, 71, 11, 31, 71, 92, 32, 93, 13, 74, 14, 34, 75, 35, 96, 16, 77, 17, 37, 98, 38, 99, 71, 11, 31, 71, 91, 31, 71, 11, 71, 91, 91, 11, 71, 31, 71, 31, 91, 11, 11, 31, 71, 92, 12, 32, 72, 92, 32, 92, 12, 12, 72, 32, 92, 12, 72, 12, 32, 33, 73, 13, 33, 73, 13, 73, 73

Offset: 1

Zak Seidov, Oct 15 2009

This is the comma transform of the primes (see A367360).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A007652, A077648, A367360.

a:= n-> parse(cat(""||(ithprime(n))[-1],""||(ithprime(n+1))[1])):
seq(a(n), n=1..99);  # Alois P. Heinz, Nov 22 2023

With[{nmax=100},Map[10Mod[#[[1]],10]+IntegerDigits[#[[2]]][[1]]&,Partition[Prime[Range[nmax+1]],2,1]]] (* Paolo Xausa, Nov 24 2023 *)

a(n) = 10 * A007652(n) + A077648(n+1). - Alois P. Heinz, Nov 23 2023

A039713 a(n) = n-th prime modulo 15.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 2, 4, 8, 14, 1, 7, 11, 13, 2, 8, 14, 1, 7, 11, 13, 4, 8, 14, 7, 11, 13, 2, 4, 8, 7, 11, 2, 4, 14, 1, 7, 13, 2, 8, 14, 1, 11, 13, 2, 4, 1, 13, 2, 4, 8, 14, 1, 11, 2, 8, 14, 1, 7, 11, 13, 8, 7, 11, 13, 2, 1, 7, 2, 4, 8, 14, 7, 13, 4, 8, 14, 7

Offset: 1

Clark Kimberling

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A039701-A039706, A038194, A007652, A039709-A039712, A039714, A039715.

[p mod(15): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 15, n=1..100); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 15], {n, 100}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 15] (* Vincenzo Librandi, May 06 2014 *)

[mod(p, 15) for p in primes(500)] # Bruno Berselli, May 05 2014

Sum_k={1..n} a(k) ~ (15/2)*n. - Amiram Eldar, Dec 12 2024

A039714 a(n) = n-th prime modulo 16.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 1, 3, 7, 13, 15, 5, 9, 11, 15, 5, 11, 13, 3, 7, 9, 15, 3, 9, 1, 5, 7, 11, 13, 1, 15, 3, 9, 11, 5, 7, 13, 3, 7, 13, 3, 5, 15, 1, 5, 7, 3, 15, 3, 5, 9, 15, 1, 11, 1, 7, 13, 15, 5, 9, 11, 5, 3, 7, 9, 13, 11, 1, 11, 13, 1, 7, 15, 5, 11, 15, 5

Offset: 1

Clark Kimberling

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A039701-A039706, A038194, A007652, A039709-A039713, A039715.

[p mod(16): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 16, n=1..100); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 16], {n, 100}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 16] (* Vincenzo Librandi, May 06 2014 *)

[mod(p, 16) for p in primes(500)] # Bruno Berselli, May 05 2014

Sum_k={1..n} a(k) ~ 8*n. - Amiram Eldar, Dec 12 2024

A072003 10's complement of final digit of n-th prime.

Original entry on oeis.org

8, 7, 5, 3, 9, 7, 3, 1, 7, 1, 9, 3, 9, 7, 3, 7, 1, 9, 3, 9, 7, 1, 7, 1, 3, 9, 7, 3, 1, 7, 3, 9, 3, 1, 1, 9, 3, 7, 3, 7, 1, 9, 9, 7, 3, 1, 9, 7, 3, 1, 7, 1, 9, 9, 3, 7, 1, 9, 3, 9, 7, 7, 3, 9, 7, 3, 9, 3, 3, 1, 7, 1, 3, 7, 1, 7, 1, 3, 9, 1, 1, 9, 9, 7, 1, 7, 1, 3, 9, 7, 3, 1, 3, 9, 1, 7, 1, 9, 7, 9, 3, 3, 7, 1, 9

Offset: 1

Roger L. Bagula, Jun 18 2002

Cf. A007652.

a[n_] := 10-Mod[Prime[n], 10];
Table[ a[n], {n, 1, 105}]

a(n) = 10 - (prime(n) mod 10).

A007652(n) + a(n) = 10.

Edited by N. J. A. Sloane and Robert G. Wilson v, Jun 20 2002

cp's OEIS Frontend

A080172 Final digit of n-th Mersenne prime A000668(n).

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A211890 Triangle read by rows, where row n starts with n-th prime, followed by n primes in arithmetic progression; T(0,0) = 1 by convention.

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A039710 a(n) = n-th prime modulo 12.

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A039711 a(n) = n-th prime modulo 13.

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PARI

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A039712 a(n) = n-th prime modulo 14.

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Sage

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A110923 Final two digits of prime(n), with leading zero omitted.

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Magma

Maple

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Sage

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Extensions

A166499 Concatenation of the rightmost digit of the n-th prime and the leftmost digit of the (n+1)th prime.

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