A007652 -id:A007652 - cp's OEIS frontend

A079950 Triangle of n-th prime modulo twice primes less n-th prime.

2, 3, 3, 1, 5, 5, 3, 1, 7, 7, 3, 5, 1, 11, 11, 1, 1, 3, 13, 13, 13, 1, 5, 7, 3, 17, 17, 17, 3, 1, 9, 5, 19, 19, 19, 19, 3, 5, 3, 9, 1, 23, 23, 23, 23, 1, 5, 9, 1, 7, 3, 29, 29, 29, 29, 3, 1, 1, 3, 9, 5, 31, 31, 31, 31, 31, 1, 1, 7, 9, 15, 11, 3, 37, 37, 37, 37, 37, 1, 5, 1, 13, 19, 15, 7, 3, 41

Offset: 1

Reinhard Zumkeller, Jan 19 2003

The right border of the triangle are the primes: T(n,n)=A000040(n); T(n,1)=A039702(n), T(n,2)=A039704(n) for n>1, T(n,3)=A007652(n) for n>2, T(n,4)=A039712(n) for n>3;

			Triangle begins:
  2;
  3, 3;
  1, 5, 5;
  3, 1, 7,  7;
  3, 5, 1, 11, 11;
  1, 1, 3, 13, 13, 13;
  1, 5, 7,  3, 17, 17, 17;
  ...

Cf. A001747, A079951, A079952, A079953.

A079950 := proc(n,k)
    modp(ithprime(n),2*ithprime(k)) ;
end proc:
seq(seq(A079950(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Sep 28 2017

T(n,k) = prime(n) % (2*prime(k));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Sep 21 2017

T(n, k) = prime(n) mod 2*prime(k), 1<=k<=n.

A089787 a(4n-3), a(4n-2), a(4n-1), and a(4n) are the units digit of the n-th prime followed by 1, 3, 7, and 9 respectively.

Original entry on oeis.org

21, 23, 27, 29, 31, 33, 37, 39, 51, 53, 57, 59, 71, 73, 77, 79, 11, 13, 17, 19, 31, 33, 37, 39, 71, 73, 77, 79, 91, 93, 97, 99, 31, 33, 37, 39, 91, 93, 97, 99, 11, 13, 17, 19, 71, 73, 77, 79, 11, 13, 17, 19, 31, 33, 37, 39, 71, 73, 77, 79, 31, 33, 37, 39, 91, 93, 97, 99, 11

Offset: 1

Roger L. Bagula, Jan 09 2004

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007652, A131712.

Cf. A089784 (essentially the same).

Flatten[Table[Mod[Prime[n], 10]*10+{1, 3, 7, 9}, {n, 1, 50}]]
a[n_] := Mod[Prime[Floor[(n + 3)/4]], 10]*10 + {1, 3, 7,
9}[[Mod[n - 1, 4] + 1]] (* Charles R Greathouse IV, Jan 02 2013 *)
Table[10*Mod[Prime[n],10]+{1,3,7,9},{n,30}]//Flatten (* Harvey P. Dale, Aug 24 2019 *)

a(n)=prime((n+3)\4)%10*10+[9,1,3,7][n%4+1] \\ Charles R Greathouse IV, Jan 02 2013

a(n) = 5+(-1)^n+3*cos(n*Pi/2)-3*sin(n*Pi/2)+10*(prime(floor((n+3)/4)) mod 10). - Wesley Ivan Hurt, May 06 2021

A101306 a(n) = Sum_{i=1..n} {last digit of prime(i)}.

Original entry on oeis.org

2, 5, 10, 17, 18, 21, 28, 37, 40, 49, 50, 57, 58, 61, 68, 71, 80, 81, 88, 89, 92, 101, 104, 113, 120, 121, 124, 131, 140, 143, 150, 151, 158, 167, 176, 177, 184, 187, 194, 197, 206, 207, 208, 211, 218, 227, 228, 231, 238, 247, 250, 259, 260, 261, 268, 271, 280

Offset: 1

Jorge Coveiro, Dec 22 2004

Asymptotically, a(n) ~ 5n by Dirichlet's theorem. - Charles R Greathouse IV, Sep 28 2008

			a(1) = 2;
a(2) = 2 + 3 = 5;
a(3) = 2 + 3 + 5 = 10;
a(4) = 2 + 3 + 5 + 7 = 17;
a(5) = 2 + 3 + 5 + 7 + 1(1) = 2 + 3 + 5 + 7 + 1 = 18.

Partial sums of A007652.

f[n_] := Sum[ Mod[ Prime[i], 10], {i, n}]; Array[ f, 60] (* Robert G. Wilson v, Dec 22 2004 *)
Rest@ FoldList[Plus, 0, Mod[Prime@ Range@ 60, 10]] (* Robert G. Wilson v, Jan 16 2011 *)

sum(k=1,n,prime(k)%10) \\ Charles R Greathouse IV, Sep 28 2008

Corrected and extended by Robert G. Wilson v, Dec 22 2004

A129750 Absolute difference of final digits of consecutive primes.

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 6, 6, 8, 6, 6, 2, 4, 4, 6, 8, 6, 6, 2, 6, 6, 6, 2, 6, 2, 4, 2, 6, 4, 6, 6, 2, 0, 8, 6, 4, 4, 4, 6, 8, 0, 2, 4, 2, 8, 2, 4, 2, 6, 6, 8, 0, 6, 4, 6, 8, 6, 6, 2, 0, 4, 6, 2, 4, 6, 6, 0, 2, 6, 6, 2, 4, 6, 6, 6, 2, 6, 8, 0, 8, 0, 2, 6, 6, 6, 2, 6, 2, 4, 2, 2, 6, 8, 6, 6, 8, 2, 2, 6, 0, 4, 6, 8, 6

Offset: 1

Giovanni Teofilatto, Jun 02 2007

Jason Yuen, Table of n, a(n) for n = 1..10000

Cf. A007652.

f[n_] := Abs[Mod[Prime[n + 1], 10] - Mod[Prime[n], 10]]; Array[f, 105] (* Ray Chandler, Jun 12 2007 *)

Corrected and extended by Ray Chandler, Jun 12 2007

A063272 Number of times most common final digit of primes appears in first n primes.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Jul 13 2001

			a(6)=2 since first six primes are 2,3,5,7,11,13, so most common final digit is 3 which has appeared twice.

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

Cf. A000040, A007652, A063272. Slightly above floor(n/4), i.e., A002265.

V:= Vector(9):
p:= 1:
for n from 1 to 100 do
  p:= nextprime(p);
  r:= p mod 10;
  V[r]:= V[r]+1;
  A[n]:= max(V)
od:
seq(A[i],i=1..100); # Robert Israel, Jul 22 2018

a(n) ~ n/phi(10) = n/4 by the Prime Number Theorem in Arithmetic Progressions. - Charles R Greathouse IV, Dec 29 2012

A089784 A nonsense sequence.

Original entry on oeis.org

81, 83, 87, 89, 71, 73, 77, 79, 51, 53, 57, 59, 31, 33, 37, 39, 11, 13, 17, 19, 31, 33, 37, 39, 71, 73, 77, 79, 91, 93, 97, 99, 31, 33, 37, 39, 91, 93, 97, 99, 11, 13, 17, 19, 71, 73, 77, 79, 11, 13, 17, 19, 31, 33, 37, 39, 71, 73, 77, 79, 31, 33, 37, 39, 91, 93, 97, 99, 11

Offset: 1

Roger L. Bagula, Jan 09 2004

Cf. A007652, A131712.

Cf. A089787 (essentially the same).

a={1, 3, 7, 9};
Flatten[Table[Mod[Abs[10-Prime[n]], 10]*10+a, {n, 50}]]

A093339 Middle digits of primes with an odd number of digits.

Original entry on oeis.org

2, 3, 5, 7, 0, 0, 0, 0, 1, 2, 3, 3, 3, 4, 5, 5, 6, 6, 7, 7, 8, 9, 9, 9, 9, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 0, 1, 1, 1, 3, 3, 4, 4, 5, 5, 6, 7, 7, 8, 8, 9, 0, 0, 1, 2, 3, 3, 3, 4, 4, 5, 6, 6, 6, 7, 8, 9, 9, 0, 0, 2, 2, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 0, 0, 1, 1, 1, 3, 4, 4, 4, 5, 5, 6, 7, 7, 8, 9, 0

Offset: 2

Cino Hilliard, Apr 25 2004

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A077648, A007652.

mdp[n_]:=Module[{idn=IntegerDigits[n],len=IntegerLength[n]},If[EvenQ[len], -1,idn[[(len-1)/2+1]]]]; Select[mdp/@Prime[Range[500]],NonNegative] (* Harvey P. Dale, Feb 26 2012 *)

midd(n) = { forprime(x=2,n, s = Str(x); ln = length(s); if(ln<2,p=1,p = (ln-1)/2+1); if(ln%2, md = eval(mid(s,p,1)); print1(md",") ) ) } \ Get a substring of length n from string str starting at position s in str. mid(str,s,n) = { v =""; tmp = Vec(str); ln=length(tmp); for(x=s,s+n-1, v=concat(v,tmp[x]); ); return(v) }

A105052 Write a(n) as a four-bit number; those bits state whether 10n+1, 10n+3, 10n+7 and 10n+9 are primes.

Original entry on oeis.org

6, 15, 5, 10, 14, 5, 10, 13, 5, 2, 15, 4, 2, 11, 1, 10, 6, 5, 8, 15, 0, 8, 7, 5, 8, 10, 5, 10, 12, 4, 2, 14, 0, 10, 3, 5, 2, 5, 5, 2, 9, 1, 8, 13, 5, 2, 14, 1, 2, 9, 5, 0, 12, 0, 10, 2, 5, 10, 2, 5, 10, 7, 0, 8, 14, 5, 8, 6, 4, 8, 9, 1, 2, 5, 4, 10, 9, 4, 2, 2, 1, 8, 15, 1, 0, 7, 4, 2, 14, 0, 2, 9, 1, 2

Offset: 0

Robert G. Wilson v, Apr 01 2005

Binary encoding of the prime-ness of the 4 integers r+10*n with remainder r=1, 3, 7 or 9. Classify the 4 integers 10n+r with r= 1, 3, 7, or 9, as nonprime or prime and associate bit positions 3=MSB,2,1,0=LSB with the 4 remainders in that order. Raise the bit if 10n+r is prime, erase it if 10n+r is nonprime. The sequence interprets the 4 bits as a number in base 2. a(n) is the decimal representation, obviously in the range 0<=a(n)<16. - Juri-Stepan Gerasimov, Jun 10 2008

			For n=2, the 4 numbers 21 (r=1), 23 (r=3), 27 (r=7), 29 (r=9) are nonprime, prime, nonprime, prime, which is rendered into 0101 = 2^0 + 2^2 = 5 = a(2).
These two hexadecimal lines represent the primes between 10 and 1010:
  F5AE5AD52F 42B1A658F0 8758A5AC42 E0A3525529 18D52E1295
  0C0A25A25A 708E586489 1254A94221 8F10742E02 912A42A4A1

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000040, A007652, A010051.
Cf. A030430, A030431, A030432, A030433.
Indices of 0,...,15: A032352, A216298, A216297, A216308, A216296, A216307, A216306 (also 0), A216299, A216295, A216305, A216304, A216300, A216303, A216301, A216302, A007811.
Cf. A140891 (analog in base 14, prime = bit 0, remainder 1 = LSB), A140387 (analog in base 30, prime = bit 0, remainder 1 = LSB).

f[n_] := FromDigits[ PrimeQ[ Drop[ Range[10n + 1, 10n + 9, 2], {3, 3}]] /. {True -> 1, False -> 0}, 2]; Table[ f[n], {n, 2, 93}]
f[n_] := If[ GCD[n, 10] == 1, If[PrimeQ@ n, 1, 0], -1]; FromDigits[#, 2] & /@ Partition[ DeleteCases[ Array[f, 940], -1], 4] (* Robert G. Wilson v, Jun 22 2012 *)
Table[FromDigits[Boole[PrimeQ[10n+{1,3,7,9}]],2],{n,0,100}] (* Harvey P. Dale, Nov 07 2016 *)

f(n)={s=0;if(isprime(10*n+1),s+=8);if(isprime(10*n+3), s+= 4);if(isprime(10*n+7),s+=2);if(isprime(10*n+9),s+=1); return(s)};for(n=0,93,print1(f(n),", ")) \\ Washington Bomfim, Jan 18 2011

Edited by Don Reble, Nov 08 2005
Further edited by R. J. Mathar, Jun 18 2008
Further edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A253721 Triprimes modulo 10.

Original entry on oeis.org

8, 2, 8, 0, 7, 8, 0, 2, 4, 5, 0, 2, 3, 6, 8, 0, 5, 6, 8, 2, 8, 9, 2, 5, 0, 4, 6, 7, 4, 5, 0, 8, 7, 8, 3, 4, 4, 5, 0, 1, 2, 4, 5, 2, 6, 8, 0, 5, 7, 2, 2, 0, 1, 6, 8, 2, 4, 5, 6, 5, 8, 1, 6, 8, 3, 5, 9, 2, 4, 5, 6, 0, 2, 0, 6, 8, 2, 5, 2, 3, 8, 3, 5, 4, 6, 7

Offset: 1

Wesley Ivan Hurt, May 02 2015

Last digit of triprimes (A014612).

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

Cf. A010879 (final digit of n), A014612 (triprimes).
Cf. A007652 (primes mod 10), A106146 (semiprimes mod 10).
Cf. A255646 (subsequence).

a253721 = flip mod 10 . a014612  -- Reinhard Zumkeller, May 05 2015

with(numtheory): A253721:=n->`if`(bigomega(n) = 3, n mod 10, NULL): seq(A253721(n), n=1..500);

Mod[#, 10] & /@ Select[Range[500], PrimeOmega[#] == 3 &]

do(x)=my(v=List(), t); forprime(p=2, x\4, forprime(q=2, min(x\(2*p), p), t=p*q; forprime(r=2, min(x\t, q), listput(v, t*r)))); Set(v)%10 \\ Charles R Greathouse IV, Aug 30 2017

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A253721(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m%10 # Chai Wah Wu, Aug 17 2024

a(n) = A010879(A014612(n)). - Michel Marcus, May 03 2015

A386964 a(1) = prime(1) = 2, a(n) = 10*a(n-1) + (prime(n) mod 10).

Original entry on oeis.org

2, 23, 235, 2357, 23571, 235713, 2357137, 23571379, 235713793, 2357137939, 23571379391, 235713793917, 2357137939171, 23571379391713, 235713793917137, 2357137939171373, 23571379391713739, 235713793917137391, 2357137939171373917, 23571379391713739171, 235713793917137391713

Offset: 1

Michael S. Branicky, Aug 11 2025

Michael S. Branicky, Table of n, a(n) for n = 1..1000

Cf. A007652, A276481.

a:= proc(n) option remember; `if`(n<1, 0, a(n-1)*10+irem(ithprime(n), 10)) end:
seq(a(n), n=1..21);  # Alois P. Heinz, Aug 12 2025

a[1]=2;a[n_]:=10a[n-1]+Mod[Prime[n],10];Array[a,21] (* James C. McMahon, Aug 12 2025 *)

from sympy import nextprime
from itertools import islice
def A386964(): # generator of terms
    an = pn = 2
    while True:
        yield an
        an = 10*an + (pn:=nextprime(pn))%10
print(list(islice(A386964(), 21)))

a(n) = concatenation of A007652(1)..A007652(n).

cp's OEIS Frontend

A079950 Triangle of n-th prime modulo twice primes less n-th prime.

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A089787 a(4n-3), a(4n-2), a(4n-1), and a(4n) are the units digit of the n-th prime followed by 1, 3, 7, and 9 respectively.

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A101306 a(n) = Sum_{i=1..n} {last digit of prime(i)}.

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A129750 Absolute difference of final digits of consecutive primes.

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A063272 Number of times most common final digit of primes appears in first n primes.

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A089784 A nonsense sequence.

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A093339 Middle digits of primes with an odd number of digits.

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A105052 Write a(n) as a four-bit number; those bits state whether 10n+1, 10n+3, 10n+7 and 10n+9 are primes.

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