A007652 -id:A007652 - cp's OEIS frontend

A083786 Composite numbers mod 10.

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

Offset: 1

Roger L. Bagula, Aug 21 2003

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A002808, A007652.

Mod[Rest@ Select[Range@140, !PrimeQ@# &], 10] (* Robert G. Wilson v, Mar 17 2006 *)

lista(nn) = forcomposite(n=2, nn, print1(n % 10, ", ")); \\ Michel Marcus, Oct 30 2017

a(n) = A002808(n) mod 10. - Ray Chandler, Sep 04 2003

More terms from Ray Chandler, Sep 04 2003

Entries checked by Robert G. Wilson v, Mar 17 2006

A093336 Second digit of prime(n).

Original entry on oeis.org

1, 3, 7, 9, 3, 9, 1, 7, 1, 3, 7, 3, 9, 1, 7, 1, 3, 9, 3, 9, 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

Offset: 5

Cino Hilliard, Apr 25 2004

Matthew House, Table of n, a(n) for n = 5..10000

Cf. A077648, A007652.

IntegerDigits[#][[2]]&/@Prime[Range[5,110]] (* Harvey P. Dale, Dec 22 2013 *)

second(n) = { forprime(x=11,n, sd = mid(Str(x),2,1); print1(sd",") ) } \ 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) }

Offset corrected by Matthew House, Aug 09 2015

A106727 Triangle T(n,k) = (f(n+1)*f(k+1) mod 10), where f(j) = 10 - (prime(j+3) mod 10), read by rows.

Original entry on oeis.org

9, 7, 1, 1, 3, 9, 9, 7, 1, 9, 3, 9, 7, 3, 1, 1, 3, 9, 1, 7, 9, 3, 9, 7, 3, 1, 7, 1, 7, 1, 3, 7, 9, 3, 9, 1, 9, 7, 1, 9, 3, 1, 3, 7, 9, 7, 1, 3, 7, 9, 3, 9, 1, 7, 1, 1, 3, 9, 1, 7, 9, 7, 3, 1, 3, 9, 9, 7, 1, 9, 3, 1, 3, 7, 9, 7, 1, 9, 1, 3, 9, 1, 7, 9, 7, 3, 1, 3, 9, 1, 9

Offset: 0

Roger L. Bagula, May 14 2005

			Triangle begins:
  9;
  7, 1;
  1, 3, 9;
  9, 7, 1, 9;
  3, 9, 7, 3, 1;
  1, 3, 9, 1, 7, 9;
  3, 9, 7, 3, 1, 7, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007652, A072003.

f[n_]:= 10 - Mod[Prime[n+3], 10];
Table[Mod[f[n+1]*f[k+1], 10], {n,0,15}, {k,0,n}]//Flatten

def f(n): return 10 - (nth_prime(n+3)%10)
def A106727(n,k): return (f(n+1)*f(k+1))%10
flatten([[A106727(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Sep 10 2021

T(n, k) = (f(n+1)*f(k+1) mod 10) where f(j) = 10 - (prime(j+3) mod 10).

A137589 a(n) is the integer that results after deletion of all digits of n-th prime, except the initial digit and the final digit.

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, 11, 13, 17, 19, 13, 17, 11, 17, 19, 19, 11, 17, 13, 17, 13, 19, 11, 11, 13, 17, 19, 21, 23, 27, 29, 23, 29, 21, 21, 27, 23, 29, 21, 27, 21, 23, 23, 37, 31, 33, 37, 31, 37, 37, 39, 33

Offset: 1

Ctibor O. Zizka, Apr 26 2008

The plot of this sequence shows number of primes on the x-axis and the split of primes into 9 groups according to their first digit on the y-axis. The plot of a(n+1)/a(n) oscillates around 1 with decreasing amplitude. Log-periodic growth is seen on the plot of partial sums b(n)= Sum_(i=1..n) a(i).

			a(100) = 51 as prime(100) = 541. Concatenating the first and last digit gives 51. - _David A. Corneth_, Mar 23 2018

David A. Corneth, Table of n, a(n) for n = 1..10000

Another version of A138840 which is older.

Cf. A000040, A007652, A010879.

fdld[n_]:=Module[{idn=IntegerDigits[n]},FromDigits[{First[idn], Last[ idn]}]]; Join[Prime[Range[25]],fdld/@Prime[Range[26,100]]] (* Harvey P. Dale, Oct 10 2012 *)

a(n) = my(p = prime(n), d); if(n<=4, return(p)); d = digits(p); 10*d[1] + d[#d] \\ David A. Corneth, Mar 23 2018

a(n) = A138840(n) if n >= 5. - Omar E. Pol, Mar 23 2018

New name from Omar E. Pol, Mar 24 2018

A137727 Final digit of prime(n)*prime(n+1).

Original entry on oeis.org

6, 5, 5, 7, 3, 1, 3, 7, 7, 9, 7, 7, 3, 1, 1, 7, 9, 7, 7, 3, 7, 7, 7, 3, 7, 3, 1, 3, 7, 1, 7, 7, 3, 1, 9, 7, 1, 1, 1, 7, 9, 1, 3, 1, 3, 9, 3, 1, 3, 7, 7, 9, 1, 7, 1, 7, 9, 7, 7, 3, 9, 1, 7, 3, 1, 7, 7, 9, 3, 7, 7, 3, 1, 7, 7, 7, 3, 7, 9, 1, 9, 1, 3, 7, 7, 7, 3, 7, 3, 1, 3, 3, 7, 9, 7, 7, 9, 3, 3, 7, 9, 1, 7, 9, 7

Offset: 1

Alexander Adamchuk, Feb 08 2008

a(n) is 1, 3, 7, or 9 for n > 3. I conjecture that 1 and 9 appear 17/66 of the time and 3 and 7 appear 8/33 of the time. - Charles R Greathouse IV, Jan 03 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A006094 (Products of 2 successive primes), A007652 (Final digit of prime(n)), A010879 (final digit of n), A110923 (final two digits of prime(n) (with leading zero omitted)), A137728 (second digit from the end of product of first n primes).

Table[ Mod[ Prime[n]*Prime[n+1], 10 ], {n,1,1000} ]
Mod[Times@@@Partition[Prime[Range[110]],2,1],10] (* Harvey P. Dale, Oct 05 2014 *)

a(n)=prime(n)*prime(n+1)%10 \\ Charles R Greathouse IV, Dec 29 2012

from sympy import prime
def a(n): return (prime(n)*prime(n+1))%10
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Jun 05 2021

# much faster alternate for initial segment of sequence
from sympy import nextprime
def aupton(terms):
    p1, p2, alst = 2, 3, []
    while len(alst) < terms:
        p1, p2, alst = p2, nextprime(p2), alst + [(p1*p2)%10]
    return alst
print(aupton(105)) # Michael S. Branicky, Jun 05 2021

a(n) = A010879(A006094(n)). - Felix Fröhlich, Jun 05 2021

A137728 Second digit from the end of product of first n primes.

Original entry on oeis.org

0, 0, 3, 1, 1, 3, 1, 9, 7, 3, 3, 1, 1, 3, 1, 3, 7, 7, 9, 9, 7, 3, 9, 1, 7, 7, 1, 7, 3, 9, 3, 3, 1, 9, 1, 1, 7, 1, 7, 1, 9, 9, 9, 7, 9, 1, 1, 3, 1, 9, 7, 3, 3, 3, 1, 3, 7, 7, 9, 9, 7, 1, 7, 7, 1, 7, 7, 9, 3, 7, 1, 9, 3, 9, 1, 3, 7, 9, 9, 1, 9, 9, 9, 7, 3, 9, 1, 7, 7, 1, 7, 3, 1, 1, 9, 7, 3, 3, 9, 9, 3, 1, 3, 7, 7

Offset: 1

Alexander Adamchuk, Feb 08 2008

a(1) = a(2) = 0 because prime(1) = 2 and prime(1)*prime(2) = 6 are one-digit numbers.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007652 = Final digit of prime(n).
Cf. A110923 = Final two digits of prime(n).
Cf. A137727 = Final digit of prime(n)*prime(n+1).
Cf. A002110 = Primorial numbers, p#.

a[1]:= 0: a[2]:= 0: a[3]:= 3: p:= 5:
for n from 4 to 1000 do
  p:= nextprime(p);
  a[n]:= (a[n-1] * p) mod 10:
od: # Robert Israel, Nov 22 2018

a(1) = a(2) = 0, for n>2 Table[ Mod[ Product[ Prime[n], {n,1,k} ], 100 ]/10, {k,3,1000} ]

a(n) = A002110(n)/10 mod 10 for n > 2; a(1) = a(2) = 0.

A244191 a(n) = most common final digit for a prime < 10^n, or 0 if there is a tie.

Original entry on oeis.org

0, 3, 7, 3, 7, 3, 3, 7, 3, 3, 7, 7, 3, 3

Offset: 1

Derek Orr, Jun 22 2014

			For all 25 primes < 100 (10^2), we see that the last digit that appears the most is 3. Thus a(2) = 3.

Index entries for sequences related to final digits of numbers

Cf. A007652, A095178, A244192.

import sympy
from sympy import isprime
def prend(d,n):
  lst = []
  for k in range(10**n):
    if isprime(k):
      lst.append((k%10**d))
  new = 0
  newlst = []
  for i in range(10**(d-1),10**d):
    new = lst.count(i)
    newlst.append(new)
  newlst1 = newlst.copy()
  a = max(newlst1)
  newlst1[newlst1.index(a)] = 0
  b = max(newlst1)
  if a == b:
    return 0
  else:
    return newlst.index(max(a,b)) + 10**(d-1)
n = 2
while n < 10:
  print(prend(1,n),end=', ')
  n += 1

a(9)-a(14) from Hiroaki Yamanouchi, Sep 27 2014

A276481 Numbers k such that b(k) is prime, where b(1) = prime(1) = 2, b(n) = 10*b(n-1) + (prime(n) mod 10).

Original entry on oeis.org

1, 2, 4, 13, 16, 17, 28, 34, 90, 100, 132, 331, 534, 7923, 10157, 40197

Offset: 1

Thomas Ordowski, Sep 05 2016

Primes in the sequence b(n) are 2, 23, 2357, 2357137939171, ...

a(17) > 50000. - Michael S. Branicky, Aug 12 2025

Cf. A007652, A069151, A386964.

Res:= NULL: p:= 0: b:= 0:
for n from 1 to 600 do
  p:= nextprime(p);
  b:= 10*b + (p mod 10);
  if isprime(b) then Res:= Res, n fi
od:
Res; # Robert Israel, Sep 05 2016

b[1] = Prime@ 1; b[n_] := b[n] = 10 b[n - 1] + Mod[Prime@ n, 10]; Select[Range[10^3], PrimeQ@ b[#] &] (* Michael De Vlieger, Sep 06 2016 *)

b(n) = if (n==1, 2, 10*b(n-1) + (prime(n) % 10));
isok(n) = isprime(b(n)); \\ Michel Marcus, Sep 05 2016

list(lim)=my(v=List(),s,n); forprime(p=2,, if(n++>lim, return(Vec(v))); if(ispseudoprime(s=10*s+p%10), listput(v, n))) \\ Charles R Greathouse IV, Sep 05 2016

{k | A386964(k) is prime}. - Michael S. Branicky, Aug 12 2025

a(7)-a(13) from Michel Marcus, Sep 05 2016
a(14) from Robert Israel, Sep 05 2016
a(15)-a(16) from Michael S. Branicky, Aug 11 2025

A354589 Primes p starting a sequence of 4 consecutive primes whose final digits are 1,3,7,9 (in any order).

Original entry on oeis.org

11, 23, 47, 53, 67, 83, 89, 101, 109, 149, 167, 191, 193, 197, 199, 211, 251, 257, 263, 383, 443, 449, 461, 487, 557, 563, 587, 593, 599, 613, 647, 659, 739, 757, 761, 821, 859, 983, 991, 1061, 1063, 1069, 1117, 1217, 1223, 1283, 1301, 1303, 1367, 1433, 1439, 1447, 1481, 1553, 1567, 1571, 1579

Offset: 1

J. M. Bergot and Robert Israel, Aug 18 2022

			a(3) = 47 is in the sequence because the 4 consecutive primes starting with 47 are 47, 53, 59, 61, and their final digits 7,3,9,1 are a permutation of 1,3,7,9.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007652, A007811, A354590.

P:= select(isprime, [seq(i,i=3..2000,2)]):
P1:= P mod 10:
P[select(i -> convert(P1[i..i+3],set) = {1,3,7,9}, [$1..nops(P)-3])];

Select[Partition[Prime[Range[300]], 4, 1], Sort[Mod[#, 10]] == {1, 3, 7, 9} &][[;; , 1]] (* Amiram Eldar, Aug 19 2022 *)

from sympy import nextprime
from itertools import islice
def agen(): # generator of terms
    p = [2, 3, 5, 7]
    while True:
        if set(map(lambda x: x%10, p)) == {1, 3, 7, 9}: yield p[0]
        p = p[1:] + [nextprime(p[-1])]
print(list(islice(agen(), 60))) # Michael S. Branicky, Aug 18 2022

A371390 Numbers k such that prime(k), prime(k+1), prime(k+2), prime(k+3) and prime(k+4) all have the same last digit.

Original entry on oeis.org

11582, 17385, 19317, 20579, 22931, 42098, 51895, 52252, 55259, 60393, 62192, 62193, 62680, 64050, 65324, 71483, 76391, 76773, 76805, 77052, 81139, 86711, 95661, 100208, 102032, 113646, 113892, 113954, 115251, 124227, 125218, 125586, 144165, 144299, 147619, 147620

Offset: 1

Michel Lagneau, Mar 20 2024

			11582 is a term because prime(11582) = 123229, prime(11583) = 123239, prime(11584) = 123259, prime(11585) = 123269 with the same last digit 9.

Robert Israel, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI program

Cf. A000040, A007652, A107730, A129750.

nn:=15*10^4:d:=array(1..5):
for n from 1 to nn do:
 for k from 1 to 5 do:
   d[k]:=irem(ithprime(n+k-1),10):
 od:
  if d[1]=d[2] and d[1]=d[3] and
d[1]=d[4] and d[1]=d[5]
    then
     printf(`%d, `,n):
    else
  fi:
od:

PARI
```
\\ See PARI link
```

from itertools import count, islice
from sympy import nextprime
def A371390_gen(): # generator of terms
    xlist, p = [2, 3, 5, 7, 1], 11
    for k in count(1):
        if len(set(xlist)) == 1:
            yield k
        p = nextprime(p)
        xlist = xlist[1:]+[p%10]
A371390_list = list(islice(A371390_gen(),10)) # Chai Wah Wu, Apr 13 2024

cp's OEIS Frontend

A083786 Composite numbers mod 10.

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A093336 Second digit of prime(n).

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A106727 Triangle T(n,k) = (f(n+1)*f(k+1) mod 10), where f(j) = 10 - (prime(j+3) mod 10), read by rows.

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A137589 a(n) is the integer that results after deletion of all digits of n-th prime, except the initial digit and the final digit.

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A137727 Final digit of prime(n)*prime(n+1).

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A137728 Second digit from the end of product of first n primes.

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A244191 a(n) = most common final digit for a prime < 10^n, or 0 if there is a tie.

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