A007652 -id:A007652 - cp's OEIS frontend

A048199 Distance of primes to next odd multiple of 5 (where n mod 10 = 5).

3, 2, 0, 8, 4, 2, 8, 6, 2, 6, 4, 8, 4, 2, 8, 2, 6, 4, 8, 4, 2, 6, 2, 6, 8, 4, 2, 8, 6, 2, 8, 4, 8, 6, 6, 4, 8, 2, 8, 2, 6, 4, 4, 2, 8, 6, 4, 2, 8, 6, 2, 6, 4, 4, 8, 2, 6, 4, 8, 4, 2, 2, 8, 4, 2, 8, 4, 8, 8, 6, 2, 6, 8, 2, 6, 2, 6, 8, 4, 6, 6, 4, 4, 2, 6, 2, 6, 8, 4, 2, 8, 6, 8, 4, 6, 2, 6, 4, 2, 4, 8, 8, 2, 6, 4

Offset: 1

Enoch Haga

			Take first prime 2, subtract from 5: 5-2=3, distance to 5; take 5-5 = 0; or 4th term, 15-7=8. Sequence may be converted to distance to nearest value of n (absolute value, where n mod 10 = 5) by changing all 8's to 2's and all 6's to 4's.

Cf. A048198, A007652.

a(n) = {p = prime(n); k = 0; while ((p+k) % 10 != 5, k++); k;} \\ Michel Marcus, Aug 25 2013

A122754 a(n) = 10*n - A101306(n).

Original entry on oeis.org

8, 15, 20, 23, 32, 39, 42, 43, 50, 51, 60, 63, 72, 79, 82, 89, 90, 99, 102, 111, 118, 119, 126, 127, 130, 139, 146, 149, 150, 157, 160, 169, 172, 173, 174, 183, 186, 193, 196, 203, 204, 213, 222, 229, 232, 233, 242, 249, 252, 253

Offset: 1

Roger L. Bagula, Sep 21 2006

Cf. A101306.

Partial sums of A072003.

Table[Sum[10 - Mod[Prime[n], 10], {n, 1, m}], {m, 1, 50}]

a(n) = Sum_{i=1..n} (10 - A007652(n)).

Definition simplified by the Assoc. Eds. of the OEIS, Mar 27 2010

A161625 Sum of all numbers from 1 up to the final digit of prime(n).

Original entry on oeis.org

3, 6, 15, 28, 1, 6, 28, 45, 6, 45, 1, 28, 1, 6, 28, 6, 45, 1, 28, 1, 6, 45, 6, 45, 28, 1, 6, 28, 45, 6, 28, 1, 28, 45, 45, 1, 28, 6, 28, 6, 45, 1, 1, 6, 28, 45, 1, 6, 28, 45, 6, 45, 1, 1, 28, 6, 45, 1, 28, 1, 6, 6, 28, 1, 6, 28, 1, 28, 28, 45, 6, 45, 28, 6, 45, 6, 45, 28, 1, 45

Offset: 1

Juri-Stepan Gerasimov, Jun 15 2009

			a(1)=3=1+2. a(2)=6=1+2+3. a(3)=15=1+2+3+4+5. a(4)=28=1+2+3+4+5+6+7. a(5)=1=1. a(6)=3=1+2.

Cf. A000027, A000040, A161570.

A007652 := proc(n) ithprime(n) mod 10; end: A000217 := proc(n) n*(n+1)/2 ; end:
A161625 := proc(n) A000217(A007652(n)) ; end: seq(A161625(n),n=1..80) ; # R. J. Mathar, Jun 16 2009

a(n)= A000217(A007652(n)). - R. J. Mathar, Jun 16 2009

Definition reworded and a 16 split in 1,6 by R. J. Mathar, Jun 16 2009

A175348 Last digit of p^p, where p is the n-th prime.

Original entry on oeis.org

4, 7, 5, 3, 1, 3, 7, 9, 7, 9, 1, 7, 1, 7, 3, 3, 9, 1, 3, 1, 3, 9, 7, 9, 7, 1, 7, 3, 9, 3, 3, 1, 7, 9, 9, 1, 7, 7, 3, 3, 9, 1, 1, 3, 7, 9, 1, 7, 3, 9, 3, 9, 1, 1, 7, 7, 9, 1, 7, 1, 7, 3, 3, 1, 3, 7, 1, 7, 3, 9, 3, 9, 3, 3, 9, 7, 9, 7, 1, 9, 9, 1, 1, 3, 9, 7, 9, 7, 1, 7, 3, 9, 3, 1, 9, 7, 9, 1, 7, 1, 3, 7, 7, 9, 1

Offset: 1

Charles R Greathouse IV, Apr 19 2010

Euler and Sadek ask whether the sequence, interpreted as the decimal expansion N = 0.47531..., is rational or irrational.

Dickson's conjecture implies that each finite sequence with values in {1,3,7,9} occurs as a substring. In particular, this implies that the above N is irrational. - Robert Israel, Jan 26 2017

			prime(4) = 7 and 7^7 = 823543, so a(4) = 3.

R. Euler and J. Sadek, A number that gives the unit digit of n^n. Journal of Recreational Mathematics, 29:3 (1998), pp. 203-204.

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

Cf. A000040, A007652, A056849, A137807.

R:= [seq(i &^ i mod 10, i=1..20)]:
seq(R[ithprime(i) mod 20],i=1..100); # Robert Israel, Jan 26 2017

Table[PowerMod[n,n,10],{n,Prime[Range[110]]}] (* Harvey P. Dale, Mar 24 2024 *)

a(n)=[1,4,7,0,5,0,3,0,9,0,1,0,3,0,0,0,7,0,9][prime(n)%20]

a(n) = A056849(A000040(n)). - Robert Israel, Jan 26 2017

A253936 a(n) = prime(n + (prime(n) mod 10)).

Original entry on oeis.org

5, 11, 19, 31, 13, 23, 43, 59, 37, 67, 37, 67, 43, 59, 79, 67, 101, 67, 101, 73, 89, 127, 101, 137, 131, 103, 113, 149, 163, 137, 163, 137, 173, 191, 193, 157, 193, 179, 199, 191, 229, 191, 193, 211, 239, 257, 223, 233, 263, 277, 251, 283, 251, 257, 293, 277, 317, 277, 317, 283, 311, 313, 349, 313, 337, 367, 337, 379, 383

Offset: 1

Zak Seidov, Jan 23 2015

			n=1: p=prime(1)=2; p mod 10 = 2, prime(1+2) = 5 = a(1);
n=6: p=prime(6)=13; p mod 10 = 3, prime(6+3) = 23 = a(6).

Zak Seidov, Color graph for first 100 terms

Cf. A000040, A007652, A010879.

Table[Prime[n+Mod[Prime[n],10]],{n,100}]

PARI
```
a(n)=prime(n+prime(n)%10)
```

a(n) = A000040(n + A010879(A000040(n))),

a(n) = A000040(n + A007652(n)).

A290630 a(n+1) = a(n) + (final digit of greatest prime < a(n)); a(1)=3.

Original entry on oeis.org

3, 5, 8, 15, 18, 25, 28, 31, 40, 47, 50, 57, 60, 69, 76, 79, 82, 91, 100, 107, 110, 119, 122, 125, 128, 135, 136, 137, 138, 145, 154, 155, 156, 157, 158, 165, 168, 175, 178, 181, 190, 191, 192, 193, 194, 197, 200, 209, 218, 219, 220, 221, 222, 223, 224, 227, 230, 239, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252

Offset: 1

David James Sycamore, Aug 07 2017

Let a(n) be composite, and gp(n) be the greatest prime less than a(n), with final digit d(n). Then sp(n), the smallest prime greater than a(n) is in the sequence if and only if d(n) divides (sp(n) - a(n)), in which case (sp(n) - a(n))/d(n) is the integer m, and a(n + m) = sp(n).
If a(n) is prime, with final digit d'(n) then sp(n) (the next prime after a(n)) is in the sequence if and only if sp(n) = a(n)+ d(n) + r * d'(n) for some r >= 1, in which case
a(n + r + 1) = sp(n).
Primes appearing in the sequence are 3, 5, 31, 47, 79, 107, 137, 157, 181, 191, 193, 197, ... If a prime occurs at a(n), a(n+2) = a(n+1) + d(n) only if there is no prime between a(n) and a(n+1).
Primes in the sequence whose final digit is not contributed to a subsequent term include 3, 5, 31, 107, 197, ...
Primes not appearing in the sequence but which contribute a final digit include 2, 7, 13, 17, 23, 29, 37, ...

			a(2) = a(1) + gp(1) = 3 + 2 = 5.
a(59) = 242 (composite), gp(59) = 241, and d(59) = 1. sp(59) = 251 is in the sequence because (sp(59) - a(59))/d(59) = (251 - 242)/1 = 9 (= m). Therefore a(59 + 9) = a(68) = 251.
a(40) = 181 (prime), d'(40) = 1, gp(40) = 179, d(40) = 9. Then sp(40) = 191 is in the sequence because with r = 1,
a(40) + d(40) + r*d'(40) = 181 + 9 + 1*1 = 191 = a(40+1+1) = a(42).

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

Cf. A000040, A007652.

A[1]:= 3:
for i from 2 to 100 do
  A[i]:= A[i-1] + (prevprime(A[i-1]) mod 10)
od:
seq(A[i],i=1..100); # Robert Israel, Aug 13 2019

NestList[# + Mod[NextPrime[#, -1], 10] &, 3, 68] (* Michael De Vlieger, Aug 19 2017 *)

lista(nn) = {print1(a = 3, ", "); for (n=2, nn, a = a + precprime(a-1) % 10; print1(a, ", "););} \\ Michel Marcus, Aug 19 2017

a(n+1) = a(n) + d(n) where d(n) = A007652(gp(n)); gp(n) = greatest prime < a(n).

A302660 a(n) = (prime(n) mod 9) + (prime(n) mod 10).

Original entry on oeis.org

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

Offset: 1

Dimitris Valianatos, Apr 11 2018

The sum (prime(n) mod 9 + prime(n) mod 10) gives numbers between 2 and 17.
For large n the distribution is displayed in the diagram below.
                                                           .
    ^
    |
  3y| .. . . . . . . . . .. o        o
    |                      /:\      /:\
    |                     / : \    / : \
  2y| .. . . . . . o      / :  o--o  : \      o
    |             /:\    /  :  :  :  :  \    /:\
    |            / | \   /  :  |  |  :  \   / | \
   y| ..  o--o--o  :  o--o  :  :  :  :  o--o  :  o--o--o
    |    /.  .  .  |  .  .  :  |  |  :  .  .  |  .  .  .\
    |   / .  .  .  :  .  .  :  :  :  :  .  .  :  .  .  . \
    |_o__o__o__o__o__o__o__o__o__o__o__o__o__o__o__o__o__o_\
    0  1  2  3  4  5  6  7  8  9  10 11 12 13 14 15 16 17 18 /
                                                             .
If y is the quantity for {2, 3, 4, 6, 7, 12, 13, 15, 16, 17} (same)
then 2y is the quantity of {5, 9, 10, 14} (same) and
3y is the quantity for {8, 11} (same).
Example: For primes less than 10^10, the distribution of frequencies of a(n) from 2 to 17 is {18960677, 18960726, 18960712, 37920181, 18959991, 18960427, 56880630, 37923467, 37921201, 56882003, 18960991, 18960869, 37920879, 18960270, 18959802, 18959685}.

			For n=7, prime(7) = 17, 17 mod 9 = 8 and 17 mod 10 = 7. So a(7) = 8 + 7 = 15.

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

Cf. A007652, A038194.

[(NthPrime(n) mod 9) + (NthPrime(n) mod 10): n in [1..100]]; // Vincenzo Librandi, Jun 10 2018

map(t -> (t mod 9)+(t mod 10), [seq(ithprime(i),i=1..100)]); # Robert Israel, Jun 10 2018

Array[Mod[#, 9] + Mod[#, 10] &@ Prime@ # &, 95] (* Michael De Vlieger, Apr 21 2018 *)

{forprime(n = 2, 1000, s = n%9 + n%10; print1(s", "))}

a(n) = A038194(n) + A007652(n).

A308572 a(n) = Fibonacci(2*prime(n)).

Original entry on oeis.org

3, 8, 55, 377, 17711, 121393, 5702887, 39088169, 1836311903, 591286729879, 4052739537881, 1304969544928657, 61305790721611591, 420196140727489673, 19740274219868223167, 6356306993006846248183, 2046711111473984623691759, 14028366653498915298923761, 4517090495650391871408712937

Offset: 1

Christopher Hohl, Jun 08 2019

nonn

This sequence is noteworthy in light of the congruence relation shared by a(n) and prime(n).  Namely, for n > 2, a(n) == prime(n) (mod 10).  That is, the last digit of prime(n) is 'preserved' as the last digit of a(n). See A007652.
As well, extending the notion, one notes that for k == 1 (mod 4), Fibonacci(2^k * prime(n)) == prime(n) (mod 10).
For any prime number p, the Fibonacci number F_(2p) == -(2p/5) (mod p), where -(2p/5) is the Legendre or Jacobi symbol. - Yike Li, Aug 30 2022

			a(4) = 377, because prime(4) = 7, 2*7 = 14, and Fibonacci(14) is 377.

Robert Israel, Table of n, a(n) for n = 1..355

Cf. A000045, A100484, A007652, A054452.

f:= n -> combinat:-fibonacci(2*ithprime(n)):
map(f, [$1..30]); # Robert Israel, Oct 23 2019

a(n) = fibonacci(2*prime(n)); \\ Michel Marcus, Jun 08 2019

a(n) = A000045(A100484(n)). - Michel Marcus, Jun 08 2019

More terms from Michel Marcus, Jun 08 2019

A354590 a(n) is the first prime that is the start of a sequence of exactly n consecutive primes that are in A354589.

Original entry on oeis.org

11, 47, 251, 9431, 191, 19457, 280627, 2213, 1006253, 9129563, 66945301, 184171621, 726512053, 2732087209, 10206934519, 59883612989, 25650350371

Offset: 1

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

			a(3) = 251 because the 4 primes starting with 251 are 251, 257, 263, 269, and 251, 257 and 263 are in A354589 but 269 is not; and 251 is the least prime for which this works.

Cf. A007652, A354589.

P:= select(isprime, [seq(i,i=1..10^7,2)]):
nP:= nops(P);
P10:= P mod 10:
s:= 0: V:= Vector(10):
for i from 1 to nP-3 do
  if convert(P10[i..i+3],set) = {1,3,7,9} then s:= s+1
  else
    if s > 0 and s <= 10 and V[s] = 0 then V[s]:= P[i-s] fi;
    s:= 0
  fi
od:
convert(V, list);

a(11)-a(14) from Michael S. Branicky, Aug 18 2022

a(15)-a(17) from Michael S. Branicky, Aug 20 2022

A381869 Smallest starting prime for which the sum of 2*n consecutive primes is 0 modulo 10, or -1 if no such prime exists.

Original entry on oeis.org

13, 11, 7, 7, 13, 17, 7, 17, 37, 3, 7, 41, 7, 7, 11, 11, 11, 11, 11, 13, 11, 13, 11, 7, 7, 17, 7, 43, 41, 3, 3, 13, 11, 7, 13, 19, 7, 11, 11, 29, 7, 43, 3, 7, 11, 13, 23, 29, 3, 7, 7, 11, 11, 11, 19, 13, 5, 5, 13, 37, 17, 3, 3, 7, 17, 17, 3, 11, 19, 13, 3, 7, 23

Offset: 1

Jean-Marc Rebert, Mar 09 2025

nonn

			a(1) = 13, because 13 and 17 are 2*1 = 2 consecutive primes such that 13 + 17 = 20 and 20 modulo 10 = 0, and no smaller prime has this property.

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

Cf. A007652, A111324.

P:= select(isprime,[2,seq(i,i=3..10^6,2)]):
S:= ListTools:-PartialSums(P):
f:= proc(n) local j,t;
  for j from 1 do
    if S[2*n+j] - S[j] mod 10 = 0 then return P[j+1] fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, May 08 2025

Do[i=1;Until[Mod[Total[Prime[Range[i,i+2*n-1]]],10]==0,i++];a[n]=Prime[i],{n,73}];Array[a,73] (* James C. McMahon, Mar 23 2025 *)

isok(p, n) = my(i=primepi(p), q=prime(2*n+i-1)); vecsum(apply(x->Mod(x,10), primes([p, q]))) == 0;
a(n) = my(p=3); while (!isok(p, n), p=nextprime(p+1)); p; \\ Michel Marcus, Mar 09 2025

from sympy import nextprime, prime, sieve
def a(n):
    plst = list(sieve.primerange(3, prime(2*n+1)+1))
    s = sum(plst)
    while s%10:
        q = nextprime(plst[-1])
        s += (q-plst[0])
        plst = plst[1:] + [q]
    return plst[0]
print([a(n) for n in range(1, 74)]) # Michael S. Branicky, Mar 09 2025

cp's OEIS Frontend

A048199 Distance of primes to next odd multiple of 5 (where n mod 10 = 5).

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A122754 a(n) = 10*n - A101306(n).

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A161625 Sum of all numbers from 1 up to the final digit of prime(n).

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A175348 Last digit of p^p, where p is the n-th prime.

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A253936 a(n) = prime(n + (prime(n) mod 10)).

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A290630 a(n+1) = a(n) + (final digit of greatest prime < a(n)); a(1)=3.

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A302660 a(n) = (prime(n) mod 9) + (prime(n) mod 10).

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