A056716 -id:A056716 - cp's OEIS frontend

A056805 Numbers k such that 6*10^k+1 is prime.

0, 1, 2, 8, 9, 15, 20, 26, 38, 45, 65, 112, 244, 303, 393, 560, 839, 1009, 1019, 1173, 1334, 2236, 2629, 4426, 8848, 20812, 37744, 72926, 86287, 231617, 281969, 488852, 522127, 655642, 758068, 879313, 1380098

Offset: 1

Robert G. Wilson v, Aug 22 2000

From the Kamada data, Edward Trice reports that 231617 and 522127 are in this sequence. But these may not be the next ones. There are no others less than 2*10^5, however. - Robert Price, Jul 09 2015

			For k=2 => (6*10^2+1)=601, which is prime.

Makoto Kamada, Prime numbers of the form 600...001.
Sabin Tabirca and Kieran Reynolds, Lacunary Prime Numbers.

Cf. A056716, A101517.

Do[ If[ PrimeQ[ 6*10^n + 1], Print[ n ]], {n, 0, 10000}]

is(n)=isprime(6*10^n+1) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = A101517(n-1) + 1.

a(22)-a(25) from Hugo Pfoertner, Feb 11 2004
a(26)=20812 from Kamada link by Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 27 2007
a(27)=37744 from Jason Earls, Mar 07 2008
a(28)-a(29) from Kamada data by Robert Price, Dec 09 2010
a(30)-a(36) from Kamada data by Tyler Busby, Apr 15 2024

A070272 Numbers n such that reverse(n) = phi(n) + sigma(n).

Original entry on oeis.org

275, 295, 2995, 299995, 2999995, 278736495, 299999995, 299999999995

Offset: 1

Joseph L. Pe, May 12 2002

For n>0 5*(6*10^A056716(n)-1) is in this sequence. In fact if p = 6*10^n-1 is prime and n>0 (p>5) then m = 5*p is in the sequence. That's because phi(m) = phi(5*p) = 4*(6*10^n-2) = 24*10^n-8 and sigma(m)= 6*6*10^n, so phi(m) + sigma(m) = 6*10^(n+1)-8 = 5.(9)(n).2 = reversal(2.(9)(n).5) = reversal (3*10^(n+1)-5) = reversal(m)(dot between numbers means concatenation and "(9)(n)" means number of 9's is n). For example 299999995 is in the sequence because 6*10^7-1 is prime and 299999995 = 5*(6*10^7-1); 299999999995 is in sequence because 6*10^10-1 is prime and 299999999995 = 5*(6*10^10-1). Next term is greater than 80000000. - Farideh Firoozbakht, Jan 11 2005
Next term is greater than 10^9. - Farideh Firoozbakht, Jan 23 2005
a(9) > 10^13. - Giovanni Resta, Feb 08 2014

			Reverse(275) = 572 = 200 + 372 = phi(275) + sigma(275).

Cf. A056716, A102284.

Select[Range[10^6], FromDigits[Reverse[IntegerDigits[ # ]]] == EulerPhi[ # ] + DivisorSigma[1, # ] &]

One more term from Farideh Firoozbakht, Jan 11 2005
More terms from Farideh Firoozbakht, Jan 23 2005
a(8) from Giovanni Resta, Nov 03 2012

A093946 Primes of the form 6*10^k - 1.

Original entry on oeis.org

5, 59, 599, 59999, 599999, 59999999, 59999999999, 59999999999999, 59999999999999999999999, 599999999999999999999999, 59999999999999999999999999999, 59999999999999999999999999999999999

Offset: 1

Rick L. Shepherd, Apr 17 2004

nonn

Equivalently, primes of the form 5*10^n + 9*R_n, where R_n is the repunit (A002275) of length n.

Makoto Kamada, Prime numbers of the form 599...99.
Index entries for primes involving repunits.

Cf. A002275, A056716.

Primes in A099151.

[a: n in [0..200] | IsPrime(a) where a is (6*10^n-1)]; // Vincenzo Librandi, May 08 2019

Select[Table[FromDigits[PadRight[{5},n,9]],{n,40}],PrimeQ] (* Harvey P. Dale, Jun 06 2016 *)

a(n) = 6*10^A056716(n) - 1 = A099151(A056716(n) + 1). - Elmo R. Oliveira, Jun 14 2025

A129990 Primes p such that the smallest integer whose sum of decimal digits is p is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 41, 43, 71, 79, 97, 173, 179, 257, 269, 311, 389, 691, 4957, 8423, 11801, 14621, 25621, 26951, 38993, 75743, 102031, 191671, 668869

Offset: 1

J. M. Bergot, Jun 14 2007

			The smallest integer whose sum of digits is 17 is 89; 89 is prime, therefore 17 is in the sequence.

Cf. A051885.

Cf. A002957, A056703, A056712, A056716, A056721, A056725.

Select[Prime[Range[1000]],PrimeQ[FromDigits[Join[{Mod[ #,9]},Table[9,{i,1,Floor[ #/9]}]]]] &]

from itertools import islice
from sympy import isprime, nextprime
def A051885(n): return ((n%9)+1)*10**(n//9)-1 # from Chai Wah Wu
def agen(startp=2):
    p = startp
    while True:
        if isprime(A051885(p)): yield p
        p = nextprime(p)
print(list(islice(agen(), 23))) # Michael S. Branicky, Jul 27 2022

sorted( filter(is_prime, sum(([9*t+k for t in oeis(seq).first_terms()] for seq,k in (('A002957',1), ('A056703',2), ('A056712',4), ('A056716',5), ('A056721',7), ('A056725',8))), [3])) ) # Max Alekseyev, Feb 05 2025

Primes p such that (p mod 9 + 1) * 10^[p/9] - 1 is prime. Therefore the sequence consists of the term 3 and the primes of the forms A002957(k)*9+1, A056703(k)*9+2, A056712(k)*9+4, A056716(k)*9+5, A056721(k)*9+7, A056725(k)*9+8. - Max Alekseyev, Nov 09 2009

Edited, corrected and extended by Stefan Steinerberger, Jun 23 2007
Extended by D. S. McNeil, Mar 20 2009
a(29)-a(33) from Max Alekseyev, Nov 09 2009

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A056805 Numbers k such that 6*10^k+1 is prime.

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A070272 Numbers n such that reverse(n) = phi(n) + sigma(n).

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A093946 Primes of the form 6*10^k - 1.

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A129990 Primes p such that the smallest integer whose sum of decimal digits is p is prime.

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