A061240 -id:A061240 - cp's OEIS frontend

A158910 First Differences of A061240.

18, 18, 18, 54, 18, 18, 18, 72, 18, 36, 18, 36, 36, 18, 18, 72, 18, 54, 36, 18, 36, 90, 18, 36, 90, 54, 36, 36, 18, 18, 54, 90, 36, 54, 18, 18, 54, 36, 18, 54, 18, 54, 18, 36, 90, 36, 54, 36, 54, 36, 18, 18, 54, 36, 72, 18, 108, 36, 36, 72, 18, 18, 126, 36, 54, 54, 54, 36, 126

Offset: 1

Paul Curtz, Mar 30 2009

Since A061240 contains prime numbers of the form 9k+5, k even, all numbers here are multiples of 18.

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

Cf. A008600.

Differences[Select[Prime[Range[500]],Mod[#,9]==5&]] (* Harvey P. Dale, May 19 2018 *)

Edited and extended by R. J. Mathar, Apr 03 2009

A007528 Primes of the form 6k-1.

Original entry on oeis.org

5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587

Offset: 1

N. J. A. Sloane

For values of k see A024898.
Also primes p such that p^q - 2 is not prime where q is an odd prime. These numbers cannot be prime because the binomial p^q = (6k-1)^q expands to 6h-1 some h. Then p^q-2 = 6h-1-2 is divisible by 3 thus not prime. - Cino Hilliard, Nov 12 2008
a(n) = A211890(3,n-1) for n <= 4. - Reinhard Zumkeller, Jul 13 2012
There exists a polygonal number P_s(3) = 3s - 3 = a(n) + 1. These are the only primes p with P_s(k) = p + 1, s >= 3, k >= 3, since P_s(k) - 1 is composite for k > 3. - Ralf Steiner, May 17 2018
From Bernard Schott, Feb 14 2019: (Start)
A theorem due to Andrzej Mąkowski: every integer greater than 161 is the sum of distinct primes of the form 6k-1. Examples: 162 = 5 + 11 + 17 + 23 + 47 + 59; 163 = 17 + 23 + 29 + 41 + 53. (See Sierpiński and David Wells.)
{2,3} Union A002476 Union {this sequence} = A000040.
Except for 2 and 3, all Sophie Germain primes are of the form 6k-1.
Except for 3, all the lesser of twin primes are also of the form 6k-1.
Dirichlet's theorem on arithmetic progressions states that this sequence is infinite. (End)
For all elements of this sequence p=6*k-1, there are no (x,y) positive integers such that k=6*x*y-x+y. - Pedro Caceres, Apr 06 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
A. Mąkowski, Partitions into unequal primes, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 125-126.
Wacław Sierpiński, Elementary Theory of Numbers, p. 144, Warsaw, 1964.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition, 1997, p. 127.

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
F. S. Carey, On some cases of the Solutions of the Congruence z^p^(n-1)=1, mod p, Proceedings of the London Mathematical Society, Volume s1-33, Issue 1, November 1900, Pages 294-312.
Amelia Carolina Sparavigna, The Pentagonal Numbers and their Link to an Integer Sequence which contains the Primes of Form 6n-1, Politecnico di Torino (Italy, 2021).
Amelia Carolina Sparavigna, Binary operations inspired by generalized entropies applied to figurate numbers, Politecnico di Torino (Italy, 2021).
Wikipedia, Dirichlet's theorem on arithmetic progressions.

Intersection of A016969 and A000040.
Cf. A003627, A010051, A117047, A132231, A214360, A057145.
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), this sequence (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), A141849 (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).
Cf. A001359 (lesser of twin primes), A005384 (Sophie Germain primes).
Cf. A048265, A324076.

Filtered(List([1..100],n->6*n-1),IsPrime); # Muniru A Asiru, May 19 2018

a007528 n = a007528_list !! (n-1)
a007528_list = [x | k <- [0..], let x = 6 * k + 5, a010051' x == 1]
-- Reinhard Zumkeller, Jul 13 2012

select(isprime,[seq(6*n-1,n=1..100)]); # Muniru A Asiru, May 19 2018

Select[6 Range[100]-1,PrimeQ]  (* Harvey P. Dale, Feb 14 2011 *)

forprime(p=2, 1e3, if(p%6==5, print1(p, ", "))) \\ Charles R Greathouse IV, Jul 15 2011

forprimestep(p=5,1000,6, print1(p", ")) \\ Charles R Greathouse IV, Mar 03 2025

A003627 \ {2}. - R. J. Mathar, Oct 28 2008
Conjecture: Product_{n >= 1} ((a(n) - 1) / (a(n) + 1)) * ((A002476(n) + 1) / (A002476(n) - 1)) = 3/4. - Dimitris Valianatos, Feb 11 2020
From Vaclav Kotesovec, May 02 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = 9*A175646/Pi^2 = 1/1.060548293.... =4/(3*A333240).
Product_{k>=1} (1 + 1/a(k)^2) = A334482.
Product_{k>=1} (1 - 1/a(k)^3) = A334480.
Product_{k>=1} (1 + 1/a(k)^3) = A334479. (End)
Legendre symbol (-3, a(n)) = -1 and (-3, A002476(n)) = +1, for n >= 1. For prime 3 one sets (-3, 3) = 0. - Wolfdieter Lang, Mar 03 2021

A061237 Prime numbers == 1 (mod 9).

Original entry on oeis.org

19, 37, 73, 109, 127, 163, 181, 199, 271, 307, 379, 397, 433, 487, 523, 541, 577, 613, 631, 739, 757, 811, 829, 883, 919, 937, 991, 1009, 1063, 1117, 1153, 1171, 1279, 1297, 1423, 1459, 1531, 1549, 1567, 1621, 1657, 1693, 1747, 1783, 1801, 1873, 1999

Offset: 1

Amarnath Murthy, Apr 23 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000

Cf. A010888, A061238, A061239, A061240, A061241, A061242.

[ p: p in PrimesUpTo(2000) | p mod 9 in {1} ]; // Vincenzo Librandi, Dec 25 2010

Select[ Range[ 2000 ], PrimeQ[ # ] && Mod[ #, 9 ] == 1 & ]
Select[Prime[Range[350]],Mod[#,9]==1&] (* Harvey P. Dale, Jan 06 2013 *)

isok(p) = { isprime(p) && p%9 == 1 } \\ Harry J. Smith, Jul 19 2009

A010888(a(n)) = 1. - Reinhard Zumkeller, Feb 25 2005

a(n) ~ 6n log n. - Charles R Greathouse IV, May 14 2025

More terms from Robert G. Wilson v, May 10 2001

A061242 Primes of the form 9*k - 1.

Original entry on oeis.org

17, 53, 71, 89, 107, 179, 197, 233, 251, 269, 359, 431, 449, 467, 503, 521, 557, 593, 647, 683, 701, 719, 773, 809, 827, 863, 881, 953, 971, 1061, 1097, 1151, 1187, 1223, 1259, 1277, 1367, 1439, 1493, 1511, 1583, 1601, 1619, 1637, 1709, 1871, 1889, 1907

Offset: 1

Amarnath Murthy, Apr 23 2001

Or, primes of the form 18k - 1. Corresponding values of k are in A138918. - Zak Seidov, Apr 03 2008
From Doug Bell, Mar 23 2009: (Start)
Conjecture: if a(n) = 9x - 1, the integer formed by the repeating digits in the decimal fraction x/a(n) is the smallest integer such that rotating the digits to the left produces a number which is (x+1)/x times larger.
Example: x = 2, a(n) = 17: 2/17 = 0.1176470588235294... repeating with a cycle of 16.
1176470588235294 * 3/2 = 1764705882352941, which is 1176470588235294 rotated to the left.
An additional conjecture is that the values of x from this sequence are the only values where rotating an integer one to the left produces a value (x+1)/x times as large. (End)
The last conjecture is false. For example, for x = 3 we have 230769*(4/3) = 307692, but 9*3-1 = 26 is not in the sequence. - Giovanni Resta, Jul 28 2015
Conjecture: Primes p such that ((x+1)^9-1)/x has 4 irreducible factors of degree 2 over GF(p). - Federico Provvedi, Jun 27 2018

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A061237, A061238, A061239, A061240, A061241 (p mod 9 = 1, 2, 4, 5 and 7), A138918 (18n - 1 is prime), A258663 (9n - 1 is prime).

Can be partitioned in disjoint subsequences A062343 (primes with sum of digits s = 8), A106758 (s = 17), A106764 (s = 26), A106770 (s = 35), A106776 (s = 44), A106782 (s = 53), A107617 (s = 62), etc.

[a: n in [0..250] | IsPrime(a) where a is 9*n - 1 ]; // Vincenzo Librandi, Jun 07 2015

select(isprime, [seq(18*i-1,i=1..1000)]); # Robert Israel, Sep 03 2014

Select[ Range[ 2500 ], PrimeQ[ # ] && Mod[ #, 9 ] == 8 & ]
Select[9*Range[300] - 1, PrimeQ]

select( {is(n)=n%9==8&&isprime(n)}, primes([1,2000])) \\ M. F. Hasler, Mar 10 2022

from sympy import prime
A061242 = [p for p in (prime(n) for n in range(1,10**3)) if not (p+1) % 18]
# Chai Wah Wu, Sep 02 2014

A010888(a(n)) = 8. - Reinhard Zumkeller, Feb 25 2005

a(n) ~ 6n log n. - Charles R Greathouse IV, May 14 2025

More terms from Robert G. Wilson v, May 10 2001
Edited by N. J. A. Sloane at the suggestion of R. J. Mathar, Apr 30 2008
Edited by M. F. Hasler, Mar 10 2022

A141849 Primes congruent to 1 mod 11.

Original entry on oeis.org

23, 67, 89, 199, 331, 353, 397, 419, 463, 617, 661, 683, 727, 859, 881, 947, 991, 1013, 1123, 1277, 1321, 1409, 1453, 1607, 1783, 1871, 2003, 2069, 2113, 2179, 2267, 2311, 2333, 2377, 2399, 2531, 2663, 2707, 2729, 2861, 2927, 2971, 3037, 3169, 3191, 3257

Offset: 1

N. J. A. Sloane, Jul 11 2008

Conjecture: Also primes p such that ((x+1)^11-1)/x has 10 distinct irreducible factors of degree 1 over GF(p). - Federico Provvedi, Apr 17 2018

Primes congruent to 1 mod 22. - Chai Wah Wu, Apr 28 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A090187, A102656.
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), A007528 (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), this sequence (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).

Filtered([1..4000],n->n mod 11=1 and IsPrime(n)); # Muniru A Asiru, Apr 19 2018

[ p: p in PrimesUpTo(5000) | p mod 11 eq 1 ]; // Vincenzo Librandi, Apr 19 2011

a:=select(n->isprime(n) and modp(n,11)=1,[$1..4000]); # Muniru A Asiru, Apr 19 2018

Select[Range[1,10000,11],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, May 18 2011 *)

is(n)=isprime(n) && n%11==1 \\ Charles R Greathouse IV, Jul 01 2016

forstep(n=2, 1e3, 2, if(isprime(p=11*n+1), print1(p, ", "))); \\ Altug Alkan, Apr 19 2018

a(n) ~ 10n log n. - Charles R Greathouse IV, Jul 02 2016

A078403 Primes whose digital root (A038194) is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 59, 61, 79, 83, 97, 101, 113, 131, 137, 149, 151, 167, 173, 191, 223, 227, 239, 241, 257, 263, 277, 281, 293, 311, 313, 317, 331, 347, 349, 353, 367, 383, 389, 401, 419, 421, 439, 443, 457, 461, 479, 491, 509, 547, 563, 569

Offset: 1

N. J. A. Sloane, Dec 26 2002

Union of A061238, A061240, A061241 and 3. - Ya-Ping Lu, Jan 03 2024

			59 is a term because 5+9=14, giving (final) iterated sum 1+4=5 and 5 is prime.

Cf. A038194, A061238, A061240, A061241, A070027, A078400.

Select[ Range[580], PrimeQ[ # ] && PrimeQ[Mod[ #, 9]] &]
Select[Prime[Range[200]],PrimeQ[Mod[#,9]]&] (* Harvey P. Dale, Aug 20 2017 *)

forprime(p=2,997,if(isprime(p%9),print1(p,",")))

from sympy import isprime, primerange; [print(p, end = ', ') for p in primerange(2, 570) if isprime(p%9)] # Ya-Ping Lu, Jan 03 2024

a(n) ~ 2n log n. - Charles R Greathouse IV, May 14 2025

Extended by Robert G. Wilson v, Klaus Brockhaus and Rick L. Shepherd, Dec 27 2002

A229324 Composite squarefree numbers n such that p + tau(n) divides n - phi(n), where p are the prime factors of n, tau(n) = A000005(n) and phi(n) = A000010(n).

Original entry on oeis.org

115, 205, 295, 565, 655, 745, 835, 1195, 1285, 1465, 1555, 1735, 1915, 2005, 2095, 2455, 2545, 2815, 2995, 3085, 3265, 3715, 3805, 3985, 4435, 4705, 4885, 5065, 5155, 5245, 5515, 5965, 6145, 6415, 6505, 6595, 6865, 7045, 7135, 7405, 7495, 7765, 7855, 8035

Offset: 1

Paolo P. Lava, Sep 20 2013

nonn

All terms are apparently multiple of 5.

It appears that a(n) = 5*A061240(n+1). - Michel Marcus, Sep 21 2013

			Prime factors of 2815 are 5, 563 and tau(2815) = 4, phi(2815) = 2248. 2815 - 2248 = 567 and  567 / (5 + 4) = 63, 567 / (563 + 4) = 1.

Paolo P. Lava, Table of n, a(n) for n = 1..500

Cf. A000005, A000010, A228299-A228302, A229274-A229276, A229321-A229323.

with (numtheory); P:=proc(q) global a, b, c, i, ok, p, n;
for n from 2 to q do  if not isprime(n) then a:=ifactors(n)[2]; ok:=1;
for i from 1 to nops(a) do if a[i][2]>1 then ok:=0; break;
else if not type((n-phi(n))/(a[i][1]+tau(n)),integer) then ok:=0; break; fi; fi; od; if ok=1 then print(n); fi; fi; od; end: P(6*10^9);

Deleted first term, changed b-file and comment by Paolo P. Lava, Sep 23 2013

A242215 a(n) = 18*n + 5.

Original entry on oeis.org

5, 23, 41, 59, 77, 95, 113, 131, 149, 167, 185, 203, 221, 239, 257, 275, 293, 311, 329, 347, 365, 383, 401, 419, 437, 455, 473, 491, 509, 527, 545, 563, 581, 599, 617, 635, 653, 671, 689, 707, 725, 743, 761, 779, 797, 815, 833, 851, 869, 887, 905, 923, 941, 959

Offset: 0

Arkadiusz Wesolowski, May 07 2014

Conjecture: there are infinitely many composite Fermat numbers such that no one of them has a divisor that belongs to this sequence.

Wikipedia, Fermat number.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Supersequence of A061240.

Cf. A229855.

Magma
```
[18*n+5: n in [0..53]];
```
Maple
```
seq(18*n+5, n=0..53);
```

Table[18*n + 5, {n, 0, 53}]
LinearRecurrence[{2,-1},{5,23},60] (* Harvey P. Dale, Aug 25 2017 *)

PARI
```
for(n=0, 53, print1(18*n+5, ", "));
```

G.f.: (5 + 13*x)/(1 - x)^2.
From Elmo R. Oliveira, Dec 08 2024: (Start)
E.g.f.: exp(x)*(5 + 18*x).
a(n) = 2*a(n-1) - a(n-2) for n > 1. (End)

A024909 Numbers k such that 9*k - 4 is prime.

Original entry on oeis.org

1, 3, 5, 7, 13, 15, 17, 19, 27, 29, 33, 35, 39, 43, 45, 47, 55, 57, 63, 67, 69, 73, 83, 85, 89, 99, 105, 109, 113, 115, 117, 123, 133, 137, 143, 145, 147, 153, 157, 159, 165, 167, 173, 175, 179, 189, 193, 199, 203, 209, 213, 215, 217, 223, 227, 235, 237, 249, 253, 257, 265, 267

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A061240 (associated primes).

[n: n in [1..300] | IsPrime(9*n-4)]; // Vincenzo Librandi, Nov 19 2010

Select[Range[400], PrimeQ[9# - 4] &] (* Vincenzo Librandi, Sep 26 2012 *)

is(n)=isprime(9*n-4) \\ Charles R Greathouse IV, Jun 06 2017

A158955 First differences of A061241.

Original entry on oeis.org

36, 18, 18, 18, 54, 72, 18, 36, 36, 18, 18, 18, 54, 18, 18, 90, 54, 18, 54, 18, 18, 18, 126, 54, 90, 36, 18, 18, 18, 36, 90, 18, 18, 54, 18, 108, 18, 36, 126, 18, 36, 36, 54, 36, 72, 18, 54, 18, 36, 126, 18, 72, 18, 18, 54, 18, 36, 36, 54, 36, 144, 54, 18, 18, 90, 36, 18, 36, 162, 18

Offset: 1

Paul Curtz, Apr 01 2009

Six prime modulo classes are A061237, A061238, A061239, A061240, A061241 and A061242

with remainders 1, 2, 4, 5, 7, and 8 (mod 9).

Cf. A158910.

Differences[Select[Prime[Range[500]],Mod[#,9]==7&]] (* Harvey P. Dale, Oct 13 2022 *)

Edited and extended by R. J. Mathar, Apr 04 2009

cp's OEIS Frontend

A158910 First Differences of A061240.

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A007528 Primes of the form 6k-1.

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A061237 Prime numbers == 1 (mod 9).

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A061242 Primes of the form 9*k - 1.

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A141849 Primes congruent to 1 mod 11.

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A078403 Primes whose digital root (A038194) is prime.

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