A031157 -id:A031157 - cp's OEIS frontend

A055728 Number of prime lucky numbers <10^n.

0, 2, 9, 43, 211, 1300, 8616, 62446, 469146, 3656784, 29285501, 239911086

Offset: 0

Robert G. Wilson v, Jun 09 2000

			a(2) = 9 since there are 9 prime numbers in the lucky number sequence A000959 that are less than 10^2 (3, 7, 13, 31, 37, 43, 67, 73, and 79).

Index entries for sequences related to numbers of primes in various ranges

Cf. A031157, A000959, A000040.

lst = Range[1, 10^8, 2]; i = 2; While[ i <= (len = Length@lst) && (k = lst[[i]]) <= len, lst = Drop[lst, {k, len, k}]; i++ ]; Table[Length@ Select[lst, PrimeQ@# && # < 10^n &], {n, 0, 8}] (* Robert G. Wilson v, May 12 2006 *)

a(7)-a(8) from Robert G. Wilson v, May 12 2006
a(9) from Donovan Johnson, Jul 06 2010
a(10) from Kevin P. Thompson, Nov 22 2021
a(11) from Dana Jacobsen, Mar 08 2023

A117708 Numbers that are both lucky numbers and Chen primes.

Original entry on oeis.org

3, 7, 13, 31, 37, 67, 127, 211, 307, 409, 487, 541, 577, 631, 769, 787, 937, 991, 1009, 1039, 1117, 1201, 1291, 1459, 1471, 1567, 1777, 1801, 2251, 2281, 2467, 2557, 2647, 2971, 3037, 3187, 3259, 3307, 3559, 3709, 3847, 3889, 4441, 4567, 4801, 4969, 4987

Offset: 1

Jani Melik, Apr 27 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A000959 and A109611.

Subsequence of A031157.

lst = Range[1, 5000, 2]; i = 2; While[ i <= (len = Length@lst) && (k = lst[[i]]) <= len, lst = Drop[lst, {k, len, k}]; i++ ]; chenQ[n_] := PrimeQ[n] && Plus @@ Last /@ FactorInteger[n + 2] < 3; Select[lst, chenQ@# &] (* Robert G. Wilson v, May 12 2006 *)

Corrected and extended by Robert G. Wilson v, May 12 2006

A129864 Numbers that are both lucky and emirp.

Original entry on oeis.org

13, 31, 37, 73, 79, 739, 769, 937, 991, 1009, 1021, 1201, 1231, 1249, 1471, 1597, 1723, 1831, 1879, 1933, 3049, 3109, 3121, 3163, 3301, 3433, 3571, 3613, 3697, 3889, 7207, 7321, 7459, 7507, 7603, 7687, 7717, 7951, 7963, 9349, 9403, 9421, 9547, 9613, 9643

Offset: 1

Jonathan Vos Post, May 23 2007

			a(9) = 1009 because 1009 is a lucky number A000959(154) and 1009 is an emirp because 1009 is prime and R(1009) = 9001 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A000959, A004086, A006567, A031157.

L = Table[2*i + 1, {i, 0,5* 10^3}]; For[n = 2, n < Length[L], r = L[[n++]]; L = ReplacePart[L, Table[r*i -> Nothing, {i, 1, Length[L]/r}]]];Select[L,PrimeQ[#]&&PrimeQ[IntegerReverse[#]]&&IntegerReverse[#]!=#&] (* James C. McMahon, Feb 02 2025 *)

Intersection of A000959 and A006567.

Corrected and extended by R. J. Mathar, Jun 12 2007

A166744 Unlucky primes: numbers which are members of both A000040 (primes) and A050505 (unlucky).

Original entry on oeis.org

2, 5, 11, 17, 19, 23, 29, 41, 47, 53, 59, 61, 71, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 157, 167, 173, 179, 181, 191, 197, 199, 227, 229, 233, 239, 251, 257, 263, 269, 271, 277, 281, 293, 311, 313, 317, 337, 347, 353, 359, 373, 379, 383, 389

Offset: 1

Gabriel Finch (salsaman(AT)xs4all.nl), Oct 21 2009

nonn

There are infinitely many unlucky prime numbers, in particular all those of the form 6n - 1, eliminated in the second step of Ulam's procedure for lucky numbers. - Davide Rotondo, Aug 31 2020

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

Cf. A007528, A031157 (lucky primes).

L:= [seq(2*i+1, i=0..10^3)]:
for n from 2 while n < nops(L) do
  r:= L[n];
  L:= subsop(seq(r*i=NULL, i=1..nops(L)/r), L);
od:
U:= {2, seq(i,i=3..2*10^3+1,2)} minus convert(L,set):
sort(convert(select(isprime,U),list)); # Robert Israel, Jul 26 2019

# Copy from  A000959 - (Robert FERREOL, Nov 19 2014)
def lucky(n):
  L=list(range(1, n+1, 2)); j=1
  while L[j] <= len(L)-1:
    L=[L[i] for i in range(len(L)) if (i+1)%L[j]!=0]
    j+=1
  return(L)
[ p for p in prime_range(1000) if p not in lucky(1000) ] # Hauke Löffler, Jul 26 2019

A225322 Lucky numbers that are prime powers.

Original entry on oeis.org

1, 9, 25, 49, 169, 289, 361, 529, 729, 841, 961, 1369, 2187, 2209, 3481, 3721, 5041, 7921, 9409, 10609, 24649, 29791, 32041, 32761, 36481, 50653, 52441, 66049, 73441, 83521, 113569, 121801, 128881, 130321, 167281, 175561, 185761, 226981, 292681, 300763, 323761

Offset: 1

Alex Ratushnyak, May 05 2013

nonn

Intersection of A025475 and A000959.

Conjecture: the sequence is infinite.

Kevin P. Thompson, Table of n, a(n) for n = 1..1577 (terms < 10^10 with terms 1..628 from Donovan Johnson)

Cf. A000959, A025475.

Cf. A031157, A031162, A057589, A118565, A128511.

A289123 Numbers n such that (n-2,n) are twin primes, and (n,n+2) are twin lucky numbers.

Original entry on oeis.org

7, 13, 31, 73, 193, 283, 619, 643, 883, 1021, 1093, 1231, 2083, 2113, 2971, 3121, 3259, 4129, 4483, 4519, 5233, 6271, 6661, 6763, 7549, 7591, 8221, 9421, 10069, 10459, 10531, 11833, 12163, 13009, 13693, 13723, 13831, 17209, 17389, 20149, 20509, 21013, 21613

Offset: 1

Amiram Eldar, Jun 25 2017

nonn

Intersection of A006512 and A031158. Subsequence of A031157. The other case in which (n-2,n) are twin lucky numbers, and (n,n+2) are twin primes has only one solution, n = 3, since twin primes are of the form (6k-1, 6k+1) (except for 3 and 5) and 6k-1 is never lucky.

			7 is in the sequence since (5,7) are twin primes, and (7,9) are twin lucky numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006512, A000959, A031157, A031158.

L = Table[2*i + 1, {i, 0, 10^5}]; For[n = 2, n < Length[L], r = L[[n++]]; L = ReplacePart[L, Table[r*i -> Nothing, {i, 1, Length[L]/r}]]]; L[[Select[Range[1, Length[L] - 1], PrimeQ[L[[#]] - 2] && PrimeQ [L[[#]]] && L[[# + 1]] == L[[#]] + 2 &]]] (* after Jean-François Alcover at A000959 *)

A309381 Lucky primes k such that k+6 is also a lucky prime.

Original entry on oeis.org

7, 31, 37, 67, 73, 613, 991, 1087, 1117, 2467, 3301, 3307, 3607, 4561, 4987, 4993, 6367, 6373, 8263, 8641, 9643, 10903, 11827, 11953, 12373, 12547, 15187, 15901, 17047, 18043, 19603, 20353, 21751, 23671, 25147, 28837, 31033, 31231, 37957, 38707, 38917, 43201, 44383, 46021, 49627

Offset: 1

Robert Israel, Jul 26 2019

nonn

A031157(k) for k such that A309334(k)=6.

The minimum gap between lucky primes (after the first) is 6.

			37 and 37+6=43 are both lucky primes, so 37 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..1615

Cf. A031157, A309334.

N:= 10^5: # for terms <= N
L:= [seq(i,i=1..N+6,2)]:
for n from 2 while n < nops(L) do
  r:= L[n];
  L:= subsop(seq(r*i=NULL, i=1..nops(L)/r), L);
od:
L:= convert(select(isprime,L),set):
A:= L intersect map(`-`,L,6):
sort(convert(A,list));

A140286 The n-th lucky number which is the product of exactly n primes (with multiplicity).

Original entry on oeis.org

3, 15, 99, 495, 2079, 4455, 36855, 70875, 280665, 1393119, 4179357, 12931731, 32417901, 161026623, 514966329, 1490692005

Offset: 1

Jonathan Vos Post, May 24 2008

This is the main diagonal of the infinite array A(k,n) = n-th n-th lucky number to be the product of exactly k primes, with multiplicity, which begins as below:
============================================================================
 k |  n=1 |  n=2 |  n=3 |  n=4 |  n=5 |  n=6 |  n=7 |  n=8 |   n=9 |  n=10 |in OEIS
 1 |    3 |   7  |  13  |  31  |  37  |  43  |  67  |  73  |   79  |  127  |A031157
 2 |    9 |  15  |  21  |  25  |  33  |  49  |  51  |  69  |   87  |   93  |A139787
 3 |   63 |  75  |  99  | 105  | 171  | 195  | 231  | 261  |  273  |  285  |
 4 |  297 | 495  | 621  | 693  | 735  | 819  | 855  | 975  | 1029  | 1107  |
 5 | 1053 |                                                                |
 6 |  729 |                                                                |
============================================================================
a(16) > 10^9. - Donovan Johnson, Oct 24 2010

			a(4) = 693 because the 113th lucky number = 693 = 3^2 * 7 * 11 is the 4th lucky number with 4 prime factors.

Cf. A000040, A000959, A001358, A014612, A031157, A139787.

a(4) corrected and 5 more terms via b000959.txt from R. J. Mathar, Oct 22 2010
a(10)-a(15) from Donovan Johnson, Oct 24 2010
a(16) from Giovanni Resta, May 10 2020

A318480 Least start of n consecutive lucky numbers that are also n consecutive prime numbers.

Original entry on oeis.org

3, 997, 3301, 9631, 62378131

Offset: 1

Amiram Eldar, Aug 27 2018

No other terms below 10^9.

Calculated using Hugo van der Sanden's Lucky numbers up to 10^9.

			The corresponding consecutive primes/lucky numbers are:
3
997, 1009
3301, 3307, 3313
9631, 9643, 9649, 9661
62378131, 62378137, 62378143, 62378149, 62378161

Cf. A000959, A031157, A057698.

A336688 Primes p such that the Wendt determinant A048954(p) has prime factors less than p.

Original entry on oeis.org

3, 7, 13, 31, 73, 127, 307, 331, 757

Offset: 1

Frank M Jackson and Michael B Rees, Jul 31 2020

Michael B Rees has conjectured that for all primes p, each fully exponentiated prime factor less than p that divides the Wendt determinant W(p), if it exists, is of the form k*p + 1.
This sequence identifies the prime index p of Wendt determinants W(p) that have prime factors less than p.
These prime indices appear to be a subset of the lucky primes A031157.

			a(3) = 13. The Wendt determinant with a prime index p = 13 has prime factors less than p. W(13) = 3^6*53^2*79^2*131^2*521^2*8191 and 3^6 = 729 is of the form k*13 + 1. It is the 3rd occurrence of such a determinant.

Gerard P. Michon, Factorization of Wendt's Determinant (table for n=1 to 114).
Eric Weisstein's World of Mathematics, Circulant matrix.
Wikipedia, Circulant matrix.

Cf. A048954, A336280.

w[n_] := Resultant[x^n-1, (1+x)^n-1, x]; getp[n_] := Module[{W=w[n], lst=Table[Prime[m], {m, 1, PrimePi[n]}], lst1={}, j, k, l}, Do[j=1; While[W>0&&IntegerQ[W/lst[[l]]^j], j++]; If[j-1>0, AppendTo[lst1, {lst[[l]], j-1}]], {l, 1, Length@lst}]; Join[{n}, lst1]]; lst = {}; Do[lst1=getp[Prime[n]]; If[Length@lst1>1, AppendTo[lst, lst1[[1]]]], {n, 1, PrimePi[331]}]; lst

a(9) from Jinyuan Wang, Sep 04 2020

cp's OEIS Frontend

A055728 Number of prime lucky numbers <10^n.

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A117708 Numbers that are both lucky numbers and Chen primes.

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A129864 Numbers that are both lucky and emirp.

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A166744 Unlucky primes: numbers which are members of both A000040 (primes) and A050505 (unlucky).

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A225322 Lucky numbers that are prime powers.

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A289123 Numbers n such that (n-2,n) are twin primes, and (n,n+2) are twin lucky numbers.

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Mathematica

A309381 Lucky primes k such that k+6 is also a lucky prime.

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Maple

A140286 The n-th lucky number which is the product of exactly n primes (with multiplicity).

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A318480 Least start of n consecutive lucky numbers that are also n consecutive prime numbers.

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