A045375 -id:A045375 - cp's OEIS frontend

A056650 a(n) is the greatest prime factor of a(n-1)^2+a(n-1)+1.

2, 7, 19, 127, 5419, 1009, 9181, 1423, 96493, 163350799, 25249969, 212520319916977, 784949209969, 145542538757017, 147660435988297, 2508855873622663, 3565137918692593, 6521735641, 11273204227, 16141059763, 125679599753438821, 240780337980146570229319, 7282590063606707136764017

Offset: 0

Henry Bottomley, Aug 09 2000

nonn

Except for the first term the initial (ten at least) terms are one more than a multiple of 6, with the result that a(n)^2+a(n)+1 is in these cases an odd multiple of 3 and appears to be 3 times the product of primes all of which are one more than a multiple of 6.

Dennis Langdeau, Table of n, a(n) for n = 0..31

Cf. A031439, A045375.

More terms from Vladeta Jovovic, Nov 26 2001

A235473 Primes whose base-3 representation is also the base-4 representation of a prime.

Original entry on oeis.org

2, 43, 61, 67, 97, 103, 127, 139, 151, 157, 199, 211, 229, 277, 283, 331, 337, 349, 373, 379, 433, 439, 463, 499, 523, 571, 601, 607, 727, 751, 787, 823, 853, 883, 919, 991, 1063, 1087, 1117, 1213, 1249, 1327, 1381, 1429, 1483, 1531, 1567, 1597, 1627, 1759, 1783, 1867, 1999

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of a two-dimensional array of sequences, given in the LINK, based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.

This is a subsequence of A045331 and A045375.

			43 = 1121_3 and 1121_4 = 89 are both prime, so 43 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235266, A235474, A152079, A235475 - A235479, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

Select[Prime[Range[400]],PrimeQ[FromDigits[IntegerDigits[#,3],4]]&] (* Harvey P. Dale, Oct 16 2015 *)

is(p,b=4,c=3)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

A260488 Numbers of the form 2^m * (6k + 1) for m, k >= 0, and 0.

Original entry on oeis.org

0, 1, 2, 4, 7, 8, 13, 14, 16, 19, 25, 26, 28, 31, 32, 37, 38, 43, 49, 50, 52, 55, 56, 61, 62, 64, 67, 73, 74, 76, 79, 85, 86, 91, 97, 98, 100, 103, 104, 109, 110, 112, 115, 121, 122, 124, 127, 128, 133, 134, 139, 145, 146, 148, 151, 152, 157

Offset: 0

Franklin T. Adams-Watters, Jul 27 2015

Alternate definition: starting with a(0) = 0, include 2n if n is in the sequence, and include 2n+1 if no two previous terms sum to 2n+1.
It suffices to prove this for odd n. If n == 3(6), n-2 == 1 (mod 6); if n == 5 (mod 6), n-4 == 1 (mod 6). However, if n == 1 (mod 6), any even k in the sequence, 0 < k < n, will have k !== 0 (mod 3), and so n-k != 1 (mod 3), so it is not in the sequence; thus n must be.
Every nonnegative integer is the sum of two members of this sequence; every positive integer is the sum of two distinct members of this sequence. For odd n, this is by the construction in the alternate definition; and for even n, by induction n/2 = i + j, and so n = 2i + 2j.
It follows that:
* No member of this sequence except 0 is a multiple of 3.
* The sequence has a density of 1/3.
* The difference between consecutive terms is always one of {1, 2, 3, 5, 6}, and each of these occurs infinitely often, with 1 having density 1/3 and the others having density 1/6.
* The sequence is closed under multiplication.
* The primes in the sequence are A045375.

			Using the alternate definition:
1 is in the sequence because it is not the sum of 2 elements from {0}.
2 is in the sequence because 2 = 2*1, and 1 is in the sequence.
3 is not in the sequence because 3 = 1 + 2, and 1 and 2 are in the sequence.
6 is not in the sequence because 6 = 2*3, and 3 is not in the sequence.

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..10000

Cf. A126684, A260489, primes A045375, A000265.

Cf. A113448, A115235, A123863.

N:= 1000: # to get all terms <= N
sort([0, seq(seq(2^m*(6*k+1), k = 0 .. floor((N/2^m - 1)/6)), m = 0 .. ilog2(N))]);  # Robert Israel, Aug 25 2015

mx=160;Join[{0},Sort@Flatten@Table[2^m*(6k+1),{m,0,Log2[mx]},{k,0,mx/(6*2^m)}]] (* Robert G. Wilson v, Aug 16 2015 *)

alist(n) = my(r=vector(n),j,k);r[1]=0;j=1;while(j

alim(n)={my(p=1,p2=p,r,j);
  for(k=1,n,
    if(if(k%2==0, polcoeff(p,k\2),polcoeff(p2,k)==0),p+=x^k;p2+=x^k*p));
  r=vector(subst(p,x,1));for(k=0,n,if(polcoeff(p,k),r[j++]=k));r}

n is in the sequence if and only if n = 0 or A000265(n) == 1 (mod 6). [Clarified by Peter Munn, Jun 11 2021]

n is in the sequence if n = 0 or b(n) is nonzero where b = A113448, A115235, or A123863. - Michael Somos, Jul 29 2015

A164979 Slowest growing sequence of primes having the semiprime-pairwise property: for any i,j, a(i)+a(j) is semiprime.

Original entry on oeis.org

2, 7, 19, 67, 127, 6619, 126127, 345979, 476407, 1658119, 15182459419, 105169832587, 287583971287

Offset: 1

Zak Seidov, Sep 03 2009

By Dirichlet's theorem and Linnik's theorem, a(n) exists for all n. - Charles R Greathouse IV, Jun 03 2025

Subsequence of A045375.

Cf. A001358, A114845, A116656.

lista(pmax) = {my(v = [2], ans); print1(v[1], ", "); forprime(p=3, pmax, ans = 1; for(i=1, #v, if(bigomega(p + v[i]) != 2, ans = 0; break)); if(ans, print1(p, ", "); v=concat(v, p)));} \\ Amiram Eldar, Jun 27 2024

a(n) = A114845(n)/2.

a(n) << A070826(n)^5. - Charles R Greathouse IV, Jun 03 2025

a(12) from Amiram Eldar, Jun 27 2024

a(13) from Jinyuan Wang, May 29 2025

A171715 Absolute value of (n-th prime of form 3m-1 minus n-th prime of form 3k+1/2+-1/2).

Original entry on oeis.org

1, 2, 2, 2, 8, 8, 2, 14, 14, 14, 8, 14, 14, 8, 20, 26, 20, 20, 14, 14, 20, 20, 20, 26, 2, 8, 32, 26, 26, 44, 44, 50, 44, 38, 50, 26, 26, 38, 26, 32, 32, 20, 26, 20, 38, 38, 56, 62, 56, 68, 68, 80, 50, 50, 50, 44, 50, 62, 56, 50, 62, 74, 74, 62, 68, 56, 50, 44, 50, 50, 32, 44, 38

Offset: 1

Juri-Stepan Gerasimov, Dec 17 2009, Feb 09 02 2010

nonn

Also, the absolute value of (n-th generalized cuban prime minus n-th generalized non-cuban prime).
Or, the absolute value of n-th prime of form 6*m-3/2+-5/2 minus n-th prime of form 6*k-2+-1.
A003627 U A007645 = A045375 U A045410 = A000040.

			a(1) = abs(3*1-1-(3*1+1/2-1/2)) = 1; a(2) = abs(3*2-1-(3*2+1/2+1/2)) = 2.

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

Cf. A000040, A003627, A007645, A045375, A045410.

A003627 := proc(n) if n <= 2 then op(n,[2,5]) ; ; else for a from procname(n-1)+2 by 2 do if isprime(a) and (a mod 3) =2 then return a ; end if; end do: end if; end proc:
A007645 := proc(n) if n <= 2 then op(n,[3,7]) ; ; else for a from procname(n-1)+2 by 2 do if isprime(a) and (a mod 3) <> 2 then return a ; end if; end do: end if; end proc:
A171715 := proc(n) abs(A003627(n)-A007645(n)) ; end proc: # R. J. Mathar, Apr 24 2010

Module[{nn=500,p1,p2,len},p1=Select[3*Range[nn]-1,PrimeQ];p2=Select[Flatten[#+{0,1}&/@ (3*Range[nn])],PrimeQ];len=Min[Length[p1],Length[p2]]; Abs[#[[1]]-#[[2]]]&/@ Thread[ {Take[p1,len],Take[p2,len]}]] (* Harvey P. Dale, Aug 29 2023 *)

a(n) = abs(A003627(n)-A007645(n)) = abs(A045375(n)-A045410(n)).

Entries checked by R. J. Mathar, Apr 24 2010

A172182 Nonprimes of the form 6k + 1 or 6k + 2.

Original entry on oeis.org

1, 8, 14, 20, 25, 26, 32, 38, 44, 49, 50, 55, 56, 62, 68, 74, 80, 85, 86, 91, 92, 98, 104, 110, 115, 116, 121, 122, 128, 133, 134, 140, 145, 146, 152, 158, 164, 169, 170, 175, 176, 182, 187, 188, 194, 200, 205, 206, 212, 217, 218, 224, 230, 235, 236, 242, 247

Offset: 1

Juri-Stepan Gerasimov, Jan 28 2010

Cf. A045375.

A172182:=func; [ n: n in [1..247] | A172182(n) ];  // Bruno Berselli, May 25 2011

a(n) ~ 3n. - Charles R Greathouse IV, May 25 2011

Entries checked by R. J. Mathar, May 22 2010

Retitled by Charles R Greathouse IV, May 25 2011

A231476 Primes whose base-3 representation is also the base-6 representation of a prime.

Original entry on oeis.org

2, 7, 13, 19, 31, 151, 163, 211, 223, 229, 241, 271, 349, 367, 439, 601, 607, 613, 631, 643, 673, 727, 733, 859, 907, 937, 997, 1021, 1033, 1039, 1051, 1093, 1117, 1123, 1129, 1153, 1321, 1327, 1399, 1423, 1429, 1609, 1627, 1657, 1669, 1741, 1747, 1759, 1777, 1789, 1831, 1867, 1933, 1951, 1993, 1999

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of a two-dimensional array of sequences, given in the LINK, based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.

Subsequence of A045375 ⊂ A045331.

			7 = 21_3 and 21_6 = 13 are both prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235469, A235265, A235266, A235473, A152079, A235461 - A235482, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924. See the LINK for further cross-references.

Select[Prime[Range[400]],PrimeQ[FromDigits[ IntegerDigits[#, 3], 6]] &] (* Harvey P. Dale, Sep 29 2016 *)

is(p,b=6,c=3)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b>c.

A279815 Numbers n such that the average of the squares of the numbers less than n that do not divide n is an integer.

Original entry on oeis.org

3, 4, 7, 13, 16, 19, 20, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 188, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613, 619, 631, 643, 661, 673, 691, 709, 727, 733, 739, 751, 757, 769, 787

Offset: 1

Ilya Gutkovskiy, Dec 19 2016

Numbers n such that A049820(n) divides A276984(n).

			7 is in the sequence because 7 has 2 divisors {1,7} therefore 5 non-divisors {2,3,4,5,6}, 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 90 and 5 divides 90.

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

Cf. A020486, A049820, A140826, A276984.

Superset of A045375(m) (m > 1) ?

Select[Range[800], Mod[#1 (#1 + 1) ((2 #1 + 1)/6) - DivisorSigma[2, #1], #1 - DivisorSigma[0, #1]] == 0 & ]

is(n)=my(f=factor(n)); n>2 && ((2*n^3+3*n^2+n)/6-sigma(f,2))%(n-numdiv(f))==0 \\ Charles R Greathouse IV, Dec 19 2016

A172181 Odd composites not of the form 6k + 1.

Original entry on oeis.org

9, 15, 21, 27, 33, 35, 39, 45, 51, 57, 63, 65, 69, 75, 77, 81, 87, 93, 95, 99, 105, 111, 117, 119, 123, 125, 129, 135, 141, 143, 147, 153, 155, 159, 161, 165, 171, 177, 183, 185, 189, 195, 201, 203, 207, 209, 213, 215, 219, 221, 225, 231, 237, 243, 245, 249, 255

Offset: 1

Juri-Stepan Gerasimov, Jan 28 2010

Cf. A005408, A045375, A045410. Odd complement is A091300.

Union[6Range[42] + 3, Select[6Range[43] - 1, Not[PrimeQ[#]] &]] (* Alonso del Arte, Jun 05 2011 *)

select(n->(n%6==3 && n>3) || (n%6==5 && !isprime(n)), vector(1000,i,i)) \\ Charles R Greathouse IV, Jun 05 2011

Entries checked by R. J. Mathar, May 19 2010

A372763 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+5))))).

Original entry on oeis.org

13, 19, 5, 31, 37, 43, 7, 11, 61, 67, 73, 79, 17, 1, 97, 103, 109, 23, 11, 127, 1, 139, 29, 151, 157, 163, 1, 1, 181, 1, 193, 199, 41, 211, 1, 223, 229, 47, 241, 1, 1, 1, 53, 271, 277, 283, 1, 59, 1, 307, 313, 1, 1, 331, 337, 1, 349, 71, 1, 367, 373, 379, 1, 1, 397, 1, 409, 83, 421

Offset: 3

Mohammed Bouras, May 12 2024

nonn

Conjecture 1: The sequence contains only 1's and the primes.
Conjecture 2: Except for 2 and 3, all primes appear in the sequence once.
Conjecture: Record values correspond to A045375(m), m > 2. - Bill McEachen, Aug 03 2024

			For n=3, 1/(2 - 3/(3 + 5)) = 8/13, so a(3)=13.
For n=4, 1/(2 - 3/(3 - 4/(4 + 5))) = 23/19, so a(4)=19.
For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(5 + 5)))) = 13/5, so a(5)=5.
For n=6, 1/(2 - 3/(3 - 4/(4 - 5/(5 - 6/(6 + 5))))) = 227/31, so a(6)=31.

Mohammed Bouras, The Distribution Of Prime Numbers And Continued Fractions, (ppt) (2022).

Cf. A051403, A356360, A369797. A370726.

a(n) = (6n - 5)/gcd(6n - 5, A051403(n-2) + 5*A051403(n-3)).

cp's OEIS Frontend

A056650 a(n) is the greatest prime factor of a(n-1)^2+a(n-1)+1.

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A235473 Primes whose base-3 representation is also the base-4 representation of a prime.

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A260488 Numbers of the form 2^m * (6k + 1) for m, k >= 0, and 0.

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A164979 Slowest growing sequence of primes having the semiprime-pairwise property: for any i,j, a(i)+a(j) is semiprime.

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A171715 Absolute value of (n-th prime of form 3*m-1 minus n-th prime of form 3*k+1/2+-1/2).

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A172182 Nonprimes of the form 6k + 1 or 6k + 2.

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A231476 Primes whose base-3 representation is also the base-6 representation of a prime.

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A279815 Numbers n such that the average of the squares of the numbers less than n that do not divide n is an integer.

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A171715 Absolute value of (n-th prime of form 3m-1 minus n-th prime of form 3k+1/2+-1/2).