A007528 -id:A007528 - cp's OEIS frontend

A117047 Primes of the form 60*n+11.

11, 71, 131, 191, 251, 311, 431, 491, 911, 971, 1031, 1091, 1151, 1451, 1511, 1571, 1811, 1871, 1931, 2111, 2351, 2411, 2531, 2591, 2711, 3011, 3191, 3251, 3371, 3491, 3671, 3851, 3911, 4091, 4211, 4271, 4391, 4451, 4691, 4751, 4871, 4931, 5051, 5171

Offset: 1

Roger L. Bagula, Apr 17 2006

nonn

a(n) = A211890(5,n-1) for n <= 6. - Reinhard Zumkeller, Jul 13 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A007528, A132231, A214360, A010051.

a117047 n = a117047_list !! (n-1)
a117047_list = [x | k <- [0..], let x = 60 * k + 11, a010051' x == 1]
-- Reinhard Zumkeller, Jul 13 2012

Flatten[Table[If[PrimeQ[60*n + 11], 60*n + 11, {}], {n, 0, 100}]]
Select[60Range[0,100]+11,PrimeQ] (* Harvey P. Dale, Feb 16 2024 *)

Wrong formula removed by Reinhard Zumkeller, Jul 13 2012

A164930 Sum of odd prime divisors of numbers with all odd prime divisors of the form 6k+5.

Original entry on oeis.org

5, 5, 11, 17, 5, 11, 23, 5, 29, 17, 5, 41, 11, 23, 47, 5, 53, 16, 29, 59, 17, 71, 5, 41, 83, 22, 11, 89, 23, 47, 5, 101, 53, 107, 16, 113, 28, 29, 59, 11, 5, 131, 17, 137, 71, 34, 149, 5, 41, 83, 167, 22, 173, 11, 89, 179, 23, 28, 47, 191, 197, 5, 101, 46, 53, 107, 16, 113, 227, 28

Offset: 1

Jonathan Vos Post, Aug 31 2009

We define a sequence b(n) = 5, 10, 11, 17, 20, 22, 23, 25, 29, 34, 40, 41, 44, 46, 47, 50, 53, 55, 58, ... to consist of those numbers where all odd prime factors are primes contained in A007528, and which have at least one prime factor in this class. a(n) is the sum of the distinct odd prime factors of b(n), where "distinct" means that the multiplicity (exponent) in the prime factorization of b(n) is ignored.
Analogous sequence for primes of form 4k+1 is A164927.
Analogous sequence for primes of form 4k+3 is A164928.
Analogous sequence for primes of form 6k+1 is A164929.
The sum of an even number of primes of form 6k+1 is even (hence composite).
The sum of 3 primes of form 6n+5 is composite because (6a+5)+(6b+5)+(6c+5) = 3*(a+b+c+3).
However, the sum of 5 primes of form 6n+5 may be prime:
The smallest number, all of whose prime factors are of form 6k+5, whose sum of distinct prime factors is prime: 881705 = 5 * 11 * 17 * 23 * 41, and 5 + 11 + 17 + 23 + 41 = 97 is prime.

			a(18) = 16 because b(18)= 55 = 5*11, and 5+11 = 16.

Cf. A000040, A007528, A164927-A164930.

isb := proc(n) fs := numtheory[factorset](n) minus {2} ; if fs = {} then RETURN(false); else for f in fs do if op(1,f) mod 6 <> 5 then RETURN(false) ; fi; od: RETURN(true) ; fi; end:
b := proc(n) if n = 1 then 5; else for a from procname(n-1)+1 do if isb(a) then RETURN(a) ; fi; od: fi; end:
A164930 := proc(n) local f; numtheory[factorset]( b(n)) minus {2} ; add(f,f=%) ; end: seq(A164930(n),n=1..120) ; # R. J. Mathar, Sep 09 2009

Edited and extended by R. J. Mathar, Sep 09 2009

A170825 a(n) is the product of the distinct primes of form 6*k-1 that divide n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 11, 1, 1, 1, 5, 1, 17, 1, 1, 5, 1, 11, 23, 1, 5, 1, 1, 1, 29, 5, 1, 1, 11, 17, 5, 1, 1, 1, 1, 5, 41, 1, 1, 11, 5, 23, 47, 1, 1, 5, 17, 1, 53, 1, 55, 1, 1, 29, 59, 5, 1, 1, 1, 1, 5, 11, 1, 17, 23, 5, 71, 1, 1, 1, 5, 1, 11, 1, 1, 5, 1, 41, 83, 1, 85, 1, 29, 11, 89, 5, 1

Offset: 1

N. J. A. Sloane, Dec 25 2009, following a suggestion from Jonathan Vos Post

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

Cf. A007528, A007947, A170819, A343431.

Cf. A140214. - R. J. Mathar, Jan 21 2010

A170825 := proc(n) a := 1 ; for p in numtheory[factorset](n) do if p mod 6 = 5 then a := a*p ; end if ; end do ; a ; end proc: seq(A170825(n),n=1..120) ; # R. J. Mathar, Jan 21 2010

Table[Times@@Select[Transpose[FactorInteger[n]][[1]],IntegerQ[(#+1)/6]&],{n,100}] (* Harvey P. Dale, Nov 01 2013 *)

a(n) = my(f=factor(n)); for(i=1, #f~, f[i,2] = (f[i,1]%6)==5); factorback(f); \\ Michel Marcus, Sep 30 2020

a(n) = A007947(A343431(n)) = A343431(A007947(n)). - Peter Munn, Jan 15 2022

More terms from R. J. Mathar, Jan 21 2010

A208296 Smallest positive nontrivial odd solution of the congruence x^2 == 1 (mod A001748(n+2)), n >= 1.

Original entry on oeis.org

11, 13, 23, 25, 35, 37, 47, 59, 61, 73, 83, 85, 95, 107, 119, 121, 133, 143, 145, 157, 167, 179, 193, 203, 205, 215, 217, 227, 253, 263, 275, 277, 299, 301, 313, 325, 335, 347, 359, 361, 383, 385, 395, 397, 421, 445, 455, 457, 467, 479, 481, 503, 515

Offset: 1

Wolfdieter Lang, Mar 14 2012

nonn

The trivial solutions of the congruence x^2 == 1 (mod 3*prime(n+2)), n>=1, with the primes prime(n+2) = A000040(n+2) have positive representatives 1 and 3*prime(n+2)-1. There are all-together four incongruent solutions due to a general theorem (see, e.g., the Hardy-Wright reference, Theorem 122, p. 96, and also A060594) and the fact that the number of incongruent solutions of this congruence with odd prime modulus p is two, namely with positive representative p and p-1 (see, e.g., Hardy-Wright, Theorem 109, p. 85). a(n) is the smallest positive odd representative >1 which solves this congruence. The other nontrivial even representative solving this congruence is 3*prime(n+2) - a(n), i.e. 4, 8, 10, 14, 16, 20, ... See 2*A207336.

a(n) solves also the congruence x^2 == 1 (Modd A001748(n+2)), n>=1. For Modd n (not to be confused with mod n) see a comment on A203571. This follows from floor(a(n)^2/3*prime(n+2)) being even, in fact it is 8*A024699(n) (see a comment there), hence a(n)^2 (Modd 3*prime(n+2)) = a(n)^2 (mod 3*prime(n+2)) = 1. For those multiplicative groups Modd 3*p with p an odd prime which are cyclic (this is not possible in the mod case, see A033949), a(n) is the representative of the only other nontrivial solution of this congruence. The representative of the trivial solution is 1 (-1 belongs to the same Modd class). (The conjecture stated here earlier is wrong, that is, the multiplicative group Modd (91=7*13) is non-cyclic. It may still be true for 3*p. - Wolfdieter Lang, Mar 15 2012)

			a(3)=23 because prime(5)=11=A007528(2), hence K(3)=11 and sqrt(8*T(11)+1)=sqrt(8*66+1)= 23. 23^2 = 529 == 1 (Modd 33), because floor(529/33)=16=8*A024699(3) is even, and 529 == 1 (mod 33).
a(4)=25 because prime(6)=13=A002476(2), hence K(4)=12 and sqrt(8*T(12)+1)=sqrt(8*78+1)=25. 25^2 = 625 == 1 (Modd 39), because floor(625/39)=16=8*A024699(4) is even, and 625 == 1 (mod 39).

H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Clarendon Press, Oxford, 2003.

Jon Maiga, Table of n, a(n) for n = 1..1000

Cf. A000217, A001748, A002476, A007528, A024699, A060594, A203571, A207336.

Table[SelectFirst[Solve[x^2==1 && x !=1,x, Modulus->3*Prime[n+2]][[All,1,2]],OddQ], {n, 53}] (* Jon Maiga, Sep 28 2019 *)

a(n) = sqrt(8*T(K(n))+1), with the triangular numbers T = A000217, and K(n) = prime(n+2)-1 if the prime prime(n+2) is of the form 6*k+1, i.e., from A002476, and K(n) = prime(n+2) if prime(n+2) is of the form 6*k-1, i.e. from A007528.
a(n)^2 == 1 (mod A001748(n+2)), n >= 1.
a(n)^2 == 1 (Modd A001748(n+2)), n >= 1.

A268928 Record (maximal) gaps between primes of the form 6k - 1.

Original entry on oeis.org

6, 12, 18, 30, 36, 42, 54, 60, 84, 126, 150, 162, 168, 246, 258, 318, 342, 354, 372, 408, 468, 534, 552, 600, 654, 762, 768, 798, 864, 894, 942, 960, 1068, 1224, 1302, 1320, 1344

Offset: 1

Alexei Kourbatov, Feb 15 2016

Dirichlet's theorem on arithmetic progressions and the GRH suggest that average gaps between primes of the form 6k - 1 below x are about phi(6)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(6)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(6)=2.
Conjecture: a(n) < phi(6)*log^2(A268930(n)) almost always.
Conjecture: phi(6)*n^2/6 < a(n) < phi(6)*n^2 almost always. - Alexei Kourbatov, Nov 27 2019

			The first two primes of the form 6k-1 are 5 and 11, so a(1)=11-5=6. The next primes of this form are 17, 23, 29; the gaps 17-11 = 23-17 = 29-23 are not records so nothing is added to the sequence. The next prime of this form is 41 and the gap 41-29=12 is a new record, so a(2)=12.

Alexei Kourbatov, On the distribution of maximal gaps between primes in residue classes, arXiv:1610.03340 [math.NT], 2016.
Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression, arXiv:1709.05508 [math.NT], 2017; Int. Math. Forum, 13 (2018), 65-78.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A007528, A268929 (primes preceding the maximal gaps), A268930 (primes at the end of the maximal gaps), A334543, A334544.

re = 0; s = 5; Reap[For[p = 11, p < 10^8, p = NextPrime[p], If[Mod[p, 6] != 5, Continue[]]; g = p - s; If[g > re, re = g; Print[g]; Sow[g]]; s = p]][[2, 1]] (* Jean-François Alcover, Dec 12 2018, from PARI *)

re=0; s=5; forprime(p=11, 1e8, if(p%6!=5, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)

a(n) = A268930(n) - A268929(n). - Alexei Kourbatov, Jun 15 2020.

Terms a(31)..a(37) from Alexei Kourbatov, Jun 15 2020

A268929 Primes 6k - 1 preceding the maximal gaps in A268928.

Original entry on oeis.org

5, 29, 113, 197, 521, 1109, 1733, 6389, 7349, 35603, 148517, 180797, 402593, 406907, 2339039, 5521721, 11157989, 20831267, 22440701, 27681263, 73451723, 241563407, 953758109, 1444257671, 1917281213, 6822753629, 15867286361, 28265029631, 40841579819, 177858259463

Offset: 1

Alexei Kourbatov, Feb 15 2016

nonn

Subsequence of A007528 and A334544.

A268928 lists the corresponding record gap sizes. See more comments there.

			The first two primes of the form 6k-1 are 5 and 11, so a(1)=5. The next primes of this form are 17, 23, 29; the gaps 17-11 = 23-17 = 29-23 are not records so nothing is added to the sequence. The next prime of this form is 41 and the gap 41-29=12 is a new record, so a(2)=29.

Alexei Kourbatov, Table of n, a(n) for n = 1..37
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A007528, A268928, A268930, A334543, A334544.

re=0; s=5; forprime(p=11, 1e8, if(p%6!=5, next); g=p-s; if(g>re, re=g; print1(s", ")); s=p)

a(n) = A268930(n) - A268928(n). - Alexei Kourbatov, Jun 15 2020.

A268930 Primes 6k - 1 at the end of the maximal gaps in A268928.

Original entry on oeis.org

11, 41, 131, 227, 557, 1151, 1787, 6449, 7433, 35729, 148667, 180959, 402761, 407153, 2339297, 5522039, 11158331, 20831621, 22441073, 27681671, 73452191, 241563941, 953758661, 1444258271, 1917281867, 6822754391, 15867287129, 28265030429, 40841580683, 177858260357

Offset: 1

Alexei Kourbatov, Feb 15 2016

nonn

Subsequence of A007528 and A334545.

A268928 lists the corresponding record gap sizes. See more comments there.

			The first two primes of the form 6k-1 are 5 and 11, so a(1)=11. The next primes of this form are 17, 23, 29; the gaps 17-11 = 23-17 = 29-23 are not records so nothing is added to the sequence. The next prime of this form is 41 and the gap 41-29=12 is a new record, so a(2)=41.

Alexei Kourbatov, Table of n, a(n) for n = 1..37
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A007528, A268928, A268929, A334543, A334545.

re=0; s=5; forprime(p=11, 1e8, if(p%6!=5, next); g=p-s; if(g>re, re=g; print1(p", ")); s=p)

a(n) = A268928(n) + A268929(n). - Alexei Kourbatov, Jun 15 2020.

A319716 Filter sequence combining the largest proper divisor of n (A032742) with modulo 6 residue of the smallest prime factor, A010875(A020639(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 7, 12, 13, 14, 5, 15, 7, 16, 17, 18, 5, 19, 20, 21, 22, 23, 5, 24, 7, 25, 26, 27, 28, 29, 7, 30, 31, 32, 5, 33, 7, 34, 35, 36, 5, 37, 38, 39, 40, 41, 5, 42, 43, 44, 45, 46, 5, 47, 7, 48, 49, 50, 51, 52, 7, 53, 54, 55, 5, 56, 7, 57, 58, 59, 60, 61, 7, 62, 63, 64, 5, 65, 66, 67, 68, 69, 5, 70, 71, 72, 73, 74, 75, 76, 7, 77, 78, 79, 5, 80, 7, 81, 82, 83, 5, 84, 7, 85, 86, 87, 5, 88, 89, 90, 91, 92, 93, 94, 43

Offset: 1

Antti Karttunen, Oct 04 2018

nonn

Restricted growth sequence transform of A286475, or equally, of A286476.
In each a(n) there is enough information to determine the modulo 6 residues of all the prime factors of n (when counted with multiplicity), thus sequences like A319690 and A319691 (which is the characteristic function of A004611) are essentially functions of this sequence. However, to determine that for all divisors of n, more information is needed. See A319717.
For all i, j:
  A319707(i) = A319707(j) => A319717(i) = A319717(j) => a(i) = a(j),
  a(i) = a(j) => A319690(i) = A319690(i) => A319691(i) = A319691(j).

			For n = 55 = 5*11 and 121 = 11*11, 55 = 121 = 1 mod 6 and 11 is their common largest proper divisor, thus they are allotted the same number by the restricted growth sequence transform, that is a(55) = a(121) = 43 (which is the number allotted). Note that such nontrivial equivalence classes may only contain numbers that are 5-rough, A007310, with no prime factors 2 or 3.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A010875, A032742, A286475, A286476, A319707, A319717.
Cf. A007528 (positions of 5's), A002476 (positions of 7's).
Cf. also A319714.
Differs from A319707 and A319717 for the first time at n=121.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A286476(n) = if(1==n,n,(6*A032742(n) + (n % 6)));
v319716 = rgs_transform(vector(up_to,n,A286476(n)));
A319716(n) = v319716[n];

A329946 Primes that are not of the form u(u+1)/2 - v(v-3)/2 for any u >= v >= 1.

Original entry on oeis.org

3, 5, 11, 17, 29, 41, 53, 59, 83, 89, 101, 107, 113, 131, 137, 149, 173, 197, 233, 257, 269, 293, 317, 353, 389, 419, 443, 449, 461, 467, 509, 557, 563, 569, 587, 593, 617, 653, 677, 761, 773, 797, 809, 827, 857, 929, 941, 947, 977, 1013, 1049, 1097, 1109

Offset: 1

Peter Luschny, Dec 16 2019

nonn

These primes were originally called "hidden primes", but since that term is already in use (see A187399) it has been replaced by an explicit definition. - Editors of OEIS, Dec 16 2019
The following is the original definition. Assume n and s are positive integers. We say a prime p is 'reachable' from n if there exists an s such that 8*(p - (s + 1)*n) + 1 is a perfect square, and that a prime p is 'hidden' if it is not reachable from any n.
Equivalently, a prime p is reachable if there exists m >= n such that p = m(m+1)/2 - n(n-3)/2.
A description of the sequence as an arithmetic training game for children was given on the Sequence Fans Mailing List. A representation as a sieve is given in the Maple script.
The game is to start at n and (cumulatively) add n, n+1, n+2, ..., m until a prime is reached, which appears to happen for all n, usually with m close to n, except for n = 3.
Conjecture: The sequence is infinite.
For comparison the number of primes < 10^n:
n              :  1   2   3    4     5      6       7        8
Ramanujan p.   :  1, 10, 72, 559, 4459, 36960, 316066, 2760321, ...
Hidden primes  :  2, 10, 49, 271, 1768, 34181, 601549,
Lesser twin p. :  2,  8, 35, 205, 1224,  8169,  58980,  440312, ...
All terms except a(1) = 3 are congruent to 5 (mod 6), i.e., in A007528. Indeed, any prime p = 6k + 1 is reached from n = 2k in 2 steps. - M. F. Hasler, Dec 16 2019
Only one prime is eliminated (for n != 3) by each (variable sized) "grid" G(n) = (2n, 3n + 1, 4n + 3, 5n + 6, ..., (m+2)n + T(m), ...), since the scan stops as soon as the first prime is found. If used as a sieve in the usual sense, the grid G(n) should also eliminate all subsequent primes of the form (m+2)n + T(m). If this were done, only Fermat primes A019434 = {3, 5, 17, 257, 65537, ?} would remain. - M. F. Hasler, Dec 17 2019

Peter Luschny, Hopping for primes, SeqFan list, Dec 13 2019.

Cf. A000040, A000217 (triangular numbers), A000096 (n(n+3)/2), A187399, A330501 (least prime m(m+1)/2 - n(n-3)/2, m >= n), A330502 (corresponding m).

aList := proc(lim) local n, p, k, L:
L := select(isprime, {$1..lim}):
for n from 1 to iquo(lim, 2) do
   p := n:
   for k from n to 10000 do
      p := p + k:
      if isprime(p)
      then L := L minus {p}: break fi;
      if p > lim then break fi;
   od:
od: sort(L) end:
aList(1111);

A329946=setminus(primes(199), Set(apply((n,p=n)->while(!isprime(p+=n), n++); p, [1..1199][^3]))) \\ M. F. Hasler, Dec 16 2019

def aSieve(lim):
    S = Set(prime_range(lim))
    for n in (1..lim//2):
        p = n
        for k in (n..10000):
            p += k
            if p > lim: break
            if is_prime(p):
                S = S.difference({p})
                break
    return sorted(S)
aSieve(1111)

Edited by M. F. Hasler and N. J. A. Sloane, Dec 16 2019

A334545 Primes of the form 6k - 1 at the end of first-occurrence gaps in A334543.

Original entry on oeis.org

11, 41, 131, 227, 383, 557, 1151, 1787, 4337, 6449, 7433, 8363, 9137, 12893, 35729, 37781, 58889, 59879, 97787, 105863, 130769, 148667, 153887, 180959, 220151, 328271, 402761, 407153, 416849, 542441, 780587, 1138367, 1294571, 1444463, 1463837, 1604951

Offset: 1

Alexei Kourbatov, May 05 2020

nonn

Subsequence of A007528. Contains A268930 as a subsequence. First differs from A268930 at a(5)=383.

A334543 lists the corresponding gap sizes; see more comments there.

			The first two primes of the form 6k-1 are 5 and 11, so we have a(1)=11. The next primes of this form are 17, 23, 29; the gaps 17-11 = 23-17 = 29-23 have size 6 which already occurred before; so nothing is added to the sequence. The next prime of this form is 41 and the gap size 41-29=12 has not occurred before, so a(2)=41.

Alexei Kourbatov, Table of n, a(n) for n = 1..160
Alexei Kourbatov and Marek Wolf, On the first occurrences of gaps between primes in a residue class, arXiv preprint arXiv:2002.02115 [math.NT], 2020.

Cf. A007528, A014320, A058320, A268930, A330855, A334543, A334544.

isFirstOcc=vector(9999,j,1); s=5; forprime(p=11,1e8,if(p%6!=5,next); g=p-s; if(isFirstOcc[g/6], print1(p", "); isFirstOcc[g/6]=0); s=p)

a(n) = A334543(n) + A334544(n).

cp's OEIS Frontend

A117047 Primes of the form 60*n+11.

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A164930 Sum of odd prime divisors of numbers with all odd prime divisors of the form 6k+5.

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A170825 a(n) is the product of the distinct primes of form 6*k-1 that divide n.

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A208296 Smallest positive nontrivial odd solution of the congruence x^2 == 1 (mod A001748(n+2)), n >= 1.

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A268928 Record (maximal) gaps between primes of the form 6k - 1.

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A268929 Primes 6k - 1 preceding the maximal gaps in A268928.

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A268930 Primes 6k - 1 at the end of the maximal gaps in A268928.

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