Chris K. Caldwell - cp's OEIS frontend

A136009 Primes p for which the least pentagonal number q=n(3n+/-1)/2 which makes p+q prime satisfies 2q > p.

3, 7, 13, 19, 23, 37, 43, 73, 103, 233, 359, 503

Offset: 1

Chris K. Caldwell, Mar 18 2008

Is the list complete?
For each prime p, the author is looking for the least generalized pentagonal number (A001318) q such that p+q is prime. The list of such q begins with: 1, 2, 2, 12, 2, 40, 2, 12, 126, 2, 12, 22, 2, 40, 12. In this list, most q's are much smaller than their corresponding p. The current sequence gives the primes p for which q is greater than p/2. - Michel Marcus, Aug 21 2013
No more terms up to 10^10. Heuristics suggest there are no more terms. - Charles R Greathouse IV, Aug 21 2013

			For p=23, q=126. For p=503, q=590.
More examples:
p=3, q=2, p+q=5.
p=7, q=12, p+q=19.
p=13, q=40, p+q=53.
p=19, q=12, p+q=31.
p=23, q=126, p+q=149.
p=37, q=22, p+q=59.
p=43, q=40, p+q=83.
p=73, q=40, p+q=113.
p=103, q=70, p+q=173.
p=233, q=126, p+q=359.
p=359, q=210, p+q=569.
p=503, q=590, p+q=1093. - _Michel Marcus_, Aug 21 2013

G. N. Watson, Math. Notes 2103. A Curious Property of the Prime Number 503, Math. Gaz. vol 33, no 306, Dec 1949, pp. 302-303.

isok(p) = my(nn = 1, ok = 0); until (ok, q = nn*(3*nn - 1)/2; if (isprime(p+q), ok = 1; if (2*q > p, return (q))); q = nn*(3*nn + 1)/2; if (isprime(p+q), ok = 1; if (2*q > p, return (q));); nn++;); 0 \\ Michel Marcus, Aug 21 2013

is(p)=my(n=1,t);while(!isprime(t=p+n*(3*n-1)/2)&&!isprime(t=p+n*(3*n+1)/2)&&2*t<=3*p,n++); 2*t>3*p && isprime(p) \\ Charles R Greathouse IV, Aug 21 2013

a(10) = 233 inserted by Michel Marcus, Aug 21 2013

A108739 Mills' constant A generates a sequence of primes via b(n)= floor(A^3^n). This sequence is a(n) = b(n+1)-b(n)^3.

Original entry on oeis.org

3, 30, 6, 80, 12, 450, 894, 3636, 70756, 97220, 66768, 300840, 1623568, 8436308

Offset: 1

Chris K. Caldwell, Jun 22 2005

This allows larger terms of A051254 (which triple in digits each entry) to be given. Like A051254, currently requires Riemann Hypothesis to show sequence continues.
Currently a(11)=66768 generates only a probable prime number. - Arkadiusz Wesolowski, May 28 2011
Likewise a(12) and a(13) generate only a probable prime numbers, as well as being conditional on a(11) and a(12) being proved primes. Minimality of a(12)-a(13) is exhaustively tested. - Serge Batalov, Aug 06 2013
a(14) = 8436308 is found by Ryan Propper and Serge Batalov, Apr 29 2024, but a few remaining gaps below this value were being double-checked. The double-check is now complete (see GitHub link). - Ryan Propper and Serge Batalov, May 24 2024.

			The Mills' primes (given in A051254) are 2, 2^3+3 = 11, (2^3+3)^3+30 = 11^3+30 = 1361, ((2^3+3)^3+30)^3+6 = 1361^3+6 = 2521008887, etc. The terms added at each step yield this sequence. They are the least positive integers which added to the cube of the preceding prime yield again a prime, cf. formula. - _M. F. Hasler_, Jul 22 2013

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 8.

Chris K. Caldwell, Mills' Theorem - a generalization.
Chris K. Caldwell, The List of Largest Known Primes, The 11th Mills' prime
Chris K. Caldwell and Yuanyou Cheng, Determining Mills' Constant and a Note on Honaker's Problem, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1.
Sergey Batalov, Mills' b(14) search report
Henri & Renaud Lifchitz, PRP Records and PRP Records, search:Mills
W. H. Mills, A prime-representing function, Bull. Amer. Math. Soc., Vol. 53 (1947), p. 604.
Eric Weisstein's World of Mathematics, Mills' Constant
Eric Weisstein's World of Mathematics, Mills' Prime
E. M. Wright, A class of representing functions, J. London Math. Soc., Vol. 29 (1954) pp. 63-71.

Cf. A051254, A051021.

B[1] = 2; B[n_] := B[n] = NextPrime[B[n - 1]^3]; Table[B[n + 1] - B[n]^3, {n, 7}] (* Robert Price, Jun 09 2019 *)

p=2; until(, np=nextprime(p^3); print1(np-p^3, ", "); p=np) \\ Jeppe Stig Nielsen, Apr 22 2020

b(1) = 2; b(n+1) = nextprime(b(n)^3); a(n) = b(n+1)-b(n)^3;

a(9)-a(11) from Caldwell and Cheng, Aug 29 2005
Corrected by T. D. Noe, Sep 24 2008
a(12) (which generates a PRP) from Serge Batalov, Jul 19 2013
a(13) (which generates a PRP) from Serge Batalov, Aug 06 2013
a(14) (which generates a PRP) from Ryan Propper and Serge Batalov, May 24 2024

A053663 Numbers k such that 2*k+1 does not divide k!+1.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Chris K. Caldwell, Feb 16 2000

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A005097, complement of A053662.

Filtered([1..100], n-> not (Factorial(n)+1) mod (2*n+1)=0) # G. C. Greubel, May 18 2019

[n: n in [1..100] | not (Factorial(n)+1) mod (2*n+1) eq 0 ]; // G. C. Greubel, May 18 2019

Select[Range[100],!Divisible[#!+1,2#+1]&] (* Harvey P. Dale, Sep 09 2014 *)

isok(n) = (n!+1) % (2*n+1) \\ Michel Marcus, Jul 12 2013

[n for n in (1..100) if not Mod(factorial(n)+1, 2*n+1)==0 ] # G. C. Greubel, May 18 2019

Data corrected by Michel Marcus, Jul 12 2013

A053662 Numbers k such that 2k+1 divides k!+1.

Original entry on oeis.org

3, 5, 9, 21, 23, 33, 39, 51, 63, 65, 81, 89, 95, 99, 113, 131, 173, 183, 191, 209, 215, 221, 239, 245, 251, 261, 281, 285, 299, 303, 309, 315, 341, 345, 363, 369, 371, 393, 411, 419, 431, 443, 473, 495, 509, 525, 543, 545, 561, 575, 593, 645, 659, 683, 711

Offset: 1

Chris K. Caldwell, Feb 16 2000

k+1 divides k!+1 gives primes-1 by Wilson's Theorem. For the present sequence, there are 309 terms below 5000, compared with 669 primes (309/669 = 0.461...). There are 553 terms below 10000, compared with 1229 primes (553/1229 = 0.449...). - Ed Pegg Jr, Dec 05 2001

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A005097, A006093, A053663.

Filtered([1..1000], n-> (Factorial(n)+1) mod (2*n+1)=0) # G. C. Greubel, May 18 2019

[n: n in [1..1000] | (Factorial(n)+1) mod (2*n+1) eq 0 ]; // G. C. Greubel, May 18 2019

A053662:=n->`if`(n!+1 mod (2*n+1) = 0, n, NULL): seq(A053662(n), n=1..1000); # Wesley Ivan Hurt, Dec 01 2015

Drop[Union[Table[If[IntegerQ[(n!+1)/(2n+1)], n], {n, 1, 1000}]], -1] (* Ed Pegg Jr, Dec 05 2001 *)
Select[Range[1000], Mod[#! +1, 2*# +1] == 0 &] (* G. C. Greubel, May 18 2019 *)

for(n=1,10^3, if((n!+1)%(2*n+1)==0, print1(n,", ")) ) \\ G. C. Greubel, May 18 2019

[n for n in (1..1000) if Mod(factorial(n)+1, 2*n+1)==0 ] # G. C. Greubel, May 18 2019

a(n) >> n log n. - Charles R Greathouse IV, Apr 16 2024

A005537 Numbers m such that 4*3^m + 1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 6, 14, 15, 39, 201, 249, 885, 1005, 1254, 1635, 3306, 3522, 9602, 19785, 72698, 233583, 328689, 537918, 887535, 980925, 1154598, 1499606, 1936890, 2016951, 2143374

Offset: 1

N. J. A. Sloane, Chris K. Caldwell

a(27) > 1.5*10^6. - Matthias Baur, Jan 16 2020
a(20) > 2*10^5. - Robert Price, Nov 23 2013
Primes resulting from a(1)-a(19) are confirmed primes (not probable primes) using BLS (N-1/N+1) test in pfgw. - Robert Price, Nov 23 2013
From Matthias Baur, Jan 16 2020: (Start)
Double checked to n=2*10^5, tested further to n=1.5*10^6 using the sieve programs newpgen and srsieve and using Jean Penné's LLR application (BLS (N-1/N+1) test).
a(20) was already known in 2005, but was not listed here until 2018 (see Prime Pages link). (End)
Because of the factorization 4*x^4 + 1 = (2*x^2 - 2*x + 1)*(2*x^2 + 2*x + 1), the only term divisible by 4 is 0. - Jeppe Stig Nielsen, Sep 12 2024

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

C. K. Caldwell, The Prime Pages
Douglas E. Iannucci, Deng Moujie and Graeme L. Cohen, On Perfect Totient Numbers, J. Integer Sequences, 6 (2003), #03.4.5.
P. Loomis, M. Plytage and J. Polhill, Summing up the Euler 'phi' function, The College Mathematics Journal, vol. 39 (2008), pp. 34-42.
H. C. Williams and C. R. Zarnke, Some prime numbers of the forms 2*3^n+1 and 2*3^n-1, Math. Comp., 26 (1972), 995-998.

Cf. A005540, A125734.

a[n_]:=If[PrimeQ[4*3^n + 1 ], n]; DeleteCases[Array[a, 40, 0], Null] (* Stefano Spezia, Nov 12 2018 *)

is_a(m) = isprime(4*3^m + 1) \\ Michel Marcus, Jul 12 2013

a(15)-a(17) from Douglas Burke (dburke(AT)nevada.edu)
a(18) from Mohammed Bouayoun (Mohammed.Bouayoun(AT)sanef.com), Jan 26 2004
a(19) from Robert Price, Nov 23 2013
a(20)-a(21) from Matthias Baur, Nov 07 2018
a(22) from Matthias Baur, Dec 06 2018
a(23)-a(24) from Matthias Baur, Jul 23 2019
a(25) from Matthias Baur, Dec 07 2019
a(26) from Matthias Baur, Jan 16 2020
a(27)-a(29) from Ryan Propper, May 08 2020

A003306 Numbers k such that 2*3^k + 1 is prime.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 9, 16, 17, 30, 54, 57, 60, 65, 132, 180, 320, 696, 782, 822, 897, 1252, 1454, 4217, 5480, 6225, 7842, 12096, 13782, 17720, 43956, 64822, 82780, 105106, 152529, 165896, 191814, 529680, 1074726, 1086112, 1175232, 1277862, 1346542, 3123036, 3648969, 5570081, 6236772, 10852677

Offset: 1

N. J. A. Sloane, Chris K. Caldwell

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

C. K. Caldwell, The Prime Pages
Hartosh Singh Bal and Gaurav Bhatnagar, Prime number conjectures from the Shapiro class structure, arXiv:1903.09619 [math.NT], 2019. See also Integers (2020) Vol. 20, Article A11.
W. Keller and J. Richstein, Solutions of the congruence a^(p-1) == 1 (mod p^r), Math. Comp. 74 (2005), 927-936.
H. C. Williams and C. R. Zarnke, Some prime numbers of the forms 2A3^n+1 and 2A3^n-1, Math. Comp., 26 (1972), 995-998.

Cf. A056802 (k such that 2*9^k + 1 is prime).

Cf. A111974 (primes of the form 2*3^k + 1), A003307 (k such that 2*3^k - 1 is prime).

lst={};Do[If[PrimeQ[2*3^n+1], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 19 2008 *)

is(n)=isprime(2*3^n+1) \\ Charles R Greathouse IV, Feb 17 2017

More terms from T. D. Noe, Aug 24 2005
More terms from David Broadhurst, Feb 14 2010
Another term from David Broadhurst, Feb 22 2010
a(42)-a(45) found by Ryan Propper and Paul S. Vanderveen, Feb 09 2020
a(46) found by Ryan Propper, Feb 14 2020
a(47)-a(48) found by Ryan Propper added by Paul S. Vanderveen, Jan 08 2023

cp's OEIS Frontend

User: Chris K. Caldwell

A136009 Primes p for which the least pentagonal number q=n(3n+/-1)/2 which makes p+q prime satisfies 2q > p.

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A108739 Mills' constant A generates a sequence of primes via b(n)= floor(A^3^n). This sequence is a(n) = b(n+1)-b(n)^3.

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A053663 Numbers k such that 2*k+1 does not divide k!+1.

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A053662 Numbers k such that 2k+1 divides k!+1.

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A005537 Numbers m such that 4*3^m + 1 is prime.

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A003306 Numbers k such that 2*3^k + 1 is prime.

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