A063980 -id:A063980 - cp's OEIS frontend

A063828 Smallest m associated with n-th Pillai prime (A063980).

14, 18, 15, 8, 18, 9, 23, 13, 86, 16, 16, 50, 102, 61, 64, 210, 97, 31, 9, 93, 40, 45, 63, 220, 91, 122, 35, 85, 198, 93, 128, 316, 366, 74, 300, 151, 290, 15, 400, 282, 22, 188, 167, 191, 360, 426, 274, 271, 456, 278, 229, 324, 135, 498, 189

Offset: 1

N. J. A. Sloane, Sep 23 2001

nonn

			14! + 1 == 0 (mod 23), while 23 != 1 (mod 14), where 23=A063980(1) so a(1)=14.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
G. E. Hardy and M. V. Subbarao, A modified problem of Pillai and some related questions, Amer. Math. Monthly 109 (2002), no. 6, 554-559.

Cf. A063980, A211411.

nn=1000; fact=1+Rest[FoldList[Times,1,Range[nn]]]; t={}; Do[p=Prime[i]; m=2; While[mT. D. Noe, Apr 22 2011 *)

first(p)=my(t=Mod(5040, p)); for(m=8, p, t*=m; if(t==-1 && p%m!=1, return(m))); 0
Pillai(p)=my(t=Mod(5040, p)); for(m=8, p-2, t*=m; if(t==-1 && p%m!=1, return(1))); 0
apply(first, select(Pillai, primes(300))) \\ Charles R Greathouse IV, Feb 10 2013

More terms from Vladeta Jovovic, Sep 27 2001

A211411 Largest m associated with n-th Pillai prime (A063980).

Original entry on oeis.org

18, 18, 43, 18, 33, 63, 55, 69, 86, 101, 16, 50, 102, 165, 64, 210, 153, 225, 259, 177, 40, 247, 252, 220, 225, 122, 343, 297, 230, 303, 375, 316, 366, 74, 300, 311, 410, 463, 400, 370, 442, 188, 395, 377, 458, 426, 274, 327, 546, 334, 383, 324, 495, 498, 457, 643, 444, 553, 506, 359, 712, 502, 681, 369, 514

Offset: 1

Charles R Greathouse IV, Feb 10 2013

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
G. E. Hardy and M. V. Subbarao, A modified problem of Pillai and some related questions, Amer. Math. Monthly 109:6 (2002), pp. 554-559.

Trivially a(n) >= A063828(n). Cf. A063980.

last(p)=if(p==23,18,my(t=Mod(1/120,p));forstep(m=p-6,8,-1,t/=m; if(t==-1,return(m-1))))
Pillai(p)=my(t=Mod(5040, p)); for(m=8, p-2, t*=m; if(t==-1 && p%m!=1, return(1))); 0
apply(last, select(Pillai, primes(300)))

19 <= a(n) <= p-7 for n > 11, where p = A063980(n).

A172034 Partial sums of Pillai primes (A063980).

Original entry on oeis.org

23, 52, 111, 172, 239, 310, 389, 472, 581, 718, 857, 1006, 1199, 1426, 1659, 1898, 2149, 2406, 2675, 2946, 3223, 3516, 3823, 4134, 4451, 4810, 5189, 5572, 5961, 6358, 6759, 7178, 7609, 8058, 8519, 8982, 9449, 9928, 10427, 10930, 11451, 12008, 12571

Offset: 1

Jonathan Vos Post, Jan 23 2010

nonn

The values alternate between odd and even. The first prime partial sum of Pillai primes is a(5) = 23 + 29 + 59 + 61 + 67 = 239. The second prime partial sum is a(7) = 389. The next such primes are a(11) = 857 (= the 72nd Pillai prime), a(23) = 3823, a(25) = 4451, a(27) = 5189. The coincidence which prompted this sequence is that the 266th Pillai prime is a(23), the sum of the first 23 Pillai primes. Curiously, 23 is the smallest Pillai prime. What are the next such Pillai primes in the partial sum?

			a(1) = 23 because 23 is the first Pillai prime A063980(1). a(2) = 52 because 23+29 = 52 is the sum of the first two Pillai primes A063980(1)+A063980(2).

a(n) = SUM[i=i..n]A063980(i) = SUM[i=i..n] {p: p prime and there exists an integer m such that m!+1 is 0 mod p and p is not 1 mod m}.

More terms from R. J. Mathar, Jan 24 2010

A225083 Non-Pillai primes: primes p such that for all m such either p is 1 mod m or m!+1 is not 0 mod p.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 43, 47, 53, 73, 89, 97, 101, 103, 107, 113, 127, 131, 151, 157, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 229, 241, 263, 281, 283, 313, 331, 337, 347, 349, 353, 367, 373, 409, 421, 433, 439, 443, 457, 487, 491, 509, 523, 541, 547, 587, 617

Offset: 1

Irina Gerasimova, Apr 27 2013

nonn

Complement of A063980 in the primes.

Sophie Germain primes in this sequence: 2, 5, 11, 41, 53, 89, 113, 131, 173, 179, 191, 281, 443, 491, 509, 641, 653, 659, 743, 761, 911, 1013, 1049, 1103, 1223, 1409, 1439, 1451, 1481, 1583, 1733, 1901, 1931, 1973, 2003,2129, 2141, 2459, 2693, 2741, 2939, 3023, 3299,...

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

is(p)=my(t=Mod(5040, p)); for(m=8, p-2, t*=m; if(t==-1 && p%m!=1, return(0))); isprime(p) \\ Charles R Greathouse IV, Mar 18 2014

New name from Charles R Greathouse IV, Mar 18 2014

A181414 Products of exactly two Pillai primes.

Original entry on oeis.org

529, 667, 841, 1357, 1403, 1541, 1633, 1711, 1769, 1817, 1909, 1943, 2059, 2291, 2407, 2507, 3161, 3481, 3599, 3721, 3953, 4087, 4189, 4331, 4489, 4661, 4757, 4819, 4897, 5041, 5063, 5293, 5561, 5609, 5893, 6241, 6431, 6557, 6649, 6889

Offset: 1

Jonathan Vos Post, Jan 28 2011

It would not be right to call these "Pillai semiprimes" as that would better describe semiprimes k such that there exists an integer m such that m!+1 is 0 mod k and k is not 1 mod m.

There are no pairs (n, n+1) in this sequence since all terms are odd. The first few n such that n and n+2 are in the sequence are 11771, 14099, 19337, 32729, 32741, 34829, 37391, 38249, 39467, 40319, 41747, ... - Charles R Greathouse IV, Jan 28 2011

			a(2) = 23*29.

Cf. A001358, A063980.

{A063980(i) * A063980(j)}.

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A063828 Smallest m associated with n-th Pillai prime (A063980).

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A211411 Largest m associated with n-th Pillai prime (A063980).

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A172034 Partial sums of Pillai primes (A063980).

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A225083 Non-Pillai primes: primes p such that for all m such either p is 1 mod m or m!+1 is not 0 mod p.

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A181414 Products of exactly two Pillai primes.

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