Mike Oakes - cp's OEIS frontend

A275747 Largest integer which cannot be written as a sum of n-th powers of odd primes.

4, 114, 1595, 24195, 190463, 4613167, 23815950, 240951477, 2448240688, 40379465440, 258630935725, 3044619003271, 26232953378021, 264066033166293, 3218396161633945, 29248021240625765, 302064442857973298, 3027736057187540859, 32207709908667683204

Offset: 1

Mike Oakes, Aug 07 2016

nonn

Except for n=6, only the powers of 3 primes are needed, namely 3^n, 5^n and 7^n, which leads to an ultra-fast O(n) execution time. I executed the algorithm in Greenberg (1988) with a PARI/GP program in only a few seconds for 1000 terms. - Mike Oakes, Aug 16 2016

			For n=2, 114 is the largest integer which cannot be written as a sum of squares of odd primes.

Mike Oakes, Table of n, a(n) for n = 1..1000
H. Greenberg, Solution to a linear diophantine equation for nonnegative integers, Journal of Algorithms, 9 (1988), 343-353.

Cf. A275746. For all primes, including 2, see A275743, A122615.

With[{prs=Prime[Range[2,5]]},Table[FrobeniusNumber[prs^n],{n,20}]] (* Harvey P. Dale, Sep 19 2018 *)

A275746 The number of integers not representable as a sum of n-th powers of odd primes.

Original entry on oeis.org

3, 62, 840, 12350, 99912, 2358913, 12173587, 120900206, 1224812108, 20439317893, 130357794430, 1534358290778, 13544972574336, 142155007727976, 1648210755327790, 14816750258004918, 151900144117560355, 1525977077403654254, 16185530587326104490

Offset: 1

Mike Oakes, Aug 07 2016

nonn

I executed the algorithm in Boecker et al. (2007) with a PARI/GP program, which needed 36 GB RAM and 2 hours to run for n=19. Results checked by an independent algorithm. - Mike Oakes, Aug 17 2016

			For n=2, there are 62 integers that cannot be written as a sum of squares of odd primes.

S. Boecker and Z. Liptak, A Fast and Simple Algorithm for the Money Changing Problem, Algorithmica, 48 (2007), 413-432.

Cf. A275747. For all primes, including 2, see A275743, A122615.

More terms from Mike Oakes, Aug 17 2016

A275743 The number of integers not representable as a sum of n-th powers of primes.

Original entry on oeis.org

1, 12, 83, 600, 3058, 18222, 130411, 783480, 3418488, 24857930, 181308931, 610675758, 6529545751, 18840405540, 102444103996, 956394012747, 3188347929974, 27288773879397, 96923972375603, 702926766993950, 2980668206142664, 20046633698159181, 88137290213079355, 502113072640777785

Offset: 1

Mike Oakes, Aug 07 2016

nonn

I executed the algorithm in Boecker et al. (2007) with a PARI/GP program, which needed 34 GB RAM and 2 hours to run for n=30. Results checked by an independent algorithm. - Mike Oakes, Aug 17 2016

			For n=2, there are 12 nonnegative integers that cannot be written as a sum of squares of primes.

Mike Oakes, Table of n, a(n) for n = 1..30
S. Boecker and Z. Liptak, A Fast and Simple Algorithm for the Money Changing Problem, Algorithmica, 48 (2007), 413-432.

Cf. A122615, A275746.

More terms from Mike Oakes, Aug 17 2016

a(16) corrected by Andrew Howroyd, Feb 23 2018

A166339 Numbers n such that Sum(n!/k!),k=0..n is prime.

Original entry on oeis.org

1, 2, 2016, 3226, 8130

Offset: 1

Mike Oakes, Oct 12 2009

nonn

No further terms up to n=20000. The sequence a[n]=Sum(n!/k!),k=0..n is A000522.

			For n=2, a[2]=5 which is a prime.

Cf. A000522 (corresponding sequence).

a=1;for(n=1,20000,a=n*a+1;if(ispseudoprime(a,0),print("n="n)));

n is an index of the sequence a[n]=Sum(n!/k!),k=0..n such that a[n] is prime.

A166342 Numbers n such that (n+2)n!Sum((-1)^k/k!),k=0..n+2 is prime.

Original entry on oeis.org

2, 3, 4, 8, 13, 42, 64, 166, 573, 1711

Offset: 1

Mike Oakes, Oct 12 2009

nonn

No further terms up to n=20000. The sequence a[n]=(n+2)*n!*Sum((-1)^k/k!),k=0..n+2 is A000255.

			For n=3, a[3]=11 which is prime.

Cf. A000255 (corresponding sequence).

z_even=1;z_odd=1;
for(n=1,20000,
if(n%2==1,
z_odd=(n+2)*(n-1)*z_odd+1; if((ispseudoprime(z_odd)),print(n="n)),
z_even=(n+2)*(n-1)*z_even-1; if((ispseudoprime(z_odd)),print(n="n))););

n is an index of the sequence a[n]=(n+2)*n!*Sum((-1)^k/k!),k=0..n+2 such that a[n] is prime.

A059803 Numbers n such that 9^n - 8^n is prime or a strong pseudoprime.

Original entry on oeis.org

2, 7, 29, 31, 67, 149, 401, 2531, 19913, 30773, 53857, 170099

Offset: 1

Mike Oakes, Feb 23 2001

Some of the larger entries may only correspond to probable primes.

In general, for any positive integers n, a and b, a>b, a necessary condition for a^n-b^n to be prime is that either a-b=1 and n be a prime or n=1 and a-b be prime (from Arturo Magidin and Hagman in Sci.Math, Sep 11, 2010). - Vincenzo Librandi, Sep 12 2010

Cf. A000043, A057468, A059801, A059802, A062572-A062574, A062576-A062666.

is(n)=ispseudoprime(9^n-8^n) \\ Charles R Greathouse IV, Jun 13 2017

Three more terms found by Jean-Louis Charton in 2004-2005: a(9) = 30773, a(10) = 53857, a(11) = 170099. - Alexander Adamchuk, Dec 08 2006

A062654 Numbers k such that 88^k - 87^k is prime or a strong pseudoprime.

Original entry on oeis.org

5, 7, 19, 241, 607, 74047

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

All terms must be prime. - M. F. Hasler, Sep 21 2013

a(7) > 10^5. - Robert Price, May 28 2015

Cf. A000043, A057468, A059801, A059802, A059803, A062572-A062652, A215536, A062655-A062666.

Select[Range[0, 1000], PrimeQ[88^# - 87^#] &] (* Robert Price, May 28 2015 *)

is(n)=ispseudoprime(88^n-87^n) \\ Charles R Greathouse IV, Jun 13 2017

Edited by M. F. Hasler, Sep 21 2013

a(6) from Robert Price, May 28 2015 (computer run by Adam Marciniec)

A074884 For a prime p, let k(p) be the least k such that 2kp+1 is prime. Sequence gives primes for which k(p) exceeds k(q) for all primes q < p.

Original entry on oeis.org

2, 7, 17, 19, 59, 167, 197, 227, 317, 457, 521, 1637, 1861, 1997, 2053, 3833, 5227, 19891, 47303, 54973, 58603, 124567, 138163, 170167, 707467, 1637429, 1940777, 3717731, 4722079, 17886697, 27507569, 73342163, 154205101, 160561133, 186668543, 429364379, 458121431, 1459411661, 2140833967, 4028983027, 5189517859, 6184586189, 10352323829, 36673176307

Offset: 1

Mike Oakes, Oct 01 2002

nonn

			a(1)=2 because k(2)=1 (2*1*2+1=5 is prime);
a(2)=7 because k(7)=2 (2*1*7+1=15 is not prime, 2*2*7+1=29 is prime).

Cf. A117673.

a(30)-a(38) from Don Reble, Jan 07 2013

a(39)-a(44) from Marco Frigerio, Mar 25 2019

A081710 n! + n# + 1 is prime, where n# is the primorial function.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 17, 18, 24, 95, 96, 142, 1022, 1120, 1580, 6942, 19255, 19401

Offset: 1

Mike Oakes, Apr 04 2003

Some of the larger entries may only correspond to probable primes.

			a(2)=2 because 2!+2#+1=5 is prime

Mike Oakes, Primo-factorials.
Mike Oakes, Primo-factorials, message 12028 in primenumbers Yahoo group, Apr 3, 2003. [Cached copy]
Joe McLean, Interesting Sources of Probable Primes

Cf. A081711, A081712, A081713.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008

A081713 n! - n# - 1 is prime, where n# is the primorial function.

Original entry on oeis.org

4, 5, 20, 92, 106, 266, 308, 343, 583, 597, 903, 1021, 1239, 1314, 2458, 6160, 9627, 10649

Offset: 1

Mike Oakes, Apr 04 2003

Some of the larger entries may only correspond to probable primes.

			a(1)=4 because 4!-4#-1=17 is prime

Mike Oakes, Primo-factorials.
Mike Oakes, Primo-factorials, message 12028 in primenumbers Yahoo group, Apr 3, 2003. [Cached copy]
Joe McLean, Interesting Sources of Probable Primes.

Cf. A081710, A081711, A081712.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008

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User: Mike Oakes

A275747 Largest integer which cannot be written as a sum of n-th powers of odd primes.

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A275746 The number of integers not representable as a sum of n-th powers of odd primes.

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A275743 The number of integers not representable as a sum of n-th powers of primes.

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A166339 Numbers n such that Sum(n!/k!),k=0..n is prime.

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A166342 Numbers n such that (n+2)*n!*Sum((-1)^k/k!),k=0..n+2 is prime.

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A059803 Numbers n such that 9^n - 8^n is prime or a strong pseudoprime.

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A062654 Numbers k such that 88^k - 87^k is prime or a strong pseudoprime.

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PARI

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A074884 For a prime p, let k(p) be the least k such that 2kp+1 is prime. Sequence gives primes for which k(p) exceeds k(q) for all primes q < p.

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A081710 n! + n# + 1 is prime, where n# is the primorial function.

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A166342 Numbers n such that (n+2)n!Sum((-1)^k/k!),k=0..n+2 is prime.