A245372 Prime-partitionable numbers a(n) for which there exists a 2-partition of the set of primes < a(n) that has a smallest subset containing four primes only.
46, 76, 96, 106, 134, 142, 146, 204, 218, 276, 310, 408, 438, 466, 518, 534, 536, 546, 580, 624, 650, 672, 680, 694, 792, 800, 896, 970, 1000, 1016, 1100, 1160, 1170, 1318, 1344, 1358, 1364, 1384, 1470, 1480
Offset: 1
Keywords
Examples
a(1) = 46 because A245602(5) = 46 and the 2-partition {3, 19, 37, 43} {2, 5, 7, 11, 13, 17, 23, 29, 31, 41} of the set of primes < 46 demonstrates it.
Links
- Christopher Hunt Gribble, Prime-partitionable numbers with #P1 = 4
- W. Holsztynski, R. F. E. Strube, Paths and circuits in finite groups, Discr. Math. 22 (1978) 263-272.
- R. J. Mathar and M. F. Hasler, Is 52 prime-partitionable?, Seqfan thread (Jun 29 2014), arXiv:1510.07997
- W. T. Trotter, Jr. and Paul Erdős, When the Cartesian product of directed cycles is Hamiltonian, J. Graph Theory 2 (1978) 137-142 DOI:10.1002/jgt.3190020206.
Programs
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PARI
prime_part(n)= { my (P = primes(primepi(n-1))); for (k1 = 2, #P - 1, for (k2 = 1, k1 - 1, for (k3 = 1, k2 - 1, for (k4 = 1, k3 - 1, mask = 2^k1 + 2^k2 + 2^k3 + 2^k4; P1 = vecextract(P, mask); P2 = setminus(P, P1); for (n1 = 1, n - 1, bittest(n - n1, 0) || next; setintersect(P1, factor(n1)[,1]~) && next; setintersect(P2, factor(n-n1)[,1]~) && next; next(2) ); print1(n, ", "); ); ); ); ); } # PP = {{2x, x = 1:1000} - {A245664(n), n = 1:145} # - {A249302(n), n = 1:77}} PP = [2, 4, 6, 8, 10, 12, 14, 18, 20, 24, 26, 28, 30, 32, \ 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, \ ... 1994, 1996, 1998, 2000]; for(m=1,#PP,prime_part(PP[m]));
Comments