A263995 Cardinality of the union of the set of sums and the set of products made from pairs of integers from {1..n}.
2, 4, 7, 11, 15, 20, 27, 32, 39, 46, 56, 63, 75, 83, 93, 102, 118, 127, 146, 156, 169, 182, 204, 215, 231, 245, 261, 274, 302, 315, 346, 361, 379, 398, 418, 432, 469, 489, 510, 527, 567, 585, 627, 647, 669, 693, 739, 756, 788, 810, 838, 862, 914, 937
Offset: 1
Keywords
Examples
a(3)=7 because the union of the set of sums {1+1, 1+2, 1+3, 2+2, 2+3, 3+3} and the set of products {1*1, 1*2, 1*3, 2*2, 2*3, 3*3} = {2,3,4,5,6} U {1,2,3,4,6,9} = {1,2,3,4,5,6,9} has cardinality 7.
References
- Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer-Verlag New York, 2004. Problem F18.
Links
- Hugo Pfoertner, Table of n, a(n) for n = 1..10000
- P. Erdős and E. Szemeredi, On sums and products of integers, Studies in Pure Mathematics, Birkhäuser, Basel, 1983, pp. 213-218. DOI:10.1007/978-3-0348-5438-2_19
- Al Zimmermann's Programming Contests, Sums and Products I, Nov 2015 - Feb 2016.
Programs
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PARI
a(n) = {my(v = [1..n]); v = setunion(setbinop((x,y)->(x+y), v), setbinop((x,y)->(x*y), v)); #v;} \\ Michel Marcus, Apr 13 2022 -
Python
from math import prod from itertools import combinations_with_replacement def A263995(n): return len(set(sum(x) for x in combinations_with_replacement(range(1,n+1),2)) | set(prod(x) for x in combinations_with_replacement(range(1,n+1),2))) # Chai Wah Wu, Apr 15 2022
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