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Positive integers that cannot be written as a sum of a practical number and a heptagonal number.
0

%I #15 Jun 10 2024 10:01:23

%S 10,14,44,45,51,53,68,69,77,86,92,110,125,131,170,173,179,182,206,233,

%T 266,284,296,303,373,377,383,425,437,444,479,524,536,584,719,845,1124,

%U 1133,1208,1325,1358,1445,1478,1499,1562,1626,1787,1947,1979,2117,2249,2252,2273,2747,2957,3209,3341,4058,4268,5609,6830,8006,8333,9284,11699,12164,14024,17432,21257,24797,25919,33773,79445

%N Positive integers that cannot be written as a sum of a practical number and a heptagonal number.

%C Somu and Tran (2024) conjectured that there are finitely many such integers. It was also conjectured that 79445 is the largest such integer. This conjecture was checked up to 10^8.

%H Sai Teja Somu and Duc Van Khanh Tran, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Somu/somu5.html">On sums of practical numbers and polygonal numbers</a>, Journal of Integer Sequences, 27(5), 2024.

%Y Cf. A000566, A005153.

%K nonn,hard

%O 1,1

%A _Duc Van Khanh Tran_, Jun 05 2024