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Numbers k such that the number of partitions of k contains k as a substring.
1

%I #13 Oct 22 2022 12:15:35

%S 1,2,3,37,47,67,87,98,252,274,450,468,556,618,662,731,789,847,866,950,

%T 971,990,1154,1163,1590,1800,1851,1954,1959,2073,2240,2775,2999,3095,

%U 3234,3520,3816,3848,3986,4193,4497,4541,4557,4661,4710,4815,4982,5123,5220

%N Numbers k such that the number of partitions of k contains k as a substring.

%C Number of partitions of a(n) ending in a(n) are 1, 2, 3, 37, 8745, ...

%H Paolo P. Lava, <a href="/A280539/b280539.txt">Table of n, a(n) for n = 1..150</a>

%e Number of partitions of 37 is 21637, and 37 is a substring.

%e Number of partitions of 468 is 380095468763120598477, and 468 is a substring.

%p with(combinat): P:=proc(q) local a,b,k,n; for n from 1 to q do

%p a:=numbpart(n); b:=ilog10(n)+1; for k from 1 to ilog10(numbpart(n))+1-ilog10(n) do

%p if n=a mod 10^b then print(n); break; fi; a:=trunc(a/10); od; od; print();end: P(50000);

%t Select[Range[6000],SequenceCount[IntegerDigits[PartitionsP[#]],IntegerDigits[#]]>0&] (* _Harvey P. Dale_, Oct 22 2022 *)

%Y Cf. A000041.

%K nonn,base

%O 1,2

%A _Paolo P. Lava_, Jan 05 2017