cp's OEIS Frontend

A242421 Fixed points of A153212: After a(1) = 1, numbers of the form p_i1^i1 * p_i2^(i2-i1) * p_i3^(i3-i2) * ... * p_ik^(ik-i_{k-1}), where p_i's are distinct primes present in the prime factorization of n, with i1 < i2 < i3 < ... < ik, and k = A001221(n) and ik = A061395(n).

%I A242421 #16 Jun 08 2014 08:58:14
%S A242421 1,2,6,9,30,45,50,125,210,294,315,350,441,686,875,2310,2401,3234,3465,
%T A242421 3630,3850,4851,5445,6050,7546,7986,9625,11979,15125,26411,29282,
%U A242421 30030,35490,42042,45045,47190,49686,50050,53235,59150,63063,65910,70785,74529,78650,98098,98865,103818,109850,115934,125125,147875,155727,161051,171366,196625,257049,274625,343343,380666,405769,510510
%N A242421 Fixed points of A153212: After a(1) = 1, numbers of the form p_i1^i1 * p_i2^(i2-i1) * p_i3^(i3-i2) * ... * p_ik^(ik-i_{k-1}), where p_i's are distinct primes present in the prime factorization of n, with i1 < i2 < i3 < ... < ik, and k = A001221(n) and ik = A061395(n).
%C A242421 This sequence is closed with respect to A122111, i.e., for any n, A122111(a(n)) is either the same as a(n) or some other term a(k) of the sequence.
%C A242421 These numbers encode partitions in whose Young diagrams all pairs of successive horizontal and vertical segments (those pairs sharing "a common convex corner") are of equal length. Cf. the example-illustration at A153212.
%C A242421 Note: The seventh primorial, 510510 (= A002110(7)) occurs here as a term a(62).
%e A242421 2 = p_1^1 is present, as the first prime index delta and exponent are equal.
%e A242421 3 = p_2^1 is not present, as 1 <> 2.
%e A242421 6 = p_1^1 * p_2^(2-1) is present.
%e A242421 9 = p_2^2 is present, as 2 = 2.
%e A242421 30 = p_1^1 * p_2^(2-1) * p_3^(3-2) is present, as all primorials are.
%e A242421 50 = p_1^1 * p_3^(3-1) is present also.
%o A242421 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A242421 (define A242421 (FIXED-POINTS 1 1 A153212)) ;; Very slow, but gives the terms in order.
%o A242421 ;; The following is a faster version, but gives the terms in non-monotone order. Note that its indexing starts from zero.
%o A242421 ;; A000975 gives the positions where primorials occur, after which only larger terms occur, use this fact for selecting a prefix sequence of appropriate length before sorting.
%o A242421 ;; E.g. (A242421_in_wrong_order 85) = 510510 = (A242421_in_wrong_order (A006068 127))
%o A242421 (define (A242421_in_wrong_order n) (let ((complist (binexp->runcount1list n))) (apply * (map (lambda (i e) (expt (A000040 i) e)) (partsums complist) complist))))
%o A242421 ;; For the function binexp->runcount1list and partsums, see for example A129594.
%Y A242421 Subsequences: A002110 (primorial numbers), A062457.
%Y A242421 Cf. A242422, A088902, A241912, A209861, A209636, A129594.
%K A242421 nonn
%O A242421 1,2
%A A242421 _Antti Karttunen_, May 16 2014