cp's OEIS Frontend

A077569 Irregular triangle read by rows: row n lists smallest numbers in increasing order of all possible prime signatures with n divisors.

%I A077569 #28 Jun 29 2025 02:22:17
%S A077569 1,2,4,6,8,16,12,32,64,24,30,128,36,256,48,512,1024,60,72,96,2048,
%T A077569 4096,192,8192,144,16384,120,210,216,384,32768,65536,180,288,768,
%U A077569 131072,262144,240,432,1536,524288,576,1048576,3072,2097152,4194304,360,420
%N A077569 Irregular triangle read by rows: row n lists smallest numbers in increasing order of all possible prime signatures with n divisors.
%C A077569 There are A001055(n) different prime signatures with n divisors.
%C A077569 If a*b*c... is a factorization of n then the corresponding prime signature is p^(a-1)*q^(b-1)*r^(c-1)... etc.
%C A077569 The corresponding term of the n-th array is obtained by arranging a>b>c>... and p<q<r<... i.e. p = 2, q = 3 and r = 5 etc.
%C A077569 The n-th row contains A001055(n) terms. Taking the first term of each row gives A005179.
%D A077569 Amarnath Murthy, A note on the Smarandache Divisor sequences, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.
%H A077569 Amiram Eldar, <a href="/A077569/b077569.txt">Table of n, a(n) for n = 1..7314</a> (rows n=1..1000, flattened; rows 1..300 from T. D. Noe)
%H A077569 Amarnath Murthy and Charles Ashbacher, <a href="http://www.gallup.unm.edu/~smarandache/MurthyBook.pdf">Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences</a>, Hexis, Phoenix; USA 2005. See Section 1.4, 1.12.
%e A077569 The row for n = 12 contains 60,72,96 and 2048, each having 12 divisors, with prime signature p^2qr, p^3q^2, p^5q, p^11.
%e A077569 The triangle begins:
%e A077569   1;
%e A077569   2;
%e A077569   4;
%e A077569   6, 8;
%e A077569   16;
%e A077569   12, 32;
%e A077569   64;
%e A077569   24, 30, 128;
%e A077569   36, 256;
%e A077569   48, 512;
%e A077569   1024;
%e A077569   60, 72, 96, 2048;
%e A077569   4096;
%e A077569   192, 8192;
%e A077569   144, 16384;
%e A077569   120, 210, 216, 384, 32768;
%e A077569   65536;
%e A077569   180, 288, 768, 131072;
%e A077569   262144;
%e A077569   240, 432, 1536, 524288;
%e A077569   576, 1048576;
%e A077569   3072, 2097152;
%e A077569   4194304;
%e A077569   ...
%t A077569 row[n_] := Module[{e = f[n] - 1}, Sort[Times @@ (Prime[Range[Length[#]]]^Reverse[#]) & /@ e]]; Table[row[n], {n, 1, 25}] // Flatten (* _Amiram Eldar_, Jun 28 2025 using the function f by _T. D. Noe_ at A162247 *)
%Y A077569 Cf. A001055, A005179, A077570, A122819, A162247.
%K A077569 nonn,tabf
%O A077569 1,2
%A A077569 _Amarnath Murthy_, Nov 11 2002
%E A077569 More terms from _Ray Chandler_, Aug 12 2003
%E A077569 Improved definition from _T. D. Noe_, Aug 31 2008
%E A077569 Edited by _N. J. A. Sloane_, Sep 05 2008
%E A077569 Name corrected by _Amiram Eldar_, Jun 28 2025