cp's OEIS Frontend

A356155 The pi-based arithmetic derivative applied to prime shift array: Square array A(n,k) = A258851(A246278(n,k)), read by falling antidiagonals.

%I A356155 #12 Jul 31 2022 08:15:31
%S A356155 1,4,2,7,12,3,12,19,30,4,11,54,41,56,5,20,26,225,79,110,6,15,87,58,
%T A356155 588,131,156,7,32,37,310,94,1815,193,238,8,33,216,69,861,162,3042,269,
%U A356155 304,9,32,140,1500,117,2156,218,6069,355,414,10,21,120,427,5488,183,3835,314,8664,491,580,11,52,44,455,1254,26620,255,6834,422,14283,629,682,12
%N A356155 The pi-based arithmetic derivative applied to prime shift array: Square array A(n,k) = A258851(A246278(n,k)), read by falling antidiagonals.
%C A356155 Each column is strictly monotonic.
%H A356155 Antti Karttunen, <a href="/A356155/b356155.txt">Table of n, a(n) for n = 1..22155; the first 210 antidiagonals</a>
%H A356155 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%e A356155 The top left corner of the array:
%e A356155    k =  1    2    3      4    5      6    7       8      9     10   11       12
%e A356155   2k =  2    4    6      8   10     12   14      16     18     20   22       24
%e A356155 -----+--------------------------------------------------------------------------
%e A356155 n= 1 |  1,   4,   7,    12,  11,    20,  15,     32,    33,    32,  21,      52,
%e A356155    2 |  2,  12,  19,    54,  26,    87,  37,    216,   140,   120,  44,     351,
%e A356155    3 |  3,  30,  41,   225,  58,   310,  69,   1500,   427,   455,  86,    2075,
%e A356155    4 |  4,  56,  79,   588,  94,   861, 117,   5488,  1254,  1022, 132,    8183,
%e A356155    5 |  5, 110, 131,  1815, 162,  2156, 183,  26620,  2561,  2717, 214,   31581,
%e A356155    6 |  6, 156, 193,  3042, 218,  3835, 255,  52728,  4828,  4316, 304,   67093,
%e A356155    7 |  7, 238, 269,  6069, 314,  6834, 373, 137564,  7695,  8075, 404,  154615,
%e A356155    8 |  8, 304, 355,  8664, 422, 10241, 457, 219488, 12098, 12426, 524,  261003,
%e A356155    9 |  9, 414, 491, 14283, 532, 17296, 609, 438012, 20909, 18653, 668,  535877,
%e A356155   10 | 10, 580, 629, 25230, 718, 27231, 787, 975560, 29388, 31552, 836, 1050409,
%o A356155 (PARI)
%o A356155 up_to = 78;
%o A356155 A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
%o A356155 A258851(n) = (n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i])); \\ From A258851
%o A356155 A356155sq(row,col) = A258851(A246278sq(row,col));
%o A356155 A356155list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A356155sq(col,(a-(col-1))))); (v); };
%o A356155 v356155 = A356155list(up_to);
%o A356155 A356155(n) = v356155[n];
%Y A356155 Cf. A000027 (column 1), A097240 (column 3), A246278, A258851.
%Y A356155 Cf. also A344027.
%K A356155 nonn,tabl
%O A356155 1,2
%A A356155 _Antti Karttunen_, Jul 29 2022