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%I #24 Nov 09 2021 15:04:44
%S 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,9,9,9,9,9,7,8,10,8,8,8,8,
%T 10,8,9,7,15,7,7,7,15,7,9,10,8,10,12,10,10,12,10,8,10,11,15,7,15,25,
%U 15,25,15,7,15,11,12,14,12,10,12,18,18,12,10,12,14,12,13,25,21,25,15,25,11,25,15,25,21,25,13
%N Symmetric square array A(n,k) = A005940(1+A156552(n)+A156552(k)), read by antidiagonals starting with A(1,1).
%C Considered as a binary operation on the positive integers, A(x, y) returns the term of the Doudna-sequence from the position that is the sum of the positions of x and y in the same sequence. (This is based on giving the Doudna-sequence an offset of 0, rather than 1 as used in A005940.) - _Peter Munn_, Feb 14 2021
%H Antti Karttunen, <a href="/A341510/b341510.txt">Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the array</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A(n, k) = A(k, n) = A005940(1 + A156552(n) + A156552(k)).
%F A(n, n) = A003961(n).
%F A(n, 2*n) = A(2*n, n) = A329603(n).
%F A(n, 2) = A(2, n) = A297165(n).
%e The top left 16x16 corner of the array:
%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
%e 2, 3, 4, 5, 6, 9, 10, 7, 8, 15, 14, 25, 22, 21, 12, 11,
%e 3, 4, 5, 6, 9, 8, 15, 10, 7, 12, 21, 18, 33, 20, 25, 14,
%e 4, 5, 6, 9, 8, 7, 12, 15, 10, 25, 20, 27, 28, 35, 18, 21,
%e 5, 6, 9, 8, 7, 10, 25, 12, 15, 18, 35, 16, 55, 30, 27, 20,
%e 6, 9, 8, 7, 10, 15, 18, 25, 12, 27, 30, 11, 42, 45, 16, 35,
%e 7, 10, 15, 12, 25, 18, 11, 16, 27, 14, 49, 20, 77, 50, 21, 24,
%e 8, 7, 10, 15, 12, 25, 16, 27, 18, 11, 24, 21, 40, 49, 14, 45,
%e 9, 8, 7, 10, 15, 12, 27, 18, 25, 16, 45, 14, 63, 24, 11, 30,
%e 10, 15, 12, 25, 18, 27, 14, 11, 16, 21, 50, 35, 70, 75, 20, 49,
%e 11, 14, 21, 20, 35, 30, 49, 24, 45, 50, 13, 36, 121, 22, 75, 32,
%e 12, 25, 18, 27, 16, 11, 20, 21, 14, 35, 36, 45, 60, 125, 30, 75,
%e 13, 22, 33, 28, 55, 42, 77, 40, 63, 70, 121, 60, 17, 98, 105, 48,
%e 14, 21, 20, 35, 30, 45, 50, 49, 24, 75, 22, 125, 98, 33, 36, 13,
%e 15, 12, 25, 18, 27, 16, 21, 14, 11, 20, 75, 30, 105, 36, 35, 50,
%e 16, 11, 14, 21, 20, 35, 24, 45, 30, 49, 32, 75, 48, 13, 50, 81,
%o (PARI)
%o up_to = 105;
%o A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
%o A156552(n) = { my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
%o A341510sq(n,k) = A005940(1+A156552(n)+A156552(k));
%o A341510list(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] = A341510sq(col,(a-(col-1))))); (v); };
%o v341510 = A341510list(up_to);
%o A341510(n) = v341510[n];
%Y Cf. A341511 (the lower triangular section).
%Y Cf. A003961 (main diagonal), A329603 (skewed diagonal).
%Y Cf. A297165 (row 2 and column 2, when started from its term a(1)).
%Y Cf. also A003056, A268715, A341515, A341520, A348041.
%K nonn,tabl
%O 1,2
%A _Antti Karttunen_, Feb 13 2021
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