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Square array A(n,k) read by falling antidiagonals, where row n gives the sum of the divisors of the {primorial inflation of k, from which all primes <= A000040(n) have been discarded}.
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%I #15 Aug 28 2020 11:07:01

%S 1,3,1,12,1,1,7,4,1,1,72,1,1,1,1,28,24,1,1,1,1,576,4,6,1,1,1,1,15,192,

%T 1,1,1,1,1,1,91,1,48,1,1,1,1,1,1,168,13,1,8,1,1,1,1,1,1,6912,24,1,1,1,

%U 1,1,1,1,1,1,60,2304,6,1,1,1,1,1,1,1,1,1,96768,4,576,1,1,1,1,1,1,1,1,1,1,1344,32256,1,96,1,1,1,1,1,1,1,1,1,1

%N Square array A(n,k) read by falling antidiagonals, where row n gives the sum of the divisors of the {primorial inflation of k, from which all primes <= A000040(n) have been discarded}.

%C Array is read by descending antidiagonals with n >= 0 and k >= 1 ranging as: (0, 1), (0, 2), (1, 1), (0, 3), (1, 2), (2, 1), (0, 4), (1, 3), (2, 2), (3, 1), ...

%H <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F A(n,k) is the sum of divisors of A108951(k) from which all primes less than A000040(n) have been removed first.

%F A(n,k) is a multiple of A(n+1,k).

%e The top left 15 x 5 corner of the array:

%e ----+------------------------------------------------------------------------

%e 0 | 1, 3, 12, 7, 72, 28, 576, 15, 91, 168, 6912, 60, 96768, 1344, 546, ...

%e 1 | 1, 1, 4, 1, 24, 4, 192, 1, 13, 24, 2304, 4, 32256, 192, 78, ...

%e 2 | 1, 1, 1, 1, 6, 1, 48, 1, 1, 6, 576, 1, 8064, 48, 6, ...

%e 3 | 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 96, 1, 1344, 8, 1, ...

%e 4 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 168, 1, 1, ...

%e etc.

%e For example, the row 1 is the sum of the {primorial inflation of k, from which all primes <= prime(1) = 2 have been discarded}, that is, it is the sum of the odd divisors of the primorial inflation of k.

%o (PARI)

%o up_to = 105-1;

%o A337205sq(n,k) = if(1==k,k, my(f=factor(k), h = #f~, prevpid=primepi(f[h,1]), e=f[h,2], p, s=1); forstep(k=h-1,0,-1, if(!k,pid=0,pid=primepi(f[k,1])); forstep(j=prevpid,(1+pid),-1, if(j<=n,return(s)); p=prime(j); s *= ((p^(1+e)-1)/(p-1))); if(pid<=n,return(s)); prevpid = pid; e += f[k,2]); (s));

%o A337205list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0, oo, for(b=1, a, i++; if(i > #v, return(v)); v[i] = A337205sq(b-1, (a-(b-1))))); (v); };

%o v337205 = A337205list(up_to);

%o A337205(n) = v337205[1+n];

%Y Cf. A337203, A337204 (rows 0 and 1).

%Y Cf. A000203, A034386, A108951.

%Y Cf. also arrays A337470, A337472.

%K nonn,tabl

%O 0,2

%A _Antti Karttunen_, Aug 22 2020