A337205 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}.

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, 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, 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

Offset: 0

Author

Antti Karttunen, Aug 22 2020

Keywords

• nonn
• tabl

Comments

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), ...

Examples

			The top left 15 x 5 corner of the array:
----+------------------------------------------------------------------------
  0 | 1, 3, 12,  7, 72, 28, 576, 15, 91, 168, 6912, 60, 96768, 1344, 546, ...
  1 | 1, 1,  4,  1, 24,  4, 192,  1, 13,  24, 2304,  4, 32256,  192,  78, ...
  2 | 1, 1,  1,  1,  6,  1,  48,  1,  1,   6,  576,  1,  8064,   48,   6, ...
  3 | 1, 1,  1,  1,  1,  1,   8,  1,  1,   1,   96,  1,  1344,    8,   1, ...
  4 | 1, 1,  1,  1,  1,  1,   1,  1,  1,   1,   12,  1,   168,    1,   1, ...
etc.
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.
		

Links

• Index entries for sequences related to primorial numbers
• Index entries for sequences related to sigma(n)

Crossrefs

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

Cf. A000203, A034386, A108951.

Cf. also arrays A337470, A337472.

Programs

PARI

up_to = 105-1;
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));
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); };
v337205 = A337205list(up_to);
A337205(n) = v337205[1+n];

Formula

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

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