A337203 -id:A337203 - cp's OEIS frontend

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

Antti Karttunen, Aug 22 2020

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), ...

			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.

Cf. A337203, A337204 (rows 0 and 1).
Cf. A000203, A034386, A108951.
Cf. also arrays A337470, A337472.

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];

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).

A337472 Square array read by falling antidiagonals, where A(n,k) = sigma(A337470(n, k)).

Original entry on oeis.org

1, 3, 1, 12, 4, 1, 7, 24, 6, 1, 72, 13, 48, 8, 1, 28, 192, 31, 96, 12, 1, 576, 78, 576, 57, 168, 14, 1, 15, 2304, 248, 1344, 133, 252, 18, 1, 91, 40, 8064, 684, 3024, 183, 360, 20, 1, 168, 403, 156, 24192, 1862, 5040, 307, 480, 24, 1, 6912, 624, 1767, 400, 60480, 3294, 8640, 381, 720, 30, 1

Offset: 0

Antti Karttunen, Aug 28 2020

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), ...

			The top left corner of the array begins as:
n/k | 1   2    3    4     5     6       7     8      9     10
----|--------------------------------------------------------------
  0 | 1,  3,  12,   7,   72,   28,    576,   15,    91,   168, ...
  1 | 1,  4,  24,  13,  192,   78,   2304,   40,   403,   624, ...
  2 | 1,  6,  48,  31,  576,  248,   8064,  156,  1767,  2976, ...
  3 | 1,  8,  96,  57, 1344,  684,  24192,  400,  7581,  9576, ...
  4 | 1, 12, 168, 133, 3024, 1862,  60480, 1464, 24339, 33516, ...
  5 | 1, 14, 252, 183, 5040, 3294, 120960, 2380, 56181, 65880, ...
etc.

Cf. A000203, A003961.
Cf. A337203 (row 0).
Cf. also arrays A337205, A337470.

up_to = 105-1; \\ Or 78-1.
Ashifted_primorial(n,d) = prod(i=1, primepi(n), prime(i+d));
A337470sq(n, k) = { my(f=factor(k)); prod(i=1, #f~, Ashifted_primorial(f[i, 1], n)^f[i, 2]); };
A337472sq(n, k) = sigma(A337470sq(n, k));
A337472list(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] = A337472sq(b-1, (a-(b-1))))); (v); };
v337472 = A337472list(up_to);
A337472(n) = v337472[1+n];

A(n,k) = A000203(A337470(n, k)).

A344698 a(n) = A344696(A108951(n)).

Original entry on oeis.org

1, 1, 1, 7, 1, 7, 1, 5, 91, 7, 1, 5, 1, 7, 91, 31, 1, 65, 1, 5, 91, 7, 1, 31, 2821, 7, 25, 5, 1, 65, 1, 21, 91, 7, 2821, 403, 1, 7, 91, 31, 1, 65, 1, 5, 25, 7, 1, 21, 7657, 403, 91, 5, 1, 155, 2821, 31, 91, 7, 1, 403, 1, 7, 25, 127, 2821, 65, 1, 5, 91, 403, 1, 91, 1, 7, 155, 5, 7657, 65, 1, 21, 3751, 7, 1, 403, 2821

Offset: 1

Antti Karttunen, May 26 2021

nonn

Records seem to occur on certain squares: 1, 4, 9, 25, 49, 100, 121, 289, 361, 529, 841, 961, etc. See A344701.

Cf. A000203, A001615, A108951, A337203, A344696, A344699, A344701 (positions of records).

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A344696(n) = { my(u=sigma(n)); (u/gcd(u,A001615(n))); };
A344698(n) = A344696(A108951(n));

A344699 a(n) = A344697(A108951(n)).

Original entry on oeis.org

1, 1, 1, 6, 1, 6, 1, 4, 72, 6, 1, 4, 1, 6, 72, 24, 1, 48, 1, 4, 72, 6, 1, 24, 2160, 6, 18, 4, 1, 48, 1, 16, 72, 6, 2160, 288, 1, 6, 72, 24, 1, 48, 1, 4, 18, 6, 1, 16, 5760, 288, 72, 4, 1, 108, 2160, 24, 72, 6, 1, 288, 1, 6, 18, 96, 2160, 48, 1, 4, 72, 288, 1, 64, 1, 6, 108, 4, 5760, 48, 1, 16, 2592, 6, 1, 288, 2160

Offset: 1

Antti Karttunen, May 26 2021

nonn

Cf. A000203, A001615, A108951, A337203, A344697, A344698, A344701 (apparently positions of records).

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };
A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A344697(n) = { my(u=A001615(n)); (u/gcd(u,sigma(n))); };
A344699(n) = A344697(A108951(n));

A337204 Sum of the odd divisors of the primorial inflation of n.

Original entry on oeis.org

1, 1, 4, 1, 24, 4, 192, 1, 13, 24, 2304, 4, 32256, 192, 78, 1, 580608, 13, 11612160, 24, 624, 2304, 278691840, 4, 403, 32256, 40, 192, 8360755200, 78, 267544166400, 1, 7488, 580608, 3224, 13, 10166678323200, 11612160, 104832, 24, 427000489574400, 624, 18788021541273600, 2304, 240, 278691840, 901825033981132800, 4, 22971, 403

Offset: 1

Antti Karttunen, Aug 22 2020

nonn

Row 1 of A337205.
Cf. A000593, A034386, A108951, A337203.
Cf. also A322819.

Array[DivisorSum[Apply[Times, FactorInteger[#] /. {p_, e_} /; e > 0 :> Apply[Times, Prime@ Range@ PrimePi@ p]^e], # &, OddQ] &, 50] (* Michael De Vlieger, Aug 27 2020 *)

A000593(n) = sigma(n>>valuation(n, 2));
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) }; \\ From A108951
A337204(n) = A000593(A108951(n));

a(n) = A000593(A108951(n)).

cp's OEIS Frontend

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}.

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A337472 Square array read by falling antidiagonals, where A(n,k) = sigma(A337470(n, k)).

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A344698 a(n) = A344696(A108951(n)).

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A344699 a(n) = A344697(A108951(n)).

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A337204 Sum of the odd divisors of the primorial inflation of n.

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