A337472 -id:A337472 - cp's OEIS frontend

A337473 Square array read by falling antidiagonals, where A(n,k) = floor(A337472(n, k)/A337470(n, k)); Abundancy index of A337470(n, k) floored down.

1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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 11 12 13 14 15 16 17 18 19 20 21 22 23
----|----------------------------------------------------------------------
  0 | 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 3, 3, 1, 3, 2, 3, 3, 3, 3, 3,
  1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2,
  2 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2,
  3 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
  4 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
  5 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
etc.
A337474 gives the distance in each column to the first 1 in that column, being 0 for columns 1, 2, 4, 8, 16, ..., where 1 is already in the top row.

Cf. A003961, A108951, A337470, A337472, A337474, A337476.

up_to = 105858-1; \\ Or 105-1.
A337473sq(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(i=h-1,0,-1, if(!i,pid=0,pid=primepi(f[i,1])); forstep(j=prevpid,(1+pid),-1, p=prime(j+n); s *= ((p^(1+e)-1)/((p-1)*(p^e)))); if(!pid,return(floor(s))); prevpid = pid; e += f[i,2]); floor(s));
A337473list(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] = A337473sq(b-1, (a-(b-1))))); (v); };
v337473 = A337473list(up_to);
A337473(n) = v337473[1+n];

A(n,k) = floor(A337472(n, k)/A337470(n, k)).

A337203 Sum of the divisors of the primorial inflation of n.

Original entry on oeis.org

1, 3, 12, 7, 72, 28, 576, 15, 91, 168, 6912, 60, 96768, 1344, 546, 31, 1741824, 195, 34836480, 360, 4368, 16128, 836075520, 124, 2821, 225792, 600, 2880, 25082265600, 1170, 802632499200, 63, 52416, 4064256, 22568, 403, 30500034969600, 81285120, 733824, 744, 1281001468723200, 9360, 56364064623820800, 34560, 3600, 1950842880

Offset: 1

Antti Karttunen, Aug 22 2020

nonn

Row 0 of A337205, and of A337472.
Cf. A000203, A034386, A108951, A337204.
Cf. also A323173.

Array[DivisorSigma[1, Apply[Times, FactorInteger[#] /. {p_, e_} /; e > 0 :> Apply[Times, Prime@ Range@ PrimePi@ p]^e]] &, 46] (* Michael De Vlieger, Aug 27 2020 *)

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
A337203(n) = sigma(A108951(n));

A337203(n) = if(1==n,n, my(f=factor(n), h = #f~, prevpid=primepi(f[h,1]), e=f[h,2], p, s=1); forstep(i=h-1,0,-1, if(!i,pid=0,pid=primepi(f[i,1])); forstep(j=prevpid,(1+pid),-1, p=prime(j); s *= ((p^(1+e)-1)/(p-1))); if(!pid,return(s)); prevpid = pid; e += f[i,2]); (s));

a(n) = A000203(A108951(n)).

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}.

Original entry on oeis.org

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).

A337470 Square array read by falling antidiagonals, where A(n,k) = primorial inflation of k prime shifted n times with A003961.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 4, 15, 5, 1, 30, 9, 35, 7, 1, 12, 105, 25, 77, 11, 1, 210, 45, 385, 49, 143, 13, 1, 8, 1155, 175, 1001, 121, 221, 17, 1, 36, 27, 5005, 539, 2431, 169, 323, 19, 1, 60, 225, 125, 17017, 1573, 4199, 289, 437, 23, 1, 2310, 315, 1225, 343, 46189, 2873, 7429, 361, 667, 29, 1, 24, 15015, 1925, 5929, 1331, 96577, 5491, 12673, 529, 899, 31

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,  2,   6,   4,   30,   12,   210,    8,    36,    60, ...
  1 | 1,  3,  15,   9,  105,   45,  1155,   27,   225,   315, ...
  2 | 1,  5,  35,  25,  385,  175,  5005,  125,  1225,  1925, ...
  3 | 1,  7,  77,  49, 1001,  539, 17017,  343,  5929,  7007, ...
  4 | 1, 11, 143, 121, 2431, 1573, 46189, 1331, 20449, 26741, ...
  5 | 1, 13, 221, 169, 4199, 2873, 96577, 2197, 48841, 54587, ...
etc.

Cf. A108951 (row 0), A337471 (row 1).
Cf. A003961, A242378.
Cf. also A337205, A337472.

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]); };
A337470list(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] = A337470sq(b-1, (a-(b-1))))); (v); };
v337470 = A337470list(up_to);
A337470(n) = v337470[1+n];

A(n,k) = A242378(n,A108951(k)).

cp's OEIS Frontend

A337473 Square array read by falling antidiagonals, where A(n,k) = floor(A337472(n, k)/A337470(n, k)); Abundancy index of A337470(n, k) floored down.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A337203 Sum of the divisors of the primorial inflation of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A337470 Square array read by falling antidiagonals, where A(n,k) = primorial inflation of k prime shifted n times with A003961.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula