cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Antti Karttunen, Aug 28 2020

Keywords

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 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.

Links

Crossrefs

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

Programs

  • PARI
    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];

Formula

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

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.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 28 2020

Keywords

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 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.

Links

Crossrefs

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

Programs

  • PARI
    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];

Formula

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

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

Views

Author

Antti Karttunen, Aug 22 2020

Keywords

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

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).

A337471 Primorial inflation of n prime shifted once: a(n) = A003961(A108951(n)).

Original entry on oeis.org

1, 3, 15, 9, 105, 45, 1155, 27, 225, 315, 15015, 135, 255255, 3465, 1575, 81, 4849845, 675, 111546435, 945, 17325, 45045, 3234846615, 405, 11025, 765765, 3375, 10395, 100280245065, 4725, 3710369067405, 243, 225225, 14549535, 121275, 2025, 152125131763605, 334639305, 3828825, 2835, 6541380665835015, 51975, 307444891294245705, 135135
Offset: 1

Views

Author

Antti Karttunen, Aug 28 2020

Keywords

Links

Crossrefs

Row 1 of A337470.
Cf. A000265, A002110, A003961, A034386, A070826, A108951.

Programs

Formula

a(n) = A003961(A108951(n)).
a(n) = A000265(A108951(A003961(n))).
Completely multiplicative with a(prime(i)) = A003961(A002110(i)) = A070826(1+i). - Antti Karttunen, Nov 17 2020
Sum_{n>=1} 1/a(n) = 1 / Product_{k>=2} (1 - 1/A070826(k)) = 1.6241170949... . - Amiram Eldar, Dec 08 2022
Showing 1-4 of 4 results.

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