A122819 -id:A122819 - cp's OEIS frontend

A077569 Irregular triangle read by rows: row n lists smallest numbers in increasing order of all possible prime signatures with n divisors.

1, 2, 4, 6, 8, 16, 12, 32, 64, 24, 30, 128, 36, 256, 48, 512, 1024, 60, 72, 96, 2048, 4096, 192, 8192, 144, 16384, 120, 210, 216, 384, 32768, 65536, 180, 288, 768, 131072, 262144, 240, 432, 1536, 524288, 576, 1048576, 3072, 2097152, 4194304, 360, 420

Offset: 1

Amarnath Murthy, Nov 11 2002

There are A001055(n) different prime signatures with n divisors.
If a*b*c... is a factorization of n then the corresponding prime signature is p^(a-1)*q^(b-1)*r^(c-1)... etc.
The corresponding term of the n-th array is obtained by arranging a>b>c>... and pThe n-th row contains A001055(n) terms. Taking the first term of each row gives A005179.

			The row for n = 12 contains 60,72,96 and 2048, each having 12 divisors, with prime signature p^2qr, p^3q^2, p^5q, p^11.
The triangle begins:
  1;
  2;
  4;
  6, 8;
  16;
  12, 32;
  64;
  24, 30, 128;
  36, 256;
  48, 512;
  1024;
  60, 72, 96, 2048;
  4096;
  192, 8192;
  144, 16384;
  120, 210, 216, 384, 32768;
  65536;
  180, 288, 768, 131072;
  262144;
  240, 432, 1536, 524288;
  576, 1048576;
  3072, 2097152;
  4194304;
  ...

Amarnath Murthy, A note on the Smarandache Divisor sequences, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.

Amiram Eldar, Table of n, a(n) for n = 1..7314 (rows n=1..1000, flattened; rows 1..300 from T. D. Noe)
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4, 1.12.

Cf. A001055, A005179, A077570, A122819, A162247.

row[n_] := Module[{e = f[n] - 1}, Sort[Times @@ (Prime[Range[Length[#]]]^Reverse[#]) & /@ e]]; Table[row[n], {n, 1, 25}] // Flatten (* Amiram Eldar, Jun 28 2025 using the function f by T. D. Noe at A162247 *)

More terms from Ray Chandler, Aug 12 2003
Improved definition from T. D. Noe, Aug 31 2008
Edited by N. J. A. Sloane, Sep 05 2008
Name corrected by Amiram Eldar, Jun 28 2025

A122812 Numbers k where A046523(A005179(k)) differs from A046523(A038547(k)).

Original entry on oeis.org

8, 24, 48, 64, 72, 80, 108, 112, 128, 144, 160, 162, 176, 192, 208, 216, 224, 243, 256, 272, 288, 304, 320, 324, 352, 368, 384, 416, 432, 448, 464, 480, 486, 496, 512, 544, 576, 592, 608, 640, 648, 656, 672, 688, 704, 729, 736, 752, 768, 832, 848, 864, 896

Offset: 1

Ray Chandler, Sep 22 2006

nonn

Where the prime signature of the least number with exactly k divisors differs from the prime signature of the least odd number with exactly k divisors.
Also where A122375(k) differs from A122810(k).
Also where A122376(k) differs from A122811(k).

Amiram Eldar, Table of n, a(n) for n = 1..2000

Cf. A005179, A038547, A046523, A122375, A122376, A122810-A122819.

A131802 Sequence related to factorizations and prime signatures: a(1) = 1; for n>1, a(n) = A057567(n) - 2*A001055(n).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 1, 2, 0, 3, 0, 3, 1, 1, 0, 7, 0, 1, 1, 3, 0, 5, 0, 5, 1, 1, 1, 8, 0, 1, 1, 7, 0, 5, 0, 3, 3, 1, 0, 14, 0, 3, 1, 3, 0, 7, 1, 7, 1, 1, 0, 14, 0, 1, 3, 8, 1, 5, 0, 3, 1, 5, 0, 20, 0, 1, 3, 3, 1, 5, 0, 14, 2, 1, 0, 14, 1, 1, 1, 7

Offset: 1

Alford Arnold, Jul 18 2007

			A001055(12) = 4 and A057567(12) = 11 so a(12) = 11 - 2*4 = 3

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001055, A025487, A057567, A077569, A108464, A122819.

fcnt(n, m) = {local(s); s=0; if(n == 1, s=1, fordiv(n, d, if(d > 1 & d <= m, s=s+fcnt(n/d, d)))); s}
A001055(n) = fcnt(n, n) \\ This function from Michael B. Porter, Oct 29 2009
A057567(n) = sumdiv(n, d, A001055(d));
A131802(n) = if(1==n,n,A057567(n) - 2*A001055(n)); \\ Antti Karttunen, May 25 2017

A131886 Rearrange the rows of array A077569 by sorting the values on each row by noting the odd part of each signature and sorting as in A131822.

Original entry on oeis.org

1, 2, 4, 8, 6, 16, 32, 12, 64, 128, 24, 30, 256, 36, 512, 48, 1024, 2048, 96, 72, 60, 4096, 8192, 192, 16384, 144, 32768, 384, 120, 216, 210, 65536, 131072, 768, 288, 180, 262144, 524288, 1536, 240, 432, 1048576, 576, 2097152, 3072, 4194304, 8388608

Offset: 1

Alford Arnold, Aug 15 2007

The shape sequence for both arrays is A001055.

			A077569 begins
  1
  2
  4
  6, 8
  ...
so this triangle begins
    1
    2
    4
    8,  6
   16
   32, 12
   64
  128, 24, 30
  ...

Cf. A001055, A025487, A077569, A131822, A122819, A131801.

A131997 Array read by rows in which the n-th row contains odd numbers of all possible prime signatures with n divisors.

Original entry on oeis.org

1, 3, 9, 27, 15, 81, 243, 45, 729, 2187, 135, 105, 6561, 225, 19683, 405, 59049, 177147, 1215, 315, 675

Offset: 1

Alford Arnold, Aug 18 2007

			Row 12 of A131886 is 2048 96 72 60 so here row 12 is 177147 1215 315 675.

Cf. A001055, A025487, A050322, A057567, A077569, A108464, A122819, A130839, A131802, A131886.

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A077569 Irregular triangle read by rows: row n lists smallest numbers in increasing order of all possible prime signatures with n divisors.

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A122812 Numbers k where A046523(A005179(k)) differs from A046523(A038547(k)).

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A131802 Sequence related to factorizations and prime signatures: a(1) = 1; for n>1, a(n) = A057567(n) - 2*A001055(n).

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A131886 Rearrange the rows of array A077569 by sorting the values on each row by noting the odd part of each signature and sorting as in A131822.

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A131997 Array read by rows in which the n-th row contains odd numbers of all possible prime signatures with n divisors.

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