A077564 -id:A077564 - cp's OEIS frontend

A077565 Number of factorizations of n where each factor has a different prime signature.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 4, 1, 1, 2, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 1, 4, 1, 4, 1, 3, 3, 1, 1, 6, 1, 3, 1, 3, 1, 4, 1, 4, 1, 1, 1, 5, 1, 1, 3, 4, 1, 4, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 1, 4, 1, 6, 2, 1, 1, 5, 1, 1, 1, 4, 1, 5, 1, 3, 1, 1, 1, 9, 1, 3, 3, 3, 1, 4, 1, 4, 4

Offset: 1

Amarnath Murthy, Nov 11 2002

nonn

In contrast to A001055 this sequence excludes from the count all such factorizations of n that include two such factors, f and g, for which it would hold that A046523(f) = A046523(g), or equally A101296(f) = A101296(g). - Antti Karttunen, Nov 24 2017

			a(24) = 4, 24 = 12*2 = 8*3 = 6*4. The factorizations 2*3*4, 2*2*2*3 etc. are not counted.
From _Antti Karttunen_, Nov 24 2017: (Start)
For n = 30 the solutions are 30, 2*15, 3*10, 5*6, thus a(30) = 4.
For n = 36 the solutions are 36, 2*18, 3*12, thus a(36) = 3.
For n = 60 the solutions are 60, 2*30, 3*20, 4*15, 5*12, thus a(60) = 5.
For n = 72 the solutions are 72, 2*36, 3*24, 4*18, 6*12, 8*9, 3*4*6, thus a(72) = 7.
(End)

Amarnath Murthy, Generalization of partition function. Introducing Smarandache Factor Partition. Smarandache Notions Journal, Vol. 11, 1-2-3,2000.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 6143 terms from Antti Karttunen, computed with the given Scheme-program)
Antti Karttunen, Scheme-program for computing this sequence
Index entries for sequences computed from exponents in factorization of n

Cf. A000009, A001055, A025487, A077563, A077564, A077566, A046523, A101296.

Table[1 + Count[Subsets[Rest@ Divisors@ n, {2, Infinity}], ?(And[Times @@ # == n, UnsameQ @@ Map[Sort[FactorInteger[#][[All, -1]], Greater] &, #]] &)], {n, 105}] (* _Michael De Vlieger, Nov 24 2017 *)

a(n) <= A001055(n). - Antti Karttunen, Nov 24 2017

a(p^e) = A000009(p^e). - David A. Corneth, Nov 24 2017

Corrected and extended by Ray Chandler, Aug 26 2003

Name improved by Antti Karttunen and David A. Corneth, Nov 24 2017

A077563 Number of partitions into two parts which have different prime signatures.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 5, 4, 4, 5, 6, 4, 8, 6, 8, 6, 8, 8, 11, 7, 10, 10, 12, 10, 12, 10, 13, 10, 15, 12, 15, 10, 17, 16, 17, 13, 18, 16, 18, 16, 19, 18, 21, 13, 20, 19, 25, 20, 23, 19, 24, 20, 25, 24, 27, 19, 24, 26, 28, 21, 28, 25, 30, 26, 31, 26, 32, 19, 30, 30, 33, 30

Offset: 0

Amarnath Murthy, Nov 11 2002

nonn

The 'prime signature' of n is the sorted list of exponents in the prime factorization of n.

Does lim n->infinity a(n)/n exist? If not, what are the limsup and liminf of a(n)/n?

			a(9) = 3; the partitions are 8+1, 6+3 and 5+4.

Sean A. Irvine, Table of n, a(n) for n = 0..10000

Cf. A077564.

sig[n_] := Sort[Last/@FactorInteger[n]]; a[n_] := Length[Select[Range[Floor[n/2]], sig[ # ]!=sig[n-# ]&]]

Edited by Dean Hickerson, Nov 11 2002

A077566 Smallest number which can be factored into parts with distinct prime signatures in n different ways, or 0 if no such number exists.

Original entry on oeis.org

1, 8, 12, 24, 60, 48, 72, 512, 96, 1024, 120, 144, 0, 0, 420, 2310, 240, 8192, 432, 0, 0, 768, 360, 0, 576, 480, 32768, 0, 0, 1536, 0, 864, 0, 0, 720, 1152, 0, 3072, 1080, 960, 0, 0, 1800, 0, 0, 262144, 0, 1728, 0, 6144, 2304, 1680, 0, 524288, 0, 0, 1920, 0, 4620, 0

Offset: 1

Amarnath Murthy, Nov 11 2002

nonn

Amarnath Murthy, Generalization of partition function, Introducing Smarandache Factor Partition, Smarandache Notions Journal, Vol. 11, 1-2-3, 2000.

Cf. A077563, A077564, A077565.

Index of the first occurrence of n in A077565.

Corrected and extended by Ray Chandler, Sep 07 2003

A088044 Number of partitions of n with distinct prime signatures.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 8, 10, 10, 11, 14, 14, 19, 22, 22, 25, 30, 34, 38, 40, 46, 54, 60, 63, 74, 81, 85, 91, 108, 118, 126, 134, 153, 173, 180, 186, 214, 236, 245, 262, 301, 326, 338, 355, 395, 437, 461, 476, 540, 592, 610, 649, 717, 768, 811, 853, 943, 1039, 1078

Offset: 1

Amarnath Murthy, Sep 20 2003

nonn

Essentially the same as A077564. [From R. J. Mathar, Sep 11 2008]

			a(9) = 6 and the partitions are (9), (1,8), (3,6),(4,5), (1,2,6),(2,3,4).
The partitions (2,7), (1,3,5),(1,4,4), etc. are not considered for the count as 2 and 7 both are prime and so on.

Cf. A088045.

ps(n) = local(f); f = factor(n); vecsort(f[,2]);
b(n, lim, used, cUsed) = local(s, v, found, j, used2); if (n == 0, return(1)); if (n > lim*(lim + 1)/2, return(0)); s = 0; for (i = 1, lim, v = ps(i); found = 0; j = 1; while (!found && j <= cUsed, if (v == used[j], found = 1, j++)); if (!found, used2 = used; used2[1 + cUsed] = v; s += b(n - i, min (n - i, i - 1), used2, cUsed + 1))); s;
a(n) = b(n, n, vector(12), 0); \\ David Wasserman, Jun 21 2005

More terms from David Wasserman, Jun 21 2005

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A077565 Number of factorizations of n where each factor has a different prime signature.

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A077563 Number of partitions into two parts which have different prime signatures.

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A077566 Smallest number which can be factored into parts with distinct prime signatures in n different ways, or 0 if no such number exists.

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A088044 Number of partitions of n with distinct prime signatures.

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