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 1-3 of 3 results.

A077564 Number of partitions of n into parts with distinct prime signatures.

Original entry on oeis.org

1, 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
Offset: 0

Views

Author

Amarnath Murthy, Nov 11 2002

Keywords

Comments

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

Examples

			a(7) = 4. The partitions are 7, 6+1, 4+3, 4+2+1. (5+2, 3+2+2, ... are not counted.)

Links

Crossrefs

Cf. A077563.

Programs

  • Mathematica
    sig[n_] := Sort[Last/@FactorInteger[n]]; f[n_, m_] := Module[{sm}, If[n>m(m+1)/2||n<0, Return[{}]]; If[n==0, Return[{{}}]]; sm=sig[m]; f[n, m]=Union[f[n, m-1], Prepend[ #, m]&/@Select[f[n-m, m-1], !MemberQ[sig/@#, sm]&]]]; a[n_] := Length[f[n, n]] (* f[n, m] is list of partitions of n into parts <= m with distinct prime signatures *)

Extensions

Edited by Dean Hickerson, Nov 11 2002

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

Original entry on oeis.org

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

Views

Author

Amarnath Murthy, Nov 11 2002

Keywords

Comments

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

Examples

			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)

References

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

Links

Crossrefs

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

Programs

Formula

a(n) <= A001055(n). - Antti Karttunen, Nov 24 2017
a(p^e) = A000009(p^e). - David A. Corneth, Nov 24 2017

Extensions

Corrected and extended by Ray Chandler, Aug 26 2003
Name improved by Antti Karttunen and David A. Corneth, Nov 24 2017

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

Views

Author

Amarnath Murthy, Nov 11 2002

Keywords

References

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

Crossrefs

Cf. A077563, A077564, A077565.

Formula

Index of the first occurrence of n in A077565.

Extensions

Corrected and extended by Ray Chandler, Sep 07 2003
Showing 1-3 of 3 results.

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