login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A286852 Number of partitions of n into unitary prime divisors of n. 3
1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 1, 0, 2, 0, 1, 1, 21, 1, 0, 2, 2, 2, 0, 1, 2, 2, 1, 1, 28, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 5, 1, 2, 1, 0, 2, 42, 1, 1, 2, 43, 1, 0, 1, 2, 1, 1, 2, 49, 1, 1, 0, 2, 1, 5, 2, 2, 2, 1, 1, 10, 2, 1, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..2309

Eric Weisstein's World of Mathematics, Unitary Divisor

Index entries for related partition-counting sequences

FORMULA

a(n) = [x^n] Product_{p|n, p prime, gcd(p, n/p) = 1} 1/(1 - x^p).

a(n) = 0 if n is a powerful number (A001694).

EXAMPLE

a(6) = 2 because 6 has 4 divisors {1, 2, 3, 6} among which 2 are unitary prime divisors {2, 3} therefore we have [3, 3] and [2, 2, 2].

MATHEMATICA

Join[{1}, Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[GCD[n/d[[k]], d[[k]]] == 1 && PrimeQ[d[[k]]]] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 1, 95}]]

PROG

(PARI)

A055231(n) = {my(f=factor(n)); for (k=1, #f~, if (f[k, 2] > 1, f[k, 2] = 0); ); factorback(f); } \\ From A055231

unitary_prime_factors(n) = { my(ufs = factor(A055231(n))); ufs[, 1]~; };

partitions_into(n, parts, from=1) = if(!n, 1, my(k = #parts, s=0); for(i=from, k, if(parts[i]<=n, s += partitions_into(n-parts[i], parts, i))); (s));

A286852(n) = if(n<2, 1-n, partitions_into(n, vecsort(unitary_prime_factors(n), , 4))); \\ Antti Karttunen, Jul 02 2018

CROSSREFS

Cf. A018818, A055231, A056169, A066874, A066882, A225244, A284289.

Sequence in context: A050326 A335452 A056169 * A125070 A125071 A177207

Adjacent sequences:  A286849 A286850 A286851 * A286853 A286854 A286855

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Aug 01 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 4 09:03 EDT 2020. Contains 336201 sequences. (Running on oeis4.)