A131371 Number of anagrams of n that are semiprimes.

0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 2, 0, 1, 0, 1, 0, 0, 1, 1, 0, 2, 0, 2, 1, 1, 0, 1, 1, 1, 2, 1, 0, 0, 2, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 2, 1, 1, 0, 0, 0, 1, 0, 2, 2, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1

Offset

1,15

Comments

An anagram of a k-digit number is one of the k! permutations of the digits that does not begin with 0.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(123) = 3 because 123 = 3 * 41 is semiprime, 213 = 3 * 71 is semiprime, 321 = 3 * 107 is semiprime, while the other anagrams 132, 231 and 312 have respectively 3, 3 and 5 prime factors with multiplicity.
a(129) = 4 because 129 = 3 * 43 is semiprime, 219 = 3 * 73 is semiprime, 291 = 3 * 97 is semiprime, 921 = 3 * 307 is semiprime, while 192 and 912 have 7 and 6 prime factors with multiplicity.
a(134) = 5 because 134 = 2 * 67 and 143 = 11 * 13 and 314 = 2 * 157 and 341 = 11 * 31 and 413 = 7 * 59 are semiprimes, while 431 is prime.

Maple

f:= proc(n) local L, m, t, i;
  L:= convert(n, base, 10); m:= nops(L);
  nops(select(t -> t[-1] <> 0 and numtheory:-bigomega(add(t[i]*10^(i-1), i=1..m))=2, combinat:-permute(L)));
end proc:
map(f, [$1..200]); # Robert Israel, Jun 11 2023

Prog

(Python)
from sympy import factorint
from sympy.utilities.iterables import multiset_permutations as mp
def c(n):
    return sum(factorint(n).values()) == 2
def a(n):
    return sum(1 for p in mp(str(n)) if p[0]!="0" and c(t:=int("".join(p))))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Jun 11 2023

Crossrefs

Cf. A000040, A001358, A002113, A046810, A097393.

Sequence in context: A248911 A116681 A359911 * A319195 A003475 A248639

Adjacent sequences: A131368 A131369 A131370 * A131372 A131373 A131374

Keyword

base,easy,less,nonn

Author

Jonathan Vos Post, Sep 30 2007

Status

approved