OFFSET

1,2

COMMENTS

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239, length A001222.

A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.

These are the Heinz numbers of integer partitions with all odd parts and all odd multiplicities, counted by A117958.

LINKS

David A. Corneth, Table of n, a(n) for n = 1..10000

FORMULA

EXAMPLE

The terms together with their prime indices begin:

1 = 1

2 = prime(1)

5 = prime(3)

8 = prime(1)^3

10 = prime(1) prime(3)

11 = prime(5)

17 = prime(7)

22 = prime(1) prime(5)

23 = prime(9)

31 = prime(11)

32 = prime(1)^5

34 = prime(1) prime(7)

40 = prime(1)^3 prime(3)

MATHEMATICA

Select[Range[100], #==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]

PROG

(Python)

from itertools import count, islice

from sympy import primepi, factorint

def A352142_gen(startvalue=1): # generator of terms >= startvalue

return filter(lambda k:all(map(lambda x:x[1]%2 and primepi(x[0])%2, factorint(k).items())), count(max(startvalue, 1)))

CROSSREFS

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 18 2022

STATUS

approved