%I #32 Nov 02 2020 20:38:25

%S 2,3,4,5,7,11,13,16,17,19,23,27,29,30,31,37,41,43,47,53,59,60,61,67,

%T 70,71,72,73,79,83,84,89,97,101,103,105,107,109,113,127,131,137,139,

%U 149,150,151,157,163,167,173,179,180,181,191,193,197,199,211,220,223

%N Numbers k such that k / sopfr(k) is an integer, where sopfr = sum-of-prime-factors, A001414.

%C Union of A046346 and the primes. - _T. D. Noe_, Feb 20 2007

%C These are the Heinz numbers of the partitions counted by A330953. - _Gus Wiseman_, Jan 17 2020

%C Alladi and Erdős (1977) noted that sopfr(k) = k if k is a prime or k = 4. They called the terms for which k/sopfr(k) > 1 "special numbers", and proved that there are infinitely many such terms that are squarefree. - _Amiram Eldar_, Nov 02 2020

%D Amarnath Murthy, Generalization of Partition function and introducing Smarandache Factor Partition, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring-2000.

%D Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 89.

%H Amiram Eldar, <a href="/A036844/b036844.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)

%H Krishnaswami Alladi and Paul Erdős, <a href="https://projecteuclid.org/euclid.pjm/1102811427">On an additive arithmetic function</a>, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294, <a href="https://msp.org/pjm/1977/71-2/pjm-v71-n2-p01-s.pdf">alternative link</a>.

%H Mohan Lal, <a href="http://dx.doi.org/10.1090/S0025-5718-1969-0242765-9">Iterates of a number-theoretic function</a>, Math. Comp., Vol. 23, No. 105 (1969), pp. 181-183.

%F A238525(a(n)) = 0. - _Reinhard Zumkeller_, Jul 21 2014

%e a(12) = 27 because sopfr(27) = 3 + 3 + 3 = 9 and 27 is divisible by 9.

%t Select[Range[2, 224], Divisible[#, Plus @@ Times @@@ FactorInteger[#]] &] (* _Jayanta Basu_, Aug 13 2013 *)

%o (PARI) is_A036844(n)=n>1 && !(n%A001414(n)) \\ _M. F. Hasler_, Mar 01 2014

%o (Haskell)

%o a036844 n = a036844_list !! (n-1)

%o a036844_list = filter ((== 0). a238525) [2..]

%o -- _Reinhard Zumkeller_, Jul 21 2014

%Y sopfr(n) is defined in A001414.

%Y The version for prime indices instead of prime factors is A324851.

%Y Partitions whose Heinz number is divisible by their sum: A330950.

%Y Partitions whose Heinz number is divisible by their sum of primes: A330953.

%Y Partitions whose product is divisible by their sum of primes: A330954.

%Y Partitions whose product divides their sum of primes: A331381.

%Y Product of prime indices is divisible by sum of prime factors: A331378.

%Y Sum of prime factors is divisible by sum of prime indices: A331380.

%Y Product of prime indices equals sum of prime factors: A331384.

%Y Cf. A056239, A112798, A120383, A238525, A331379, A331382, A331383.

%K nonn

%O 1,1

%A Robert A. Stump (bee_ess107(AT)yahoo.com), Jan 09 2002