A036844 Numbers k such that k / sopfr(k) is an integer, where sopfr = sum-of-prime-factors, A001414.

2, 3, 4, 5, 7, 11, 13, 16, 17, 19, 23, 27, 29, 30, 31, 37, 41, 43, 47, 53, 59, 60, 61, 67, 70, 71, 72, 73, 79, 83, 84, 89, 97, 101, 103, 105, 107, 109, 113, 127, 131, 137, 139, 149, 150, 151, 157, 163, 167, 173, 179, 180, 181, 191, 193, 197, 199, 211, 220, 223

Offset: 1

Author

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

Keywords

• nonn

Comments

Union of A046346 and the primes. - T. D. Noe, Feb 20 2007

These are the Heinz numbers of the partitions counted by A330953. - Gus Wiseman, Jan 17 2020

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

Examples

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

References

• Amarnath Murthy, Generalization of Partition function and introducing Smarandache Factor Partition, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring-2000.
• Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 89.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294, alternative link.
• Mohan Lal, Iterates of a number-theoretic function, Math. Comp., Vol. 23, No. 105 (1969), pp. 181-183.

Crossrefs

sopfr(n) is defined in A001414.

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

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

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

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

Partitions whose product divides their sum of primes: A331381.

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

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

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

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

Programs

Haskell

a036844 n = a036844_list !! (n-1)
a036844_list = filter ((== 0). a238525) [2..]
-- Reinhard Zumkeller, Jul 21 2014

Mathematica

Select[Range[2, 224], Divisible[#, Plus @@ Times @@@ FactorInteger[#]] &] (* Jayanta Basu, Aug 13 2013 *)

PARI

is_A036844(n)=n>1 && !(n%A001414(n)) \\ M. F. Hasler, Mar 01 2014

Formula

A238525(a(n)) = 0. - Reinhard Zumkeller, Jul 21 2014