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

1,1

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

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.

Formula

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

Example

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

Mathematica

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

Prog

(PARI) is_A036844(n)=n>1 && !(n%A001414(n)) \\ M. F. Hasler, Mar 01 2014
(Haskell)
a036844 n = a036844_list !! (n-1)
a036844_list = filter ((== 0). a238525) [2..]
-- Reinhard Zumkeller, Jul 21 2014

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.

Sequence in context: A062972 A231878 A285304 * A284696 A033070 A211781

Adjacent sequences: A036841 A036842 A036843 * A036845 A036846 A036847

Keyword

nonn

Author

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

Status

approved