A046346 -id:A046346 - cp's OEIS frontend

A159237 Composite numbers A046346 divided by the sum of their prime factors (counted with multiplicity).

1, 2, 3, 3, 5, 5, 6, 6, 7, 10, 12, 11, 11, 15, 16, 11, 18, 14, 21, 20, 25, 17, 24, 30, 28, 32, 28, 27, 26, 19, 17, 36, 26, 28, 40, 23, 40, 26, 23, 45, 45, 48, 48, 40, 44, 54, 44, 34, 54, 29, 45, 48, 29, 54, 52, 44, 51, 31, 38, 65, 37, 29, 57, 52, 77, 78, 43, 43, 88, 69, 105, 105

Offset: 1

Zak Seidov, Apr 06 2009

nonn

			n=1: A046346(1)=4, A046346(4)=4, a(1)=4/4=1;
n=2: A046346(2)=16, A046346(16)=8, a(2)=16/8=2;
n=3: A046346(3)=27, A046346(27)=9, a(3)=27/9=3;
n=4: A046346(4)=30, A046346(30)=10, a(4)=30/10=3;
n=10: A046346(10)=150, A046346(150)=15, a(10)=150/15=10;
n=100: A046346(100)=3840, A046346(3840)=24, a(100)=3840/24=160;
n=1000: A046346(1000)=102860, A046346(102860)=185, a(1000)=102860/185=556.

Zak Seidov, Table of n, a(n) for n=1..1000

Cf. A046346 Composite numbers that are divisible by the sum of their prime factors (counted with multiplicity), A001414 Integer log of n: sum of primes dividing n (with repetition), sopfr(n), A036844 Numbers n such that n / sopfr(n) is an integer, where sopr() = sum-of-prime-factors, A001414.

a(n) = A046346(n)/sopfr(A046346(n)) = A046346(n)/A001414(A046346(n)).

A266955 Intersection of A046346 (numbers that are divisible by the sum of their prime factors, counted with multiplicity) and A097889 (numbers that are products of at least two consecutive primes).

Original entry on oeis.org

30, 105, 15015, 9699690, 37182145, 215656441, 955049953, 33426748355, 247357937827, 1448810778701, 3710369067405, 304250263527210, 102481630431415235, 1086305282573001491, 261682369333342226303, 37420578814667938361329, 241532826894674874877669

Offset: 1

Michel Marcus, Jan 07 2016

nonn

Alladi and Erdős ask if this sequence is infinite and give 3 terms: 2*3*5, 2*3*5*7*11*13*17*19 and 2*3*5*7*11*13*17*19*23*29*31*37*41, that is, a(1), a(4) and a(12).

This sequence contains A159578(n) for all values of n > 1. - Altug Alkan, Jan 07 2016

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..500
K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294.

Cf. A046346, A097889, A159578.

sopfr(n) = {my(f=factor(n)); sum(k=1, #f~, f[k, 1]*f[k, 2]); }
list(lim)= {my(v=List(), p, t); for(e=2, log(lim+.5)\log(2), p=1; t=prod(i=1, e-1, prime(i)); forprime(q=prime(e), lim, t*=q/p; if(t>lim, next(2)); if (! (t % sopfr(t)), listput(v, t)); p=nextprime(p+1))); vecsort(Vec(v));} \\ adapted from A097889

a(13)-a(17) from Hiroaki Yamanouchi, Jan 12 2016

A309310 Intersection of A046346 and (A046346-2).

Original entry on oeis.org

70, 286, 646, 648, 18048, 26752, 39128, 40000, 55648, 60760, 64798, 72928, 73726, 164736, 167440, 174018, 298298, 324478, 332748, 352798, 361788, 373246, 434928, 649798, 719998, 862750, 871198

Offset: 1

J. M. Bergot and Robert Israel, Jul 22 2019

nonn

Numbers k such that both k and k+2 are composite and each is divisible by the sum of its prime factors (counted with multiplicity).
There are at least two cases where k, k+2 and k+4 are all in A046346: k=646 and k=38104990.  Are there more?
Up to 3*10^12 there is only one other such triple for k=590269019100. - Giovanni Resta, Jul 24 2019

			a(1)=70 is a term because 70=2*5*7 is divisible by 2+5+7=14 and 72=2^3*3^2 is divisible by 2*3+3*2=12.

Amiram Eldar, Table of n, a(n) for n = 1..2000 (terms 1..200 from Robert Israel)

Cf. A046346.

filter:= proc(n) local E,t;
  if isprime(n) then return false fi;
  E:= ifactors(n)[2];
  n mod add(t[1]*t[2],t=E) = 0
end proc:
A046346:= select(filter, {$2..10^6}):
sort(convert(A046346 intersect map(`-`,A046346,2),list));

Select[Partition[Select[Range[2, 10^6], And[! PrimeQ[#], IntegerQ[#/Total[Times @@@ FactorInteger[#]]]] &], 2, 1], Subtract @@ # == -2 &][[All, 1]] (* Michael De Vlieger, Jul 22 2019 *)
cdsQ[n_]:=CompositeQ[n]&&Divisible[n,Total[Times@@@FactorInteger[n]]]; SequencePosition[Table[If[cdsQ[n],1,0],{n,872000}],{1,,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jul 23 2019 *)

A134330 Incorrect version of A046346.

Original entry on oeis.org

1, 4, 16, 27, 30, 60, 70, 72, 84, 105, 150, 180, 220, 231, 240, 256, 286, 288, 308, 378, 440, 450, 476, 528, 540, 560, 576, 588, 594, 624, 627, 646, 648, 650, 728, 800, 805, 840, 884, 897, 900

Offset: 1

dead

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

Original entry on oeis.org

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

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

nonn

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

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

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.

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.

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.

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

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

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

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

A134333 Numbers n whose number of prime factors (counted with multiplicity) is a prime factor of n.

Original entry on oeis.org

4, 6, 10, 12, 14, 18, 22, 26, 27, 30, 34, 38, 42, 45, 46, 58, 62, 63, 66, 74, 75, 78, 80, 82, 86, 94, 99, 102, 105, 106, 114, 117, 118, 120, 122, 134, 138, 142, 146, 147, 153, 158, 165, 166, 171, 174, 178, 180, 186, 194, 195, 200, 202, 206, 207, 214, 218, 222, 226

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

			a(1) = 4, since 4 has 2 prime factors and 2 is a prime factor of 4.
a(4) = 12, since 12 = 2*2*3 has 3 prime factors, and 3 is a prime factor of 12.
a(21) = 75, since 75 = 3*3*5 has 3 prime factors. and 3 is a prime factor of 75.
9 = 3*3 is not a term, since the number of prime factors (=2) is not a divisor of 9.
28 = 2*2*7 is not a term, since the number of prime factors (=3) is not a divisor of 28.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A100118, A046363, A133620, A133621, A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134334, A134344, A134376, A063989.

fQ[n_] := Module[{d = Total[Transpose[FactorInteger[n]][[2]]]}, PrimeQ[d] && Mod[n, d] == 0]; Select[Range[2, 226], fQ] (* T. D. Noe, Apr 05 2013 *)

a(n)=my(t=bigomega(n)); n%t==0 && isprime(t) \\ Charles R Greathouse IV, Sep 14 2015

a(n) << n log n/(log log n)^k for any fixed k. - Charles R Greathouse IV, Sep 14 2015

Sequence definition corrected and examples added by Hieronymus Fischer, Apr 05 2013

A134344 Composite numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is prime.

Original entry on oeis.org

4, 8, 9, 16, 20, 21, 25, 27, 32, 33, 44, 49, 57, 60, 64, 68, 69, 81, 85, 93, 105, 112, 116, 121, 125, 128, 129, 133, 145, 156, 169, 177, 180, 188, 195, 205, 212, 213, 217, 220, 231, 237, 243, 249, 253, 256, 265, 272, 275, 289, 297, 309, 332, 336, 343, 356, 361

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

Originally, the definition started with "Nonprime numbers ...". This may be misleading, since 1 is also nonprime, but has no prime factors. - Hieronymus Fischer, May 05 2013

			a(1) = 4, since 4 = 2*2 and the arithmetic mean (2+2)/2 = 2 is prime.
a(5) = 20, since 20 = 2*2*5 and the arithmetic mean (2+2+5)/3 = 3 is prime.

Harvey P. Dale and Hieronymus Fischer, Table of n, a(n) for n = 1..10000 (first 1000 terms from _Harvey P. Dale_)

Cf. A000040, A001222, A100118, A046363, A133620, A133621, A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134334.

ampfQ[n_]:=PrimeQ[Mean[Flatten[Table[#[[1]],{#[[2]]}]&/@FactorInteger[ n]]]]; nn=400;Select[Complement[Range[nn],Prime[Range[ PrimePi[nn]]]], ampfQ] (* Harvey P. Dale, Nov 06 2012 *)

is(n)=if(n<4,return(0)); my(f=factor(n),s=sum(i=1,#f~,f[i,1]*f[i,2])/sum(i=1,#f~,f[i,2])); (#f~>1 || f[1,2]>1) && denominator(s)==1 && isprime(s) \\ Charles R Greathouse IV, Sep 14 2015

Definition clarified by Hieronymus Fischer, May 05 2013

A134376 Numbers whose sum of prime factors (counted with multiplicity) is not prime.

Original entry on oeis.org

1, 4, 8, 9, 14, 15, 16, 18, 20, 21, 24, 25, 26, 27, 30, 32, 33, 35, 36, 38, 39, 42, 44, 46, 49, 50, 51, 55, 57, 60, 62, 64, 65, 66, 68, 69, 70, 72, 74, 77, 78, 81, 84, 85, 86, 87, 91, 92, 93, 94, 95, 98, 100, 102, 105, 106, 110, 111, 112, 114, 115, 116, 119, 120, 121, 122

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

The first term is 1, since 1 has no prime factors and so the sum of prime factors evaluates to zero.

Conjecture: a(n) ~ n. - Charles R Greathouse IV, Apr 28 2015

			a(2) = 4, since 4 = 2*2 and 2+2 = 4 is not prime.
a(5) = 14, since 14 = 2*7 and 2+7 = 9 is not prime.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A100118, A046363, A133620, A133621, A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134334, A134344, A078177.

Select[Range[150],!PrimeQ[Total[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ #]]]]&] (* Harvey P. Dale, Jul 05 2021 *)

sopfr(n)=my(f=factor(n)); sum(i=1,#f~,f[i,1]*f[i,2])
is(n)=!isprime(sopfr(n)) \\ Charles R Greathouse IV, Apr 28 2015

Edited by the author at the suggestion of T. D. Noe, May 20 2013

A134334 Numbers which are not divisible by the number of their prime factors (counted with multiplicity).

Original entry on oeis.org

8, 9, 15, 20, 21, 25, 28, 32, 33, 35, 39, 44, 48, 49, 50, 51, 52, 54, 55, 57, 64, 65, 68, 69, 70, 72, 76, 77, 81, 85, 87, 90, 91, 92, 93, 95, 98, 108, 110, 111, 112, 115, 116, 119, 121, 123, 124, 125, 126, 128, 129, 130, 133, 135, 141, 143, 145, 148, 150, 154, 155, 159

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

The asymptotic density of this sequence is 1 (Erdős and Pomerance, 1990). - Amiram Eldar, Jul 10 2020

			a(1) = 8, since 8 = 2*2*2 has 3 prime factors and 8 is not divisible by 3.
a(3) = 15, since 15 = 3*5 has 2 prime factors and 15 is not divisible by 2.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
Paul Erdős and Carl Pomerance, On a theorem of Besicovitch: values of arithmetic functions that divide their arguments, Indian J. Math., Vol. 32 (1990), pp. 279-287.

Cf. A000040, A001222, A074946 (complement), A100118, A046363, A133620, A133621.

Cf. A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134335, A134344, A134376.

Select[Range[2,200],Mod[#,PrimeOmega[#]]!=0&] (* Harvey P. Dale, May 13 2023 *)

isok(n) = (n % bigomega(n)) \\ Michel Marcus, Jul 15 2013

A046347 Odd composite numbers divisible by the sum of their prime factors (counted with multiplicity).

Original entry on oeis.org

27, 105, 231, 627, 805, 897, 945, 1581, 1755, 2079, 2625, 2967, 3055, 3125, 3861, 4185, 4543, 5355, 5445, 5487, 5967, 6075, 6461, 6525, 6745, 7881, 8085, 8505, 8883, 9555, 9717, 10125, 10647, 10707, 11375, 11385, 12675, 12789, 13005, 13275, 13475

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

			897 is a term since 897 = 3 * 13 * 23 and 3 + 13 + 23 = 39 and 897 / 39 = 23.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A001414, A046346, A046348.

Select[Range[9,13500,2],!PrimeQ[#]&&IntegerQ[#/Total[Times@@@FactorInteger[#]]]&] (* Jayanta Basu, Jun 04 2013 *)

sopfr(n) = {my(f=factor(n)); sum(k=1, #f~, f[k, 1]*f[k, 2]); } \\ A001414
lista(nn) = forcomposite(n=2, nn, if ((n%2) && !(n % sopfr(n)), print1(n, ", ")); ); \\ Michel Marcus, Feb 06 2019

Name clarified by Michel Marcus, Feb 06 2019

cp's OEIS Frontend

A159237 Composite numbers A046346 divided by the sum of their prime factors (counted with multiplicity).

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A266955 Intersection of A046346 (numbers that are divisible by the sum of their prime factors, counted with multiplicity) and A097889 (numbers that are products of at least two consecutive primes).

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A309310 Intersection of A046346 and (A046346-2).

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Maple

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A134330 Incorrect version of A046346.

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

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Haskell

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A134333 Numbers n whose number of prime factors (counted with multiplicity) is a prime factor of n.

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A134344 Composite numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is prime.

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Mathematica

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A134376 Numbers whose sum of prime factors (counted with multiplicity) is not prime.

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