A100118 -id:A100118 - cp's OEIS frontend

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

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

A133911 Number of prime factors (counted with multiplicity) of the period numbers defined by A133900.

Original entry on oeis.org

0, 2, 2, 4, 2, 5, 2, 6, 4, 6, 2, 8, 2, 6, 5, 8, 2, 9, 2, 8, 5, 7, 2, 10, 4, 7, 6, 8, 2, 12, 2, 10, 6, 8, 5, 12, 2, 8, 6, 11, 2, 12, 2, 9, 8, 8, 2, 13, 4, 10, 6, 9, 2, 12, 5, 11, 6, 8, 2, 14, 2, 8, 8, 12, 5, 13, 2, 10, 6, 13, 2, 14, 2, 9, 8, 10, 5, 13, 2, 13, 8, 10, 2, 17, 5, 9, 7, 11, 2, 16, 5, 10, 7, 9

Offset: 1

Hieronymus Fischer, Oct 20 2007

nonn

			a(6)=5, since A133900(6)=72=2*2*2*3*3.
a(12)=8, since A133900(12)=864=2*2*2*2*2*3*3*3.

Cf. A000040, A001222, A100118, A046363, A133620, A133621, A133623, A133630, A133635.

Cf. A133872, A133880, A133890, A133900, A133910, A134331.

a(n)=A001222(A133900(n)).

A134331 Sum of prime factors (counted with multiplicity) of the period numbers defined by A133900.

Original entry on oeis.org

0, 4, 6, 8, 10, 12, 14, 12, 12, 18, 22, 19, 26, 22, 19, 16, 34, 22, 38, 22, 23, 32, 46, 23, 20, 36, 18, 26, 58, 37, 62, 20, 34, 46, 29, 29, 74, 50, 38, 31, 82, 38, 86, 36, 30, 58, 94, 30, 28, 32, 46, 40, 106, 30, 37, 37, 50, 70, 118, 41, 122, 74, 36, 24, 41, 48, 134, 50, 58, 50

Offset: 1

Hieronymus Fischer, Oct 20 2007

nonn

			a(6)=12, since A133900(6)=72=2*2*2*3*3 and 2+2+2+3+3=12.
a(12)=19, since A133900(12)=864=2*2*2*2*2*3*3*3 and 2+2+2+2+2+3+3+3=19.

Cf. A000040, A100118, A046363, A133620, A133621, A133623, A133630, A133635.

Cf. A133872, A133880, A133890, A133900, A133910, A133911.

A134332 Integer part of the arithmetic mean of the prime factors (counted with multiplicity) of the period numbers defined by A133900.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 3, 11, 2, 13, 3, 3, 2, 17, 2, 19, 2, 4, 4, 23, 2, 5, 5, 3, 3, 29, 3, 31, 2, 5, 5, 5, 2, 37, 6, 6, 2, 41, 3, 43, 4, 3, 7, 47, 2, 7, 3, 7, 4, 53, 2, 7, 3, 8, 8, 59, 2, 61, 9, 4, 2, 8, 3, 67, 5, 9, 3, 71, 2, 73, 9, 4, 5, 8, 4, 79, 2, 3, 9, 83, 3, 9, 11, 10, 3, 89, 2, 9, 6, 11, 12

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

			a(6)=2, since floor(A134331(6)/A133911(6))=floor(12/5)=2.
a(7)=7, since floor(A134331(7)/A133911(7))=floor(14/2)=7.

Cf. A000040, A100118, A046363, A133620, A133621, A133623, A133630, A133635.

Cf. A133872, A133880, A133890, A133900, A133910, A133911, A134331.

a(n)=floor(A134331(n)/A133911(n)) for n>1, defining a(1):=1.

a(n)=n, if n is a prime or 1.

A078177 Composite numbers with an integer arithmetic mean of all prime factors.

Original entry on oeis.org

4, 8, 9, 15, 16, 20, 21, 25, 27, 32, 33, 35, 39, 42, 44, 49, 50, 51, 55, 57, 60, 64, 65, 68, 69, 77, 78, 81, 85, 87, 91, 92, 93, 95, 105, 110, 111, 112, 114, 115, 116, 119, 121, 123, 125, 128, 129, 133, 140, 141, 143, 145, 155, 156, 159, 161, 164, 169, 170, 177, 180

Offset: 1

Reinhard Zumkeller, Nov 20 2002

nonn

That is, composite numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is an integer.

			60 = 2*2*3*5: (2+2+3+5)/4 = 3, therefore 60 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A078175, A001222, A100118, A046363, A133620, A133621.

Cf. A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134334, A134344, A134376.

Select[Range[200], CompositeQ[#] && IntegerQ[Mean[Flatten[Table[#[[1]], #[[2]]]& /@ FactorInteger[#]]]]&] (* Jean-François Alcover, Aug 03 2018 *)

lista(nn) = {forcomposite(n=1, nn, my(f = factor(n)); if (! (sum(k=1, #f~, f[k,1]*f[k,2]) % vecsum(f[,2])), print1(n, ", ")););} \\ Michel Marcus, Feb 22 2016

A001414(a(n)) == 0 modulo A001222(a(n)).

Edited by N. J. A. Sloane, May 30 2008 at the suggestion of R. J. Mathar

A316092 Heinz numbers of integer partitions of prime numbers into prime parts.

Original entry on oeis.org

3, 5, 11, 15, 17, 31, 33, 41, 45, 59, 67, 83, 93, 109, 127, 153, 157, 177, 179, 191, 211, 241, 275, 277, 283, 297, 327, 331, 353, 367, 369, 375, 401, 405, 425, 431, 459, 461, 509, 537, 547, 563, 587, 599, 603, 605, 617, 709, 739, 773, 775, 797, 825, 831, 837

Offset: 1

Gus Wiseman, Jun 24 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of integer partitions of prime numbers into prime parts together with their Heinz numbers begins:
   3: (2)
   5: (3)
  11: (5)
  15: (2,3)
  17: (7)
  31: (11)
  33: (2,5)
  41: (13)
  45: (2,2,3)

Cf. A000041, A000607, A056239, A056768, A076610, A100118, A112798, A215366, A296150, A300383, A316091.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[900],And[PrimeQ[Total[primeMS[#]]],And@@PrimeQ/@primeMS[#]]&]

A316219 Number of triangles of weight prime(n) in the multiorder of integer partitions of prime numbers into prime parts.

Original entry on oeis.org

1, 1, 3, 6, 15, 31, 92, 161, 464, 2347, 3987, 18202, 50136, 81722, 214976, 903048, 3684567, 5842249, 23206424, 57341256, 89938662, 343306266, 829972421, 3084219358, 17375700038, 40920517008, 62656899579, 146415515992, 223442878751, 518427758704, 9544240589455, 21746920337606

Offset: 1

Gus Wiseman, Jun 26 2018

nonn

A prime partition is an integer partition of a prime number into prime parts. Then a(n) is the number of sequences of prime partitions whose sums are weakly decreasing and sum to the n-th prime number.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Gus Wiseman, Comcategories and Multiorders
Gus Wiseman, Illustration of the first five terms of A316219.

Cf. A000040, A000041, A000607, A056768, A063834, A100118, A269134, A281113, A316220.

nn=20;
pen[n_]:=pen[n]=SeriesCoefficient[Product[1/(1-x^p),{p,Select[Range[n],PrimeQ]}],{x,0,n}]
Table[Sum[Times@@pen/@p,{p,Select[IntegerPartitions[Prime[n]],And@@PrimeQ/@#&]}],{n,nn}]

P(n,f)={1/prod(k=1, n, 1 - f(k)*x^prime(k) + O(x*x^prime(n)))}
seq(n)={my(p=P(n, i->1), q=P(n, i->polcoef(p, prime(i)))); vector(n, k, polcoef(q, prime(k)))} \\ Andrew Howroyd, Jan 16 2023

Terms a(16) and beyond from Andrew Howroyd, Jan 16 2023

A316091 Heinz numbers of integer partitions of prime numbers.

Original entry on oeis.org

3, 4, 5, 6, 8, 11, 14, 15, 17, 18, 20, 24, 26, 31, 32, 33, 35, 41, 42, 44, 45, 50, 54, 56, 58, 59, 60, 67, 69, 72, 74, 80, 83, 92, 93, 95, 96, 106, 109, 114, 119, 122, 124, 127, 128, 141, 143, 145, 152, 153, 157, 158, 161, 170, 174, 177, 179, 182, 188, 191

Offset: 1

Gus Wiseman, Jun 24 2018

nonn

Also the union of prime-indexed rows of A215366.

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			Sequence of all integer partitions of prime numbers begins (2), (1, 1), (3), (2, 1), (1, 1, 1), (5), (4, 1), (3, 2), (7), (2, 2, 1), (3, 1, 1), (2, 1, 1, 1), (6, 1).

Cf. A000041, A000607, A056239, A056768, A076610, A100118, A112798, A215366, A296150, A300383, A316092.

primeMS[n_] := If[n == 1, {}, Flatten[Cases[FactorInteger[n],{p_, k_} :> Table[PrimePi[p], {k}]]]]; Select[Range[100], PrimeQ[Total[primeMS[#]]] &]

A213015 Numbers n such that the sum of prime factors of n (counted with multiplicity) is 2 times a prime.

Original entry on oeis.org

4, 8, 9, 21, 25, 30, 32, 33, 36, 49, 57, 69, 70, 84, 85, 93, 100, 102, 120, 121, 128, 129, 133, 135, 144, 145, 162, 169, 174, 177, 182, 190, 205, 213, 217, 228, 237, 238, 246, 249, 253, 260, 265, 286, 289, 308, 309, 310, 312, 318, 340, 351, 361, 372, 393, 406

Offset: 1

Michel Lagneau, Jun 02 2012

nonn

The numbers A100118(n)^2 are in the sequence.

			36 is in the sequence because 36 = 2^2 * 3^2 => sum of prime factors = 2*2+3*2 = 10 = 2*5 where 5 is prime.

Cf. A001414, A100118.

with(numtheory):A:= proc(n) local e, j; e := ifactors(n)[2]: add (e[j][1]*e[j][2], j=1..nops(e)) end: for m from 1 to 3000 do: if type(A(m)/2,prime)= true then printf(`%d, `,m):else fi:od:

L = {}; Do[ww = Transpose[FactorInteger[k]]; w = ww[[1]].ww[[2]]; If[PrimeQ[w/2], AppendTo[L, k]], {k, 2, 500}]; L

sopfr(n) = 2*p, p prime.

A276493 Perfect numbers whose sum of prime factors is prime.

Original entry on oeis.org

6, 28, 8128, 14474011154664524427946373126085988481573677491474835889066354349131199152128

Offset: 1

Juri-Stepan Gerasimov, Sep 05 2016

nonn

The next term is too large to include.
Numbers (2^n - 1)*2^(n - 1) such that both 2^n - 1 and 2^n + 2*n - 3 are prime.
Conjectures (defining x = 170141183460469231731687303715884105727 = A007013(4)):
(1) (2^x - 1)*2^(x - 1) is a term because 2^x - 1 and 2^x + 2*x - 3 are primes;
(2) a(n) is equal to (2^A007013(k) - 1)*2^(A007013(k) - 1) such that 2^A007013(k) - 1 and 2^A007013(k) + 2*A007013(k) - 3 are primes for some prime value of A007013(k) where k => 0;
(3) primes of A007013 are Mersenne prime exponents A000043, i.e. x is new exponent in A000043.

			a(1) = (2^2-1)*2^(2-1) = 6 because both 2^2-1 = 3 and 2^2+2*2-3 = 5 are primes.
a(2) = (2^3-1)*2^(3-1) = 28 because both 2^3-1 = 7 and 2^3+2*3-3 = 11 are primes.
a(3) = (2^7-1)*2^(7-1) = 8128 because both 2^7-1 = 127 and 2^7+2*7-3 = 139 are primes.

Subsequence of A000396. Subsequence of A100118.

Cf. A000043, A000668, A007013, A192436, A276511.

[(2^p-1)*2^(p-1): p in PrimesUpTo(2000) | IsPrime(2^p+2*p-3)];

[(2^n-1)*2^(n-1): n in [1..200] | IsPrime(n) and IsPrime(2^n-1) and IsPrime(2^n+2*n-3)]; // Vincenzo Librandi, Sep 06 2016

A276493:=n->`if`(isprime(n) and isprime(2^n-1) and isprime(2^n+2*n-3), (2^n-1)*2^(n-1), NULL): seq(A276493(n), n=1..10^3); # Wesley Ivan Hurt, Sep 07 2016

Select[PerfectNumber[Range[12]],PrimeQ[Total[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[#]]]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 06 2020 *)

cp's OEIS Frontend

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

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A133911 Number of prime factors (counted with multiplicity) of the period numbers defined by A133900.

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A134331 Sum of prime factors (counted with multiplicity) of the period numbers defined by A133900.

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A134332 Integer part of the arithmetic mean of the prime factors (counted with multiplicity) of the period numbers defined by A133900.

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A078177 Composite numbers with an integer arithmetic mean of all prime factors.

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A316092 Heinz numbers of integer partitions of prime numbers into prime parts.

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A316219 Number of triangles of weight prime(n) in the multiorder of integer partitions of prime numbers into prime parts.

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A316091 Heinz numbers of integer partitions of prime numbers.

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Mathematica

A213015 Numbers n such that the sum of prime factors of n (counted with multiplicity) is 2 times a prime.

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