A276687 -id:A276687 - cp's OEIS frontend

A100118 Numbers whose sum of prime factors is prime (counted with multiplicity).

2, 3, 5, 6, 7, 10, 11, 12, 13, 17, 19, 22, 23, 28, 29, 31, 34, 37, 40, 41, 43, 45, 47, 48, 52, 53, 54, 56, 58, 59, 61, 63, 67, 71, 73, 75, 76, 79, 80, 82, 83, 88, 89, 90, 96, 97, 99, 101, 103, 104, 107, 108, 109, 113, 117, 118, 127, 131, 136, 137, 139, 142, 147, 148, 149

Offset: 1

Carlos Alves, Dec 26 2004

nonn

Numbers n such that integer log of n is a prime number.
As in A001414, denote sopfr(n) the integer log of n. Since sopfr(p)=p, the sequence includes all prime numbers.
See A046363 for the analog excluding prime numbers. - Hieronymus Fischer, Oct 20 2007
These numbers may be arranged in a family of posets of triangles of multiarrows (see link and example). - Gus Wiseman, Sep 14 2016

			40 = 2^3*5 and 2*3 + 5 = 11 is a prime number.
These numbers correspond to multiarrows in the multiorder of partitions of prime numbers into prime parts. For example: 2:2<=(2), 3:3<=(3), 6:5<=(2,3), 5:5<=(5), 12:7<=(2,2,3), 10:7<=(2,5), 7:7<=(7), 48:11<=(2,2,2,2,3), 52:11<=(2,3,3,3), 40:11<=(2,2,2,5), 45:11<=(3,3,5), 28:11<=(2,2,7), 11:11<=(11). - _Gus Wiseman_, Sep 14 2016

Jayanta Basu, Table of n, a(n) for n = 1..10000
Gus Wiseman, Lattice form posets indexed by A100118

Cf. A001414, A046363, A056768, A276687.

for n from 1 to 200 do
    if isprime(A001414(n)) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Sep 09 2015

L = {}; Do[ww = Transpose[FactorInteger[k]];w = ww[[1]].ww[[2]]; If[PrimeQ[w], AppendTo[L, k]], {k, 2, 500}];L
Select[Range[150], PrimeQ[Total[Times @@@ FactorInteger[#]]] &] (* Jayanta Basu, Aug 11 2013 *)

is(n)=my(f=factor(n)); isprime(sum(i=1,#f~,f[i,1]*f[i,2])) \\ Charles R Greathouse IV, Sep 21 2013

A056768 Number of partitions of the n-th prime into parts that are all primes.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 23, 40, 87, 111, 219, 336, 413, 614, 1083, 1850, 2198, 3630, 5007, 5861, 9282, 12488, 19232, 33439, 43709, 49871, 64671, 73506, 94625, 221265, 279516, 394170, 441250, 766262, 853692, 1175344, 1608014, 1975108, 2675925

Offset: 1

Brian Galebach, Aug 16 2000

nonn

			a(4) = 3 because the 4th prime is 7 which can be partitioned using primes in 3 ways: 7, 5 + 2, and 3 + 2 + 2.
In connection with the 6th prime 13, for instance, we have the a(6) = 9 prime partitions: 13 = 2 + 2 + 2 + 2 + 2 + 3 = 2 + 2 + 2 + 2 + 5 = 2 + 2 + 2 + 7 = 2 + 2 + 3 + 3 + 3 = 2 + 3 + 3 + 5 = 2 + 11 = 3 + 3 + 7 = 3 + 5 + 5.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 4000 terms from Alois P. Heinz)

Cf. A000041, A000607, A100118, A276687, A070215 (distinct parts).

a056768 = a000607 . a000040  -- Reinhard Zumkeller, Aug 05 2012

b:= proc(n, i) option remember; `if`(n=0 or i=2
       and n::even, 1, `if`(i=2 or n=1, 0,
       b(n, prevprime(i)))+`if`(i>n, 0, b(n-i, i)))
    end:
a:= n-> b(ithprime(n)$2):
seq(a(n), n=1..50);  # Alois P. Heinz, Sep 15 2016

Table[Count[IntegerPartitions[n],?(AllTrue[#,PrimeQ]&)],{n,Prime[ Range[ 40]]}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Mar 07 2015 *)
n=40;ser=Product[1/(1-x^Prime[i]),{i,1,n}];Table[SeriesCoefficient[ser,{x,0,Prime[i]}],{i,1,n}] (* Gus Wiseman, Sep 14 2016 *)

a(n) = A000607(prime(n)).
a(n) = A168470(n) + 1. - Alonso del Arte, Feb 15 2014, restating the corresponding formula given by R. J. Mathar for A168470.
a(n) = [x^prime(n)] Product_{k>=1} 1/(1 - x^prime(k)). - Ilya Gutkovskiy, Jun 05 2017

More terms from James Sellers, Aug 25 2000

cp's OEIS Frontend

A100118 Numbers whose sum of prime factors is prime (counted with multiplicity).

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