A100118 -id:A100118 - cp's OEIS frontend

A276511 Primes that are equal to the sum of the prime factors of some perfect number.

5, 11, 139, 170141183460469231731687303715884105979

Offset: 1

Juri-Stepan Gerasimov, Sep 06 2016

Primes of the form 2^n + 2*n - 3 such that 2^n - 1 is also prime.
Conjectures (defining x = 170141183460469231731687303715884105727 = A007013(4)):
(1) 2^x + 2*x - 3 is in this sequence;
(2) a(5) = 2^x + 2*x - 3 (see comments of A276493);
(3) primes of A007013 are Mersenne prime exponents A000043, i.e., x is new exponent in A000043.

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

Subsequence of A192436.

Cf. A000043, A000396, A000668, A007013, A100118, A276493.

[2^n+2*n-3: n in [1..200] | IsPrime(2^n-1) and IsPrime(2^n+2*n-3)];

A276511:=n->`if`(isprime(2^n-1) and isprime(2^n+2*n-3), 2^n+2*n-3, NULL): seq(A276511(n), n=1..10^3); # Wesley Ivan Hurt, Sep 07 2016

Name suggested by Michel Marcus, Sep 07 2016

A276687 Number of prime plane trees of weight prime(n).

Original entry on oeis.org

1, 1, 2, 4, 11, 30, 122, 336, 1412, 15129, 44561, 417542, 2479120, 7540843, 35983502, 451454834, 5313515136, 16809858904, 190077477328, 1124302066470, 3521811953565, 38563707677633, 240966297786218, 3192420711942298, 95433674596402663, 567734580765228356

Offset: 1

Gus Wiseman, Sep 13 2016

nonn

A prime plane tree is either (case 1) a prime number, or (case 2) a sequence of prime plane trees whose weights are an integer partition of a prime number, where the weight of a tree is the sum of weights of its branches. Prime plane trees are "multichains" in the multiorder of integer partitions of prime numbers into prime parts (A056768).

			The a(5) = 11 prime plane trees of weight A000040(5) = 11 are: {11, (3,3,5), (3,3,(2,3)), (2,2,7), (2,2,(2,5)), (2,2,(2,(2,3))), (2,2,(2,2,3)), (2,3,3,3), (2,2,2,5), (2,2,2,(2,3)), (2,2,2,2,3)}.

Alois P. Heinz, Table of n, a(n) for n = 1..681
Gus Wiseman, Comcategories and Multiorders, (pdf version)

Cf. A000040, A056768, A000607, A100118, A196545, A273873.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=2, 0,
       b(n, prevprime(i)))+`if`(i>n, 0, b(n-i, i)*(1+
      `if`(i>2, b(i, prevprime(i)), 0))))
    end:
a:= n-> `if`(n<3, 1, 1+b(ithprime(n), ithprime(n-1))):
seq(a(n), n=1..40);  # Alois P. Heinz, Sep 15 2016

n=20;
ser=Product[1/(1-c[Prime[i]]*x^Prime[i]),{i,1,n}];
sys=Table[c[Prime[i]]==Expand[SeriesCoefficient[ser,{x,0,Prime[i]}]-c[Prime[i]]+1],{i,1,n}];
Block[{c},Set@@@sys]

A343016 a(n) is the least nonnegative m such that m*n + A001414(n) is not prime.

Original entry on oeis.org

0, 1, 1, 0, 1, 5, 1, 0, 0, 2, 1, 4, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 3, 0, 0, 0, 1, 1, 1, 0, 1, 0, 2, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 6, 0, 0, 0, 0, 0, 2, 1, 0, 1

Offset: 1

J. M. Bergot and Robert Israel, Apr 02 2021

nonn

a(n) <= A001414(n).
a(n) = 0 iff n is not in A100118.
a(n) = 1 if n is prime.
a(n) >= 3 iff n is in A342302.

			a(6) = 5 because A001414(6) = 5 and 5, 6+5=11, 2*6+5=17, 3*6+5=23, and 4*6+5=29 are prime but 5*6+5=35 is not.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001414, A100118, A342302.

f:= proc(n) local s, t, k;
  s:= add(t[1]*t[2], t = ifactors(n)[2]);
  for k from 0 do if not isprime(k*n+s) then return k fi od;
end proc:
map(f, [$1..100]);

Array[Block[{m = 0, k = Plus @@ Times @@@ FactorInteger[#]}, While[PrimeQ[# m + k], m++]; m] &, 105] (* Michael De Vlieger, Apr 13 2021 *)

A134335 Numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is an integer, but not a prime.

Original entry on oeis.org

15, 35, 39, 42, 50, 51, 55, 65, 77, 78, 87, 91, 92, 95, 110, 111, 114, 115, 119, 123, 140, 141, 143, 155, 159, 161, 164, 170, 183, 185, 186, 187, 189, 201, 203, 204, 209, 215, 219, 221, 222, 225, 230, 235, 236, 242, 247, 258, 259, 264, 267, 284, 285, 287, 290

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

			a(1) = 15, since 15 = 3*5 and (3+5)/2 = 4 is not prime.
a(5) = 50, since 50 = 2*5*5 and (2+5+5)/3 = 4 is not prime.

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

Cf. A000040, A001222, A100118, A046363, A133620, A133621.

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

fp[{a_,b_}]:=a*b;s={};Do[If[q=Total[fp/@FactorInteger[n]]/Total[Last/@FactorInteger[n]];IntegerQ[q]&&!PrimeQ[q],AppendTo[s,n]],{n,2,290}];s (* James C. McMahon, Apr 05 2025 *)

Definition clarified by the author, May 06 2013

A213016 Numbers n such that the sum of prime factors of n (counted with multiplicity) is 3 times a prime.

Original entry on oeis.org

8, 9, 14, 20, 24, 26, 27, 38, 44, 62, 68, 74, 105, 112, 116, 125, 126, 134, 150, 160, 180, 188, 192, 195, 208, 212, 216, 218, 231, 234, 243, 254, 275, 278, 314, 330, 332, 343, 352, 356, 362, 396, 398, 422, 428, 465, 483, 490, 496, 548, 558, 575, 588, 609, 614

Offset: 1

Michel Lagneau, Jun 02 2012

nonn

The numbers A100118(n)^3 are in the sequence.

			44 is in the sequence because 44 = 2^2 * 11 => sum of prime factors = 2*2+11 = 15 = 3*5 where 5 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001414, A100118, A213015.

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)/3,prime)= true then printf(`%d, `,m):else fi:od:

L = {}; Do[ww = Transpose[FactorInteger[k]]; w = ww[[1]].ww[[2]]; If[PrimeQ[w/3], AppendTo[L, k]], {k, 2, 1000}]; L
Select[Range[700],PrimeQ[Total[Times@@@FactorInteger[#]]/3]&] (* Harvey P. Dale, Nov 23 2022 *)

sopfr(n) = 3*p, p prime.

A213020 Smallest number k such that the sum of prime factors of k (counted with multiplicity) is n times a prime.

Original entry on oeis.org

2, 4, 8, 15, 21, 35, 33, 39, 65, 51, 57, 95, 69, 115, 86, 87, 93, 155, 212, 111, 122, 123, 129, 215, 141, 235, 158, 159, 265, 371, 177, 183, 194, 427, 201, 335, 213, 219, 365, 511, 237, 395, 249, 415, 446, 267, 278, 623, 964, 291, 302, 303, 309, 515, 321, 327

Offset: 1

Michel Lagneau, Jun 02 2012

nonn

Smallest k such that sopfr(k) = n*p, p prime.

			a(19) = 212 because 212 = 2^2 * 53 => sum of prime factors = 2*2+53 = 57 = 19*3 where 3 is prime.

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A001414, A100118, A213015, A213016.

sopfr:= proc(n) option remember;
          add(i[1]*i[2], i=ifactors(n)[2])
        end:
a:= proc(n) local k, p;
      for k from 2 while irem (sopfr(k), n, 'p')>0 or
        not isprime(p) do od; k
    end:
seq (a(n), n=1..100); # Alois P. Heinz, Jun 03 2012

sopfr[n_] := Sum[Times @@ f, {f, FactorInteger[n]}];
a[n_] := For[k = 2, True, k++, If[PrimeQ[sopfr[k]/n], Return[k]]];
Array[a, 100] (* Jean-François Alcover, Nov 13 2020 *)

sopfr(n) = my(f=factor(n)); sum(k=1,#f~,f[k,1]*f[k,2]); \\ A001414
isok(k, n) = my(dr = divrem(sopfr(k), n)); (dr[2]==0) && isprime(dr[1]);
a(n) = {my(k=2); while (!isok(k, n), k++); k;} \\ Michel Marcus, Nov 13 2020

A274718 Set x = n. Then a(n) is the number of iterations of successive applications of the map x = A001414(x) that leave x composite, or a(n) = -1 if x always remains composite.

Original entry on oeis.org

-1, 0, 0, -1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 2, 0, 2, 0, 2, 1, 0, 0, 2, 1, 3, 2, 0, 0, 1, 0, 1, 3, 0, 1, 1, 0, 2, 3, 0, 0, 1, 0, 3, 0, 2, 0, 0, 3, 1, 3, 0, 0, 0, 3, 0, 1, 0, 0, 1, 0, 4, 0, 1, 3, 3, 0, 2, 4, 3, 0, 1, 0, 4, 0, 0, 3, 3, 0, 0, 1, 0, 0, 3, 1, 1, 2, 0, 0, 0, 3, 3, 1, 4, 3, 0, 0, 3, 0, 3, 0, 1, 0, 0, 3

Offset: 1

Felix Fröhlich, Jul 03 2016

sign

a(1) and a(4) are the only terms with a value of -1.

a(n) = 0 iff n is a term of A100118.

			For n = 26: A001414(26) = 15, A001414(15) = 8, A001414(8) = 6 and A001414(6) = 5. 5 is prime and so 26 remains composite through 3 iterations of the map given in the definition, therefore a(26) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001414, A100118.

lim = 10^4; Table[Length@ NestWhileList[If[# == 1, 0, Total@ Flatten[Table[#1, {#2}] & @@@ FactorInteger@ #]] &, n, ! PrimeQ@ # &, 1, lim] - 2 /. {-1 -> 0, lim - 1 -> -1}, {n, 86}] (* Michael De Vlieger, Jul 03 2016 *)

sopfr(n) = my(f=factor(n)); sum(i=1, #f[, 1], f[i, 1]*f[i, 2]) /* after Charles R Greathouse IV in A050703 */
a(n) = my(i=0, s=sopfr(n)); while(1, if(ispseudoprime(s), return(i)); if(s==sopfr(s), return(-1)); s=sopfr(s); i++)

More terms from Antti Karttunen, Mar 07 2018

A316228 Numbers whose Fermi-Dirac prime factorization sums to a Fermi-Dirac prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 28, 29, 31, 34, 36, 37, 39, 40, 41, 43, 46, 47, 48, 49, 52, 53, 55, 56, 58, 59, 61, 63, 66, 67, 71, 73, 76, 79, 81, 82, 83, 88, 89, 90, 94, 97, 100, 101, 103, 104, 107, 108, 109, 112

Offset: 1

Gus Wiseman, Jun 27 2018

nonn

A Fermi-Dirac prime (A050376) is a number of the form p^(2^k) where p is prime and k >= 0. Every positive integer has a unique factorization into distinct Fermi-Dirac primes.

			Sequence of multiarrows in the form "number: sum <= factors" begins:
   2:  2 <= {2}
   3:  3 <= {3}
   4:  4 <= {4}
   5:  5 <= {5}
   6:  5 <= {2,3}
   7:  7 <= {7}
   9:  9 <= {9}
  10:  7 <= {2,5}
  11: 11 <= {11}
  12:  7 <= {3,4}
  13: 13 <= {13}
  14:  9 <= {2,7}
  16: 16 <= {16}
  17: 17 <= {17}
  18: 11 <= {2,9}
  19: 19 <= {19}
  20:  9 <= {4,5}
  22: 13 <= {2,11}
  23: 23 <= {23}
  24:  9 <= {2,3,4}

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

Cf. A050376, A064547, A100118, A213925, A299757, A305829, A316202, A316210, A316211, A316220.

FDfactor[n_]:=If[n==1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
Select[Range[2,200],Length[FDfactor[Total[FDfactor[#]]]]==1&]

A316525 Numbers whose average of prime factors is prime.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 20, 21, 23, 25, 27, 29, 31, 32, 33, 37, 41, 43, 44, 47, 49, 53, 57, 59, 60, 61, 64, 67, 68, 69, 71, 73, 79, 81, 83, 85, 89, 93, 97, 101, 103, 105, 107, 109, 112, 113, 116, 121, 125, 127, 128, 129, 131, 133, 137, 139

Offset: 1

Gus Wiseman, Jul 05 2018

nonn

Prime factors counted with multiplicity. - Harvey P. Dale, Sep 28 2018

			60 = 2*2*3*5 has average of prime factors (2+2+3+5)/4 = 3, which is prime, so 60 belongs to the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Subsequence of A078175.

Cf. A000040, A000607, A056768, A071321, A100118, A316219, A316313.

Select[Range[100],PrimeQ[Mean[If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[p,{k}]]]]]]&]
Select[Range[200],PrimeQ[Mean[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ #]]]]&] (* Harvey P. Dale, Sep 28 2018 *)

isok(n) = {my(f=factor(n)); iferr(isprime(sum(k=1, #f~, f[k,1]*f[k,2])/sum(k=1, #f~, f[k,2])), E, 0);} \\ Michel Marcus, Jul 06 2018

A337047 Numbers k such that A001414(k) and A001414(A004086(k)) are twin primes p, p+2.

Original entry on oeis.org

405, 412, 850, 25315, 49419, 50127, 224315, 293394, 308700, 697136, 801350, 811910, 997425, 1118520, 1152000, 1177250, 1550520, 1659350, 1725332, 1739640, 1824500, 1976895, 2141150, 2580640, 2580831, 3530466, 3718376, 4050405, 4459455, 4536532, 4577732, 4832796, 5173100, 5510287, 5601570, 5603989, 5609439

Offset: 1

J. M. Bergot and Robert Israel, Aug 12 2020

			a(3)=850 is in the sequence because A001414(850)=2+5+5+17=29, A001414(58)=2+29=31, and (29,31) is a pair of twin primes.

Robert Israel, Table of n, a(n) for n = 1..150

Cf. A001097, A001414, A004086. Subsequence of A100118.

revdigs:= proc(n) local L,k;
  L:= convert(n,base,10);
  add(L[-k]*10^(k-1),k=1..nops(L))
end proc:
filter:= proc(n) local a,b;
  a:= convert(map(convert,ifactors(n)[2],`*`),`+`);
  if not isprime(a) then return false fi;
  b:= convert(map(convert,ifactors(revdigs(n))[2],`*`),`+`);
  b = a+2 and isprime(b)
end proc:
select(filter, [$1 .. 10^7]);

cp's OEIS Frontend

A276511 Primes that are equal to the sum of the prime factors of some perfect number.

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A276687 Number of prime plane trees of weight prime(n).

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A343016 a(n) is the least nonnegative m such that m*n + A001414(n) is not prime.

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A134335 Numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is an integer, but not a prime.

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Mathematica

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A213016 Numbers n such that the sum of prime factors of n (counted with multiplicity) is 3 times a prime.

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A213020 Smallest number k such that the sum of prime factors of k (counted with multiplicity) is n times a prime.

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A274718 Set x = n. Then a(n) is the number of iterations of successive applications of the map x = A001414(x) that leave x composite, or a(n) = -1 if x always remains composite.

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