A088685 -id:A088685 - cp's OEIS frontend

A052147 a(n) = prime(n) + 2.

4, 5, 7, 9, 13, 15, 19, 21, 25, 31, 33, 39, 43, 45, 49, 55, 61, 63, 69, 73, 75, 81, 85, 91, 99, 103, 105, 109, 111, 115, 129, 133, 139, 141, 151, 153, 159, 165, 169, 175, 181, 183, 193, 195, 199, 201, 213, 225, 229, 231, 235, 241, 243, 253, 259

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk), Jan 24 2000

A048974, A052147, A067187 and A088685 are very similar after dropping terms less than 13. - Eric W. Weisstein, Oct 10 2003
A117530(n,2) = a(n) for n>1. - Reinhard Zumkeller, Mar 26 2006
a(n) = A000040(n) + 2 = A008864(n) + 1 = A113395(n) - 1 = A175221(n) - 2 = A175222(n) - 3 = A139049(n) - 4 = A175223(n) - 5 = A175224(n) - 6 = A140353(n) - 7 = A175225(n) - 8. - Jaroslav Krizek, Mar 06 2010
Left edge of the triangle in A065342. - Reinhard Zumkeller, Jan 30 2012
Union of A006512 and A107986. - David James Sycamore, Jul 08 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

A139690 is a subsequence.

Cf. A040976, A000040.

Filtered([1..300], k-> IsPrime(k) ) + 2; # G. C. Greubel, May 20 2019

a052147 = (+ 2) . a000040  -- Reinhard Zumkeller, Jul 03 2015

[p+2: p in PrimesUpTo(400)]; // Vincenzo Librandi, Nov 27 2013

seq(ithprime(n)+2,n=1..55); # Muniru A Asiru, Jul 08 2018

Prime[Range[70]]+2 (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

a(n)=prime(n)+2 \\ Charles R Greathouse IV, Jan 19 2017

[nth_prime(n) +2 for n in (1..70)] # G. C. Greubel, May 20 2019

A067187 Numbers that can be expressed as the sum of two primes in exactly one way.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 12, 13, 15, 19, 21, 25, 31, 33, 39, 43, 45, 49, 55, 61, 63, 69, 73, 75, 81, 85, 91, 99, 103, 105, 109, 111, 115, 129, 133, 139, 141, 151, 153, 159, 165, 169, 175, 181, 183, 193, 195, 199, 201, 213, 225, 229, 231, 235, 241, 243, 253, 259, 265, 271

Offset: 1

Amarnath Murthy, Jan 10 2002

nonn

All primes + 2 are terms of this sequence. Is 12 the last even term? - Frank Ellermann, Jan 17 2002
A048974, A052147, A067187 and A088685 are very similar after dropping terms less than 13. - Eric W. Weisstein, Oct 10 2003
Values of n such that A061358(n)=1. - Emeric Deutsch, Apr 03 2006

			4 is a term as 4 = 2+2, 15 is a term as 15 = 13+2.

Robert Price, Table of n, a(n) for n = 1..5002
Index entries for sequences related to Goldbach conjecture

Cf. A023036, A045917, A061358.
Subsequence of A014091.
Numbers that can be expressed as the sum of two primes in k ways for k=0..10: A014092 (k=0), this sequence (k=1), A067188 (k=2), A067189 (k=3), A067190 (k=4), A067191 (k=5), A066722 (k=6), A352229 (k=7), A352230 (k=8), A352231 (k=9), A352233 (k=10).

g:=sum(sum(x^(ithprime(i)+ithprime(j)),i=1..j),j=1..80): gser:=series(g,x=0,280): a:=proc(n) if coeff(gser,x^n)=1 then n else fi end: seq(a(n),n=1..272); # Emeric Deutsch, Apr 03 2006

cQ[n_]:=Module[{c=0},Do[If[PrimeQ[n-i]&&PrimeQ[i],c++],{i,2,n/2}]; c==1]; Select[Range[4,271],cQ[#]&] (* Jayanta Basu, May 22 2013 *)
y = Select[Flatten@Table[Prime[i] + Prime[j], {i, 60}, {j, 1, i}], # < Prime[60] &]; Select[Union[y], Count[y, #] == 1 &] (* Robert Price, Apr 21 2025 *)

Edited by Frank Ellermann, Jan 17 2002

A288814 a(n) is the smallest composite number whose prime divisors (with multiplicity) sum to n.

Original entry on oeis.org

4, 6, 8, 10, 15, 14, 21, 28, 35, 22, 33, 26, 39, 52, 65, 34, 51, 38, 57, 76, 95, 46, 69, 92, 115, 184, 161, 58, 87, 62, 93, 124, 155, 248, 217, 74, 111, 148, 185, 82, 123, 86, 129, 172, 215, 94, 141, 188, 235, 376, 329, 106, 159, 212, 265, 424, 371, 118, 177, 122, 183, 244, 305, 488, 427, 134, 201, 268, 335, 142

Offset: 4

David James Sycamore, Jun 16 2017

nonn

Agrees with A056240(n) if n is composite (but not if n is prime).
For n prime, let P_n = greatest prime < n such that A056240(n-P_n) = A288313(m) for some m; then a(n) = Min{q*a(n-q): q prime, n-1 > q >= P_n}.
In most cases q is the greatest prime < p, but there are exceptions; e.g., p=211 is the smallest prime for which q (=197) is the second prime removed from 211, not the first. 541 is the next prime with this property (q=521). The same applies to p=16183, for which q=16139, the second prime removed from p. These examples all arise with q being the lesser of a prime pair.
For p prime, a(p) = q*a(p-q) for some prime q < p as described above. Then a(p-q) = 2,4,8 or 3*r for some prime r.
The subsequence of terms (4, 6, 8, 10, 14, 21, 22, 26, 34, ...), where for all m > n, a(m) > a(n) is the same as sequence A088686, and the sequence of its indices (4, 5, 6, 7, 9, 10, 13, 19, ...) is the same as A088685. - David James Sycamore, Jun 30 2017
Records are in A088685. - Robert G. Wilson v, Feb 26 2018
Number of terms less than 10^k, k=1,2,3,...: 3, 32, 246, 2046, 17053, 147488, ..., . - Robert G. Wilson v, Feb 26 2018

			a(5) = 6 = 2*3 is the smallest composite number whose prime divisors add to 5.
a(7) = 10 = 2*5 is the smallest composite number whose prime divisors add to 7.
12 = 2 * 2 * 3 is not in the sequence, since the sum of its prime divisors is 7, a value already obtained by the lesser 10. - _David A. Corneth_, Jun 22 2017

David A. Corneth, Table of n, a(n) for n = 4..10003 (terms 4..1000 from Michel Marcus)
David A. Corneth, PARI program

Cf. A046343, A056240, A088685, A288313.

N:= 100: # to get a(4)..a(N)
V:= Array(4..N): count:= 0:
for k from 4 while count < N-3 do
  if isprime(k) then next fi;
  s:= add(t[1]*t[2], t = ifactors(k)[2]);
if s <= N and V[s]=0 then
    V[s]:= k; count:= count+1;
fi
od:
convert(V,list); # Robert Israel, Feb 26 2018
# alternative
A288814 := proc(n)
    local k ;
    for k from 1 do
        if not isprime(k) and A001414(k) = n then
            return k ;
        end if;
    end do:
end proc:
seq(A288814(n),n=4..80) ; # R. J. Mathar, Apr 15 2024

Function[s, Table[FirstPosition[s, ?(# == n &)][[1]], {n, 4, 73}]]@ Table[Boole[CompositeQ@ n] Total@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger[n]], {n, 10^3}] (* _Michael De Vlieger, Jun 19 2017 *)
f[n_] := If[ PrimeQ@ n, 0, spf = Plus @@ Flatten[ Table[#1, {#2}] & @@@ FactorInteger@ n]]; t[] := 0; k = 1; While[k < 500, If[ t[f[k]] == 0, t[f[k]] = k]; k++]; t@# & /@ Range[4, 73] (* _Robert G. Wilson v, Feb 26 2018 *)

isok(k, n) = my(f=factor(k)); sum(j=1, #f~, f[j,1]*f[j,2]) == n;
a(n) = forcomposite(k=1,, if (isok(k, n), return(k))); \\ Michel Marcus, Jun 21 2017

lista(n) = {my(res = vector(n), s, todo); if(n < 4, return([]), todo = n-3); forcomposite(k=4, , f=factor(k); s = sum(j=1, #f~, f[j, 1]*f[j, 2]); if(s<=n, if(res[s]==0, res[s]=k; todo--; if(todo==0, return(vector(n-3, i, res[i+3]))))))} \\ David A. Corneth, Jun 21 2017

See PARI-link \\ David A. Corneth, Mar 23 2018

A048974 Odd numbers that are the sum of 2 primes.

Original entry on oeis.org

5, 7, 9, 13, 15, 19, 21, 25, 31, 33, 39, 43, 45, 49, 55, 61, 63, 69, 73, 75, 81, 85, 91, 99, 103, 105, 109, 111, 115, 129, 133, 139, 141, 151, 153, 159, 165, 169, 175, 181, 183, 193, 195, 199, 201, 213, 225, 229, 231, 235, 241, 243, 253, 259

Offset: 1

N. J. A. Sloane

A048974, A052147, A067187 and A088685 are very similar after dropping terms less than 13. - Eric W. Weisstein, Oct 10 2003

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

Cf. A014092, A065091.

Select[Flatten@Table[Prime[i] + Prime[j], {i, 100}, {j, 1, i}], # < Prime[100] && OddQ[#] &] (* Robert Price, Apr 21 2025 *)

One of the primes must be 2, so this is simply the odd primes + 2.

a(n) = A065091(n) + 2. - Sean A. Irvine, Jul 15 2021

A088686 Positions of the records in the sum-of-primes function sopfr(n) if sopfr(prime) is taken to be 0.

Original entry on oeis.org

1, 4, 6, 8, 10, 14, 21, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482

Offset: 1

Eric W. Weisstein, Oct 05 2003

nonn

Indranil Ghosh, Table of n, a(n) for n = 1..2765 (terms < 50000)
Eric Weisstein's World of Mathematics, Sum of Prime Factors

Cf. A001414, A088685.

Function[s, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]] ]@ Table[Total@ Flatten@ Map[ConstantArray[#1, #2] /. 1 -> 0 & @@ # &, FactorInteger@ n] - n Boole[PrimeQ@ n], {n, 500}] (* Michael De Vlieger, Jun 29 2017 *)

sopfr(k) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]);
lista(nn) = {my(record = -1); for (n=1, nn, if (! isprime(n), if ((x=sopfr(n)) > record, record = x; print1(n, ", "));););} \\ Michel Marcus, Jun 29 2017

from sympy import factorint, isprime
def sopfr(n):
    f=factorint(n)
    return sum([i*f[i] for i in f])
l=[]
record=-1
for n in range(1, 501):
    if not isprime(n):
        x=sopfr(n)
        if x>record:
            record=x
            l.append(n)
print(l) # Indranil Ghosh, Jun 29 2017

cp's OEIS Frontend

A052147 a(n) = prime(n) + 2.

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A067187 Numbers that can be expressed as the sum of two primes in exactly one way.

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A288814 a(n) is the smallest composite number whose prime divisors (with multiplicity) sum to n.

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Maple

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A048974 Odd numbers that are the sum of 2 primes.

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A088686 Positions of the records in the sum-of-primes function sopfr(n) if sopfr(prime) is taken to be 0.

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