A051034 -id:A051034 - cp's OEIS frontend

A024935 Duplicate of A051034.

1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 3, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3

Offset: 2

Clark Kimberling

a(2)-a(6) prepended by Sean A. Irvine, Jul 29 2019

A174799 Partial sums of A051034.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 12, 14, 15, 17, 18, 20, 22, 24, 25, 27, 28, 30, 32, 34, 35, 37, 39, 41, 44, 46, 47, 49, 50, 52, 54, 56, 59, 61, 62, 64, 66, 68, 69, 71, 72, 74, 76, 78, 79, 81, 83, 85, 88, 90, 91, 93, 95, 97, 100, 102, 103, 105, 106, 108, 110, 112, 115, 117, 118, 120, 122, 124, 125, 127, 128, 130, 132, 134, 137, 139, 140, 142, 144, 146, 147, 149, 151, 153, 156, 158, 159, 161, 163, 165, 168, 170, 173, 175, 176, 178, 180, 182, 183, 185, 186, 188, 190, 192, 193, 195, 196, 198, 200, 202

Offset: 2

Jonathan Vos Post, Nov 30 2010

Cf. A051034.

a(n) = Sum_{i=2..n} A051034(i).

A000607 Number of partitions of n into prime parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 30, 35, 40, 46, 52, 60, 67, 77, 87, 98, 111, 124, 140, 157, 175, 197, 219, 244, 272, 302, 336, 372, 413, 456, 504, 557, 614, 677, 744, 819, 899, 987, 1083, 1186, 1298, 1420, 1552, 1695, 1850, 2018, 2198, 2394, 2605, 2833, 3079, 3344

Offset: 0

N. J. A. Sloane

a(n) gives the number of values of k such that A001414(k) = n. - Howard A. Landman, Sep 25 2001
Let W(n) = {prime p: There is at least one number m whose spf is p, and sopfr(m) = n}. Let V(n,p) = {m: sopfr(m) = n, p belongs to W(n)}. Then a(n) = sigma(|V(n,p)|). E.g.: W(10) = {2,3,5}, V(10,2) = {30,32,36}, V(10,3) = {21}, V(10,5) = {25}, so a(10) = 3+1+1 = 5. - David James Sycamore, Apr 14 2018
From Gus Wiseman, Jan 18 2020: (Start)
Also the number of integer partitions such that the sum of primes indexed by the parts is n. For example, the sum of primes indexed by the parts of the partition (3,2,1,1) is prime(3)+prime(2)+prime(1)+prime(1) = 12, so (3,2,1,1) is counted under a(12). The a(2) = 1 through a(14) = 10 partitions are:
  1  2  11  3   22   4    32    41    33     5      43      6       44
            21  111  31   221   222   42     322    331     51      52
                     211  1111  311   321    411    421     332     431
                                2111  2211   2221   2222    422     3222
                                      11111  3111   3211    3221    3311
                                             21111  22111   4111    4211
                                                    111111  22211   22221
                                                            31111   32111
                                                            211111  221111
                                                                    1111111
(End)

			n = 10 has a(10) = 5 partitions into prime parts: 10 = 2 + 2 + 2 + 2 + 2 = 2 + 2 + 3 + 3 = 2 + 3 + 5 = 3 + 7 = 5 + 5.
n = 15 has a(15) = 12 partitions into prime parts: 15 = 2 + 2 + 2 + 2 + 2 + 2 + 3 = 2 + 2 + 2 + 3 + 3 + 3 = 2 + 2 + 2 + 2 + 2 + 5 = 2 + 2 + 2 + 2 + 7 = 2 + 2 + 3 + 3 + 5 = 2 + 3 + 5 + 5 = 2 + 3 + 3 + 7 = 2 + 2 + 11 = 2 + 13 = 3 + 3 + 3 + 3 + 3 = 3 + 5 + 7 = 5 + 5 + 5.

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; see p. 203.
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
B. C. Berndt and B. M. Wilson, Chapter 5 of Ramanujan's second notebook, pp. 49-78 of Analytic Number Theory (Philadelphia, 1980), Lect. Notes Math. 899, 1981, see Entry 29.
D. M. Burton, Elementary Number Theory, 5th ed., McGraw-Hill, 2002.
L. M. Chawla and S. A. Shad, On a trio-set of partition functions and their tables, J. Natural Sciences and Mathematics, 9 (1969), 87-96.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Hans Havermann, Table of n, a(n) for n = 0..50000 (first 1000 terms from T. D. Noe, terms 1001-20000 from Vaclav Kotesovec, terms 20001-50000 extracted from files by Hans Havermann)
K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294.
George E. Andrews, Arnold Knopfmacher, and John Knopfmacher, Engel expansions and the Rogers-Ramanujan identities, J. Number Theory 80 (2000), 273-290. See Eq. 2.1.
George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the Number of Distinct Multinomial Coefficients, Journal of Number Theory, Volume 118, Issue 1, May 2006, pp. 15-30.
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Johann Bartel, R. K. Bhaduri, Matthias Brack, and M. V. N. Murthy, Asymptotic prime partitions of integers, Phys. Rev. E, 95 (2017), 052108, arXiv:1609.06497 [math-ph], 2016-2017.
P. T. Bateman and P. Erdős, Partitions into primes, Publ. Math. Debrecen 4 (1956), 198-200.
J. Browkin, Sur les décompositions des nombres naturels en sommes de nombres premiers, Colloquium Mathematicum 5 (1958), 205-207.
Edward A. Bender, Asymptotic methods in enumeration, SIAM Review 16 (1974), no. 4, p. 509.
L. M. Chawla and S. A. Shad, Review of "On a trio-set of partition functions and their tables", Mathematics of Computation, Vol. 24, No. 110 (Apr., 1970), pp. 490-491.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see Section VIII.6, pp. 576ff.
H. Gupta, Partitions into distinct primes, Proc. Nat. Acad. Sci. India, 21 (1955), 185-187.
O. P. Gupta and S. Luthra, Partitions into primes, Proc. Nat. Inst. Sci. India. Part A. 21 (1955), 181-184.
R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12. (annotated scanned copy)
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
H. Havermann, Tables of sum-of-prime-factors sequences (including sequence lengths, i.e., number of prime-parts partitions, for the first 50000).
BongJu Kim, Partition number identities which are true for all set of parts, arXiv:1803.08095 [math.CO], 2018.
Vaclav Kotesovec, Graph - asymptotic ratio for log(a(n)), without minor asymptotic term.
Vaclav Kotesovec, Wrong asymptotics of OEIS A000607? (MathOverflow). Includes discussion of the contradiction between the results for the next-to-leading term in the asymptotic formulas by Vaughan and by Bartel et al.
John F. Loase (splurge(AT)aol.com), David Lansing, Cassie Hryczaniuk and Jamie Cahoon, A Variant of the Partition Function, College Mathematics Journal, Vol. 36, No. 4 (Sep 2005), pp. 320-321.
Ljuben Mutafchiev, A Note on Goldbach Partitions of Large Even Integers, arXiv:1407.4688 [math.NT], 2014-2015.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
N. J. A. Sloane, Transforms.
R. C. Vaughan, On the number of partitions into primes, Ramanujan J. vol. 15, no. 1 (2008) 109-121.
Eric Weisstein's World of Mathematics, Prime Partition.
Roger Woodford, Bounds for the Eventual Positivity of Difference Functions of Partitions, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.3.
Index entries for sequences related to Goldbach conjecture

G.f. = 1 / g.f. for A046675. See A046113 for the ordered (compositions) version.
Cf. A046676, A048165, A004526, A051034, A000040, A001414, A000586 (distinct parts), A000041, A070214, A192541, A112021, A056768, A128515, A319265, A319266.
Row sums of array A116865 and of triangle A261013.
Column sums of A331416.
Partitions whose Heinz number is divisible by their sum of primes are A330953.
Partitions of whose sum of primes is divisible by their sum are A331379.
Cf. A000720, A014689, A331385, A331387, A331415, A331417.

a000607 = p a000040_list where
   p _      0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 05 2012

[1] cat [#RestrictedPartitions(n,{p:p in PrimesUpTo(n)}): n in [1..100]]; // Marius A. Burtea, Jan 02 2019

with(gfun):
t1:=mul(1/(1-q^ithprime(n)),n=1..51):
t2:=series(t1,q,50):
t3:=seriestolist(t2); # fixed by Vaclav Kotesovec, Sep 14 2014

CoefficientList[ Series[1/Product[1 - x^Prime[i], {i, 1, 50}], {x, 0, 50}], x]
f[n_] := Length@ IntegerPartitions[n, All, Prime@ Range@ PrimePi@ n]; Array[f, 57] (* Robert G. Wilson v, Jul 23 2010 *)
Table[Length[Select[IntegerPartitions[n],And@@PrimeQ/@#&]],{n,0,60}] (* Harvey P. Dale, Apr 22 2012 *)
a[n_] := a[n] = If[PrimeQ[n], 1, 0]; c[n_] := c[n] = Plus @@ Map[# a[#] &, Divisors[n]]; b[n_] := b[n] = (c[n] + Sum[c[k] b[n - k], {k, 1, n - 1}])/n; Table[b[n], {n, 1, 20}] (* Thomas Vogler, Dec 10 2015: Uses Euler transform, caches computed values, faster than IntegerPartitions[] function. *)
nmax = 100; pmax = PrimePi[nmax]; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; poly[[3]] = -1; Do[p = Prime[k]; Do[poly[[j + 1]] -= poly[[j + 1 - p]], {j, nmax, p, -1}];, {k, 2, pmax}]; s = Sum[poly[[k + 1]]*x^k, {k, 0, Length[poly] - 1}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 11 2021 *)

N=66;x='x+O('x^N); Vec(1/prod(k=1,N,1-x^prime(k))) \\ Joerg Arndt, Sep 04 2014

from sympy import primefactors
l = [1, 0]
for n in range(2, 101):
    l.append(sum(sum(primefactors(k)) * l[n - k] for k in range(1, n + 1)) // n)
l  # Indranil Ghosh, Jul 13 2017

[Partitions(n, parts_in=prime_range(n + 1)).cardinality() for n in range(100)]  # Giuseppe Coppoletta, Jul 11 2016

Asymptotically a(n) ~ exp(2 Pi sqrt(n/log n) / sqrt(3)) (Ayoub).
a(n) = (1/n)*Sum_{k=1..n} A008472(k)*a(n-k). - Vladeta Jovovic, Aug 27 2002
G.f.: 1/Product_{k>=1} (1-x^prime(k)).
See the partition arrays A116864 and A116865.
From Vaclav Kotesovec, Sep 15 2014 [Corrected by Andrey Zabolotskiy, May 26 2017]: (Start)
It is surprising that the ratio of the formula for log(a(n)) to the approximation 2 * Pi * sqrt(n/(3*log(n))) exceeds 1. For n=20000 the ratio is 1.00953, and for n=50000 (using the value from Havermann's tables) the ratio is 1.02458, so the ratio is increasing. See graph above.
A more refined asymptotic formula is found by Vaughan in Ramanujan J. 15 (2008), pp. 109-121, and corrected by Bartel et al. (2017): log(a(n)) = 2*Pi*sqrt(n/(3*log(n))) * (1 - log(log(n))/(2*log(n)) + O(1/log(n))).
See Bartel, Bhaduri, Brack, Murthy (2017) for a more complete asymptotic expansion. (End)
G.f.: 1 + Sum_{i>=1} x^prime(i) / Product_{j=1..i} (1 - x^prime(j)). - Ilya Gutkovskiy, May 07 2017
a(n) = A184198(n) + A184199(n). - Vaclav Kotesovec, Jan 11 2021

A025583 Composite numbers that are not the sum of 2 primes.

Original entry on oeis.org

27, 35, 51, 57, 65, 77, 87, 93, 95, 117, 119, 121, 123, 125, 135, 143, 145, 147, 155, 161, 171, 177, 185, 187, 189, 203, 205, 207, 209, 215, 217, 219, 221, 237, 245, 247, 249, 255, 261, 267, 275, 287, 289, 291, 297, 299, 301, 303, 305, 321, 323, 325, 327, 329, 335, 341

Offset: 1

N. J. A. Sloane

Goldbach conjectured that every integer > 5 is the sum of three primes.
Conjecture: This is the sequence of odd numbers k such that (k mod x) mod 2 != 1, where x is the greatest m <= k such that m, m-1 and m-2 are all composite. Verified for first 10000 terms. - Benedict W. J. Irwin, May 06 2016
Numbers k, such that however many of k coins are placed with heads rather than tails showing, either those showing heads or those showing tails can be arranged in a rectangular pattern with multiple rows and columns. (If the Goldbach conjecture for even numbers is false this comment should be restricted to the odd terms of this sequence, as it might otherwise define a variant sequence). - Peter Munn, May 15 2017

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Partition
Eric Weisstein's World of Mathematics, Twin Composites
Index entries for sequences related to Goldbach conjecture

Cf. A051034, A001031, A002372, A002374, A071335.

Cf. A002808, A000040, A010051.

a025583 n = a025583_list !! (n-1)
a025583_list = filter f a002808_list where
   f x = all (== 0) $ map (a010051 . (x -)) $ takeWhile (< x) a000040_list
-- Reinhard Zumkeller, Oct 15 2014

f[n_] := (p = 0; pn = PrimePi[n]; Do[ If[n == Prime[i] + Prime[k], p = p + 1; If[p > 2, Break[]]], {i, 1, pn}, {k, i, pn}]; p ); Select[Range[2, 400], ! PrimeQ[#] && f[#] == 0 & ] (* Jean-François Alcover, Mar 07 2011 *)
upto=350;With[{c=PrimePi[upto]},Complement[Range[4,upto], Prime[Range[ c]], Union[Total/@Tuples[Prime[Range[c]],{2}]]]] (* Harvey P. Dale, Jul 14 2011 *)
Select[Range[400],CompositeQ[#]&&Count[IntegerPartitions[#,{2}],?(AllTrue[ #,PrimeQ]&)]==0&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Feb 21 2021 *)

A330507 a(n) is the smallest number k having for every prime p <= prime(n) at least one prime partition with least part p, and no such partition having least part > prime(n). If no such k exists then a(n) = 0 (see comments).

Original entry on oeis.org

2, 6, 10, 18, 24, 30, 51, 57, 69, 60, 99, 111, 123, 143, 147, 159, 177, 189, 201, 213, 225, 245, 255, 267, 291, 303, 309, 321, 345, 357, 381, 393, 411, 427, 447, 465, 471, 493, 507, 519, 537, 553, 573, 583, 623, 621, 633, 669, 681, 695, 707, 729, 749, 753, 783

Offset: 1

David James Sycamore, Mar 01 2020

nonn

Alternatively, a(n) is the smallest number whose product of distinct least part primes from all partitions of n into prime parts, is equal to primorial(n).
2 is the only prime term.
a(n) = 0 for n = 90, 151, 349, 352, 444, ... . - Alois P. Heinz, Mar 12 2020

			a(1) = 2 because [2] is the only prime partition of prime(1) = 2.
a(2) = 6 because [2,2,2] and [3,3] are the only possible prime partitions of 6, namely with prime(1) and prime(2) the only least parts.

Alois P. Heinz, Table of n, a(n) for n = 1..500
Michael De Vlieger, Labeled plot of p at (pi(p), n), for 2 <= n <= 240, with k in a(n) shown in red.

Cf. A000040, A051034, A331634, A332861, A002110, A333238.

b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
      add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
    end:
a:= proc(n) option remember; local f, k, p; p:= ithprime(n);
      for k to 4*p do f:= b(k, 2, x); if degree(f)<= p and andmap(
        h->0Alois P. Heinz, Mar 12 2020

With[{s = Array[Union@ Select[IntegerPartitions[#], AllTrue[#, PrimeQ] &][[All, -1]] &, 70]}, TakeWhile[Map[FirstPosition[s, #][[1]] &, Rest@ NestList[Append[#, Prime[Length@ # + 1]] &, {}, 12]], IntegerQ]] (* Michael De Vlieger, Mar 06 2020 *)
(* Second program: *)
Block[{a, m = 125, s}, a = ConstantArray[{}, m]; s = {Prime@ PrimePi@ m}; Do[If[# <= m, If[FreeQ[a[[#]], Last@ s], a = ReplacePart[a, # -> Union@ Append[a[[#]], Last@ s]], Nothing]; AppendTo[s, Last@ s], If[Last@ s == 2, s = DeleteCases[s, 2]; If[Length@ s == 0, Break[], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]]] &@ Total[s], {i, Infinity}]; TakeWhile[Map[FirstPosition[a, #][[1]] &, Rest@ NestList[Append[#, Prime[Length@ # + 1]] &, {}, Max[Length /@ a]]], IntegerQ]] (* Michael De Vlieger, Mar 11 2020 *)

a(19)-a(21) from Michael De Vlieger, Mar 12 2020

a(22)-a(55) from Alois P. Heinz, Mar 12 2020

A072491 Define f(1) = 0. For n>=2, let f(n) = n - p where p is the largest prime <= n. a(n) = number of iterations of f to reach 0, starting from n.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 3, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2

Offset: 0

Amarnath Murthy, Jul 14 2002

a(p)=1, a(p+1)=2 and a(p+4)=3 if p is an odd prime but p+2 and p+4 are composite.

Number of noncomposites (A008578) needed to sum to n using the greedy algorithm. - Antti Karttunen, Aug 09 2015

			a(27)=3 as f(27)=27-23=4, f(4)=4-3=1 and f(1)=0.

S. S. Pillai, "An arithmetical function concerning primes", Annamalai University Journal (1930), pp. 159-167.

Antti Karttunen, Table of n, a(n) for n = 0..10007

Cf. A008578, A072492. A066352(n) is the smallest k such that a(k)=n.

Not the same as A051034: a(122) = 3, but A051034(122) = 2.

f[1]=0; f[n_] := n-Prime[PrimePi[n]]; a[n_] := Module[{k, x}, For[k=0; x=n, x>0, k++; x=f[x], Null]; k]

a(n)=if(n<4,n>0,1+a(n-precprime(n))) \\ Charles R Greathouse IV, Feb 04 2013

On Cramér's conjecture, a(n) = O(log* n). - Charles R Greathouse IV, Feb 04 2013

Edited by Dean Hickerson, Nov 26 2002

a(0) = 0 prepended by Antti Karttunen, Aug 09 2015

A096436 a(n) = the number of squared primes and 1's needed to sum to n.

Original entry on oeis.org

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

Offset: 1

Tom Raes (tommy1729(AT)hotmail.com), Aug 10 2004

a(n) has a new maximum at n=1,2,3,7,24,73,266,795.

I suspect that a(n) <= 9 for all n. - Robert G. Wilson v, Sep 18 2004

			a(5) = 2 because 5=4+1.
a(17) = 3 because 17=9+4+4.
A number may have many such sums: 27=25+1+1=9+9+9, 50=25+25=49+1.

Nicholas Matteo, Table of n, a(n) for n = 1..10000

Cf. A001248, A002828, A045698, A051034, A063274.

f[n_] := Block[{d = n, k = PrimePi[ Sqrt[n]], sp = {}}, While[d > 3, While[p = Prime[k]; d >= p^2, AppendTo[sp, p]; d = d - p^2]; k-- ]; While[d != 0, AppendTo[sp, 1]; d = d - 1]; If[Position[sp, 3] != {} && sp[[ -3]] == 1, sp = Delete[Drop[sp, -3], Position[sp, 3][[1]]]; AppendTo[sp, {2, 2, 2}]]; Reverse[ Sort[ Flatten[ sp]]]]; Table[ Length[ f[n]], {n, 105}] (* Robert G. Wilson v, Sep 20 2004 *)

Edited and extended by Robert G. Wilson v, Sep 18 2004

Edited by Don Reble, Apr 23 2006

A101422 Minimal number of primes needed to sum to Fibonacci(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 3, 2, 3, 1, 2, 3, 2, 2, 3, 1, 2, 3, 3, 2, 3, 1, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 1, 3, 2, 3, 1, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 1, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 2, 2, 3

Offset: 3

Giovanni Teofilatto, Jan 17 2005

nonn

			a(5)=1 because Fibonacci(5)=5 is a prime.
a(6)=2 because Fibonacci(6)=8 = 3+5.
a(7)=1 because Fibonacci(7)=13 is a prime.
a(14)=3 because Fibonacci(14)=377 = 2+2+373.

Cf. A000045, A051034, A001605, A102106, A102107.

a(n) = A051034(A000045(n)).

Edited and extended by Ray Chandler, Jan 18 2005

A087942 Number of partitions of n into as many primes as n has prime factors.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 7, 1, 3, 7, 3, 1, 2, 1, 11, 1, 4, 0, 15, 1, 2, 1, 21, 1, 3, 1, 4, 12, 4, 1, 26, 1, 5, 0, 4, 1, 33, 1, 38, 0, 4, 1, 41, 1, 3, 19, 137, 0, 5, 1, 6, 1, 2, 1, 61, 1, 5, 22, 5, 0, 5, 1, 67, 24, 5, 1, 81, 1, 5, 0, 96, 1, 93, 1, 9, 0

Offset: 1

Reinhard Zumkeller, Oct 27 2003

nonn

Conjecture, for m>1: a(m)=0 iff n is an odd semiprime such that m-2 is not prime, i.e. m=A089268(k) for some k. - Reinhard Zumkeller, Oct 28 2003

			n=20 = 2*2*5 = 13+5+2 = 11+7+2, all other partitions into 3 primes have fewer than or more than 3 parts, therefore a(20)=2.

Eric Weisstein's World of Mathematics, Prime Partition.
Index entries for sequences related to Goldbach conjecture

Cf. A001222, A000607, A051034, A004526.

Table[Count[IntegerPartitions[n,{PrimeOmega[n]}],?(AllTrue[#,PrimeQ]&)],{n,100}] (* _Harvey P. Dale, Jul 26 2023 *)

A103765 Number of times n can be written as sum of a minimal set of primes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 3, 6, 2, 1, 3, 1, 2, 1, 4, 7, 4, 1, 2, 1, 3, 1, 4, 1, 3, 1, 4, 1, 5, 1, 4, 14, 3, 1, 5, 1, 3, 16, 4, 1, 6, 1, 3, 1, 5, 20, 6, 1, 2, 1, 5, 1, 6, 1, 5, 1, 5, 27, 7, 1, 4, 1, 5, 1, 8, 1, 5, 28, 4, 1, 9, 1, 4, 32, 5, 35, 7, 1, 3, 1, 6, 1, 8, 1

Offset: 2

Reinhard Zumkeller, Mar 29 2005

nonn

A051034 gives the minimal number of primes needed to sum to n;

a(A000040(n)) = 1; a(2*k) = A002375(k) for k>2.

			A051034(25) = 2: a(25) = #{2+23}=1;
A051034(26) = 2: a(26) = #{3+23, 7+19, 13+13} = 3;
A051034(27) = 3: a(27) = #{3+5+19, 3+7+17, 3+11+13, 5+5+17,
5+11+11, 7+7+13} = 6;
A051034(28) = 2: a(28) = #{5+23, 11+17} = 2;
A051034(29) = 1: a(29) = #{29} = 1.

cp's OEIS Frontend

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A330507 a(n) is the smallest number k having for every prime p <= prime(n) at least one prime partition with least part p, and no such partition having least part > prime(n). If no such k exists then a(n) = 0 (see comments).

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A072491 Define f(1) = 0. For n>=2, let f(n) = n - p where p is the largest prime <= n. a(n) = number of iterations of f to reach 0, starting from n.

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