%I #64 Jul 18 2022 20:18:36

%S 0,1,2,3,4,1,2,3,4,5,2,3,1,2,3,3,4,2,3,4,4,5,1,2,2,3,4,2,3,3,4,5,3,4,

%T 2,1,2,3,4,3,2,3,4,5,2,3,3,2,3,3,4,1,2,3,4,5,2,2,3,3,4,3,3,2,3,4,3,4,

%U 3,3,1,2,3,2,3,2,3,4,3,3,3,4,2,3,4,3,2,3,4,5,4,3,1,2,3,3,4,2,3,4,4,4,2,3,2

%N Smallest number of pentagonal numbers which sum to n.

%C From _Bernard Schott_, Jul 15 2022: (Start)

%C In September 1636, Fermat, in a letter to Mersenne, made the statement that every number is a sum of at most three triangular numbers, four squares, five pentagonal numbers, and so on.

%C The square case was proved by Lagrange in 1770; it is known as Lagrange's four squares theorem (see A002828). Then Gauss proved the triangular case in 1796 (see A061336).

%C In 1813, Cauchy proved this polygonal number theorem: for m >= 3, every positive integer N can be represented as a sum of m+2 (m+2)-gonal numbers, at most four of which are different from 0 and 1 (Deza reference). Hence every number is expressible as the sum of at most five positive pentagonal numbers (A000326). (End)

%D Elena Deza and Michel Marie Deza, Fermat's polygonal number theorem, Figurate numbers, World Scientific Publishing (2012), Chapter 5, pp. 313-377.

%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section D3, Figurate numbers, pp. 222-228.

%H Augustin-Louis Cauchy, <a href="https://doi.org/10.1017/CBO9780511702655.028">Démonstration du théorème général de Fermat sur les nombres polygones</a>, Extrait des Mémoires de l'Institut, 1813-15.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FermatsPolygonalNumberTheorem.html">Fermat's Polygonal Number Theorem</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Fermat_polygonal_number_theorem">Fermat polygonal number theorem</a>.

%F a(n) <= 5 (inequality proposed by Fermat and proved by Cauchy). - _Bernard Schott_, Jul 13 2022

%e a(5)=1 since 5=5, a(6)=2 since 6=1+5, a(7)=3 since 7=1+1+5, a(10)=2 since 10=5+5 with 1 and 5 pentagonal numbers.

%o (PARI) a(n) = my(nb=oo); forpart(vp=n, if (vecsum(apply(x->ispolygonal(x, 5), Vec(vp))) == #vp, nb = min(nb, #vp)),,5); nb; \\ _Michel Marcus_, Jul 15 2022

%o (PARI) a(n) = for(i = 1, oo, p = partitions(n, , [i,i]); for(j = 1, #p, if(sum(k = 1, i, ispolygonal(p[j][k],5)) == i, return(i)))) \\ _David A. Corneth_, Jul 15 2022

%Y Cf. A002828, A061336, A355717.

%Y Cf. A000326 (a(n) = 1), A003679 (a(n) = 4 or 5), A355660 (a(n) = 4), A133929 (a(n) = 5).

%K nonn

%O 0,3

%A _Franz Vrabec_, Jan 09 2005

%E More terms from _David Wasserman_, Mar 04 2008