A007527 -id:A007527 - cp's OEIS frontend

A136117 Pentagonal numbers (A000326) which are the sum of 2 other positive pentagonal numbers.

70, 92, 852, 925, 1247, 1426, 1926, 2625, 3577, 5192, 6305, 6501, 7107, 7740, 7957, 8177, 8626, 9560, 10292, 12927, 13207, 14652, 15555, 16172, 18095, 20475, 20827, 21901, 22265, 22632, 23002, 23751, 24130, 28497, 29330, 31032, 33227, 33675

Offset: 1

M. F. Hasler, Dec 15 2007; corrected Dec 25 2007

nonn

It is conjectured that every integer and hence every pentagonal number, greater than 33066, hence greater than A000326(149) = 33227, can be represented as the sum of three pentagonal numbers. - Jonathan Vos Post, Dec 18 2007

			a(1)=70=P(7) is the least pentagonal number which can be written as sum of two other pentagonal numbers, P(7)=P(5)+P(5).

Cf. A000326, A136112-A136118, A007527.

P(n)=n*(3*n-1)>>1 /* a.k.a. A000326 */
isPent(t)=P(sqrtint(t<<1\3)+1)==t
for(i=1,299,for(j=1,(i+1)\sqrt(2),isPent(P(i)-P(j)) && print1(P(i)",") || next(2)))
/* The following is much faster, at the cost of implementing sum2sqr(), cf. A133388*/
A136117next(i)=i=sqrtint(i\3*2)*6+5; until(0, for(j=2,#t=sum2sqr((i+=6)^2+1), t[j]%6==[5,5] && break(2)));i^2\24
A136117vect(n,i)=vector(n,j,i=A136117next(i)) /* 2nd arg =0 by default but allows one to start elsewhere */
A136117(n,i)=until(!n--,i=A136117next(i));i \\ M. F. Hasler, Dec 25 2007

a(n) = A000326(A136116(n)) = A000326(m)+A136114(m) where m is the index of the n-th nonzero term in A136114 or A136115.

A133929 Positive integers that cannot be expressed using four pentagonal numbers.

Original entry on oeis.org

9, 21, 31, 43, 55, 89

Offset: 1

Eric W. Weisstein, Sep 29 2007

Equivalently, integers m such that the smallest number of pentagonal numbers (A000326) which sum to m is exactly five, that is, A100878(a(n)) = 5. Richard Blecksmith & John Selfridge found these six integers among the first million, they believe that they have found them all (Richard K. Guy reference). - Bernard Schott, Jul 22 2022

			   9 =  5 +  1 + 1 + 1 + 1.
  21 =  5 +  5 + 5 + 5 + 1.
  31 = 12 + 12 + 5 + 1 + 1.
  43 = 35 +  5 + 1 + 1 + 1.
  55 = 51 +  1 + 1 + 1 + 1.
  89 = 70 + 12 + 5 + 1 + 1.

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

Eric Weisstein's World of Mathematics, Pentagonal Number

Cf. A000326, A007527, A100878.

Equals A003679 \ A355660.

A117089 Primes that are not the sum of 3 hexagonal numbers.

Original entry on oeis.org

5, 11, 19, 23, 37, 41, 53, 59, 83, 89, 113, 131, 167, 173, 179, 229, 251, 269, 293, 313, 317, 383, 389, 439, 443, 509, 599, 641, 683, 859, 929, 1031, 1033, 1049, 1163, 1193, 1283, 1301, 1303, 1307, 1439, 1493, 1499, 1543, 1619, 1733, 2143, 2153, 2333, 2687, 2693, 3083, 3089, 3533, 3719, 3989, 4003, 4583, 4673, 4703, 5387, 5651, 5849, 5903, 6173, 6389, 6449, 7481, 9293, 12113, 15803, 16433, 19763, 61403

Offset: 1

Jonathan Vos Post, Apr 18 2006

			5 is the sum of five hexagonal numbers; 11 is the sum of six hexagonal numbers; the other 72 primes are the sum of four hexagonal numbers. - _T. D. Noe_, Apr 20 2006

Legendre, Théorie des Nombres, 3rd edition, 1830.

W. Duke and R. Schulze-Pilot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Invent. Math. 99(1990), 49-57.
R. K. Guy, Every number is expressible as the sum of how many polygonal numbers?, Amer. Math. Monthly 101 (1994), 169-172.

Cf. A000040, A000384, A007527, A007536, A117065.

nn=201; hex=Table[n(2n-1), {n,0,nn-1}]; ps=Prime[Range[PrimePi[hex[[ -1]]]]]; Do[n=hex[[i]]+hex[[j]]+hex[[k]]; If[n<=hex[[ -1]]&&PrimeQ[n], ps=DeleteCases[ps,n]], {i,nn}, {j,i,nn}, {k,j,nn}]; ps (* T. D. Noe, Apr 20 2006 *)

A000040 INTERSECT A007536.

More terms from T. D. Noe, who conjectures that the list shown here is complete. His search up to 7*10^7 gave no further terms. - Apr 20 2006

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A136117 Pentagonal numbers (A000326) which are the sum of 2 other positive pentagonal numbers.

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A133929 Positive integers that cannot be expressed using four pentagonal numbers.

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A117089 Primes that are not the sum of 3 hexagonal numbers.

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