A064826 -id:A064826 - cp's OEIS frontend

A117065 Primes that are not the sum of 3 pentagonal numbers.

19, 31, 43, 67, 89, 101, 113, 131, 229, 241, 277, 359, 383, 491, 523, 619, 631, 643, 701, 761, 1321, 1381, 1621, 2221, 2861

Offset: 1

Jonathan Vos Post, Apr 17 2006

5 is the only prime pentagonal number; every greater pentagonal number A000326(n) = n(3n-1)/2 is either divisible by n/2 or (3n-1)/2. Every number is the sum of 5 pentagonal numbers, hence every prime is the sum of 5 pentagonal numbers. There are an infinite number of primes which are the sum of two pentagonal numbers, the subset of primes which are the sum of two pentagonal numbers in exactly two different ways begins {211, 853, 1259, 1427, 1571, 2297, 2351}.
The sum may include the pentagonal number 0. Hence this sequence does not have any primes that are the sum of two positive pentagonal numbers. The sequence is probably finite. There are no other primes < 59900. - T. D. Noe, Apr 19 2006
The next term, if it exists, is greater than 160000000. - Jack W Grahl, Jul 10 2018
a(26) > 10^11, if it exists. - Giovanni Resta, Jul 13 2018

J. W. Grahl, C code which was used to check for elements of this sequence up to 160,000,000.
R. K. Guy, Every number is expressible as the sum of how many polygonal numbers?, Amer. Math. Monthly 101 (1994), 169-172.

Cf. A000040, A000326, A003679, A064826.

nn=201; pen=Table[n(3n-1)/2, {n,0,nn-1}]; ps=Prime[Range[PrimePi[pen[[ -1]]]]]; Do[n=pen[[i]]+pen[[j]]+pen[[k]]; If[n<=pen[[ -1]]&&PrimeQ[n], ps=DeleteCases[ps, ?(#==n&)]], {i,nn}, {j,i,nn}, {k,j,nn}]; ps (* _T. D. Noe, Apr 19 2006 *)

A000040 INTERSECT A003679.

More terms from T. D. Noe, Apr 19 2006

Mathematica program corrected by Robert Price, Aug 25 2019

A117104 Sum of two positive heptagonal numbers A000566.

Original entry on oeis.org

2, 8, 14, 19, 25, 35, 36, 41, 52, 56, 62, 68, 73, 82, 88, 89, 99, 110, 113, 115, 119, 130, 136, 146, 149, 155, 162, 166, 167, 182, 190, 193, 196, 203, 207, 223, 224, 229, 236, 242, 244, 253, 260, 269, 270, 287, 290, 293, 296, 301, 304, 316, 320, 337, 341, 343, 347

Offset: 1

Jonathan Vos Post, Apr 18 2006

7 is the only prime heptagonal number. Primes which are sums of two positive heptagonal numbers include: {2, 19, 41, 73, 89, 113, 149, 167, 193, 223, 229, 269, 293, 337, 347, 367, 383, ...}.

Cf. A000566, A000040, A000326, A003679, A064826, A117065.

Lim=348;nl=Ceiling[Sqrt[(2/5)Lim]];Select[Union[Total/@Tuples[Table[n*(5n-3)/2,{n,nl}],{2}]],#James C. McMahon, Sep 27 2024 *)

{a(n)} = {A000566} + {A000566} = {a*(5*a-3)/2 + b*(5*b-3)/2} \ {A000566}.

Missing 343 added by Giovanni Resta, Jun 15 2016

A117105 Numbers that are the sum of three positive heptagonal numbers (A000566) in at least one way.

Original entry on oeis.org

3, 9, 15, 20, 21, 26, 32, 36, 37, 42, 43, 48, 53, 54, 57, 59, 63, 69, 70, 74, 75, 80, 83, 86, 89, 90, 91, 95, 96, 100, 102, 106, 107, 111, 114, 116, 117, 120, 122, 123, 126, 128, 131, 133, 137, 143, 144, 147, 148, 149, 150, 153, 154, 156, 162, 163, 164, 165

Offset: 1

Jonathan Vos Post, Apr 18 2006

7 is the only prime heptagonal number. Primes which are sums of two positive heptagonal numbers include: {2, 19, 41, 73, 89, 113, 149, 167, 193, 223, 229, 269, 293, 337, 347, 367, 383, ...}. Primes which are sums of three positive heptagonal numbers include: {3, 37, 43, 53, 59, 83, 89, 107, 131, 137, 149, 163, 167, 173, 191, 197, 211, 227, 241, 251, 257, 263, 271, ...}.

By definition this does not contain any repeated terms. - N. J. A. Sloane, Aug 15 2020

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

Cf. A000566, A000040, A000326, A003679, A064826, A117065.

With[{nn=10},Select[Union[Total/@Tuples[PolygonalNumber[7,Range[ nn]],3]], #<=PolygonalNumber[7,nn]-2&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 16 2020 *)

{a(n)} = {A000566} + {A000566} + {A000566} = {a*(5*a-3)/2 + b*(5*b-3)/2 + c*(5*c-3)/2} \ {A000566}.

Missing 106 and 131 added by Giovanni Resta, Jun 15 2016

Corrected (deleting duplicates) and extended by Harvey P. Dale, Aug 16 2020

A117111 Sum of four positive heptagonal numbers A000566.

Original entry on oeis.org

4, 10, 16, 21, 22, 27, 28, 33, 37, 38, 39, 43, 44, 49, 50, 54, 55, 58, 60, 61, 64, 66, 70, 71, 72, 75, 76, 77, 81, 82, 84, 87, 88, 90, 91, 92, 93, 96, 97, 98, 101, 102, 103, 104, 107, 108, 109, 112, 113, 114, 115, 117, 118, 120, 121, 123, 124, 125, 127, 129, 130, 132

Offset: 1

Jonathan Vos Post, Apr 18 2006

Fermat discovered, Gauss, Legendre and [1813] Cauchy proved that every integer is the sum of 7 heptagonal numbers (and there are some numbers which require all 7, the smallest being 13). 7 is the only prime heptagonal number. Primes which are sums of two positive heptagonal numbers include: {19, 41, 73, 89, 113, 149, 167, 193, 223, 229, 269, 293, 337, 347, 367, 383, 521, ...}. Primes which are sums of three positive heptagonal numbers include: {3, 37, 43, 53, 59, 83, 89, 107, 131, 137, 149, 163, 167, 173, 191, 197, 211, 227, 241, 251, 257, 263, 271, ...}. Primes which are sums of four positive heptagonal numbers include: {37, 43, 61, 71, 97, 101, 103, 107, 109, 113, 127, 149, 151, 167, 181, 191, 197, 199, 211, 223, 229, 239, 251, ...}.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A000566, A000040, A000326, A003679, A064826, A117065.

Module[{upto=150,max},max=Ceiling[(3+Sqrt[9+40upto])/10];Select[Total/@
Tuples[PolygonalNumber[7,Range[max]],4]//Union,#<=upto&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 15 2016 *)

{a(n)} = {A000566} + {A000566} + {A000566} + {A000566} = {a*(5*a-3)/2 + b*(5*b-3)/2 + c*(5*c-3)/2 + d*(5*d-3)/2 such that every term is positive}.

A120536 Semiprimes which are the sum of two pentagonal numbers (A000326) in exactly two different ways.

Original entry on oeis.org

215, 381, 447, 766, 807, 1457, 1622, 1639, 1927, 2047, 2245, 2302, 2497, 3027, 3173, 3437, 3715, 3787, 4359, 4369, 4577, 4594, 4677, 4681, 5029, 5277, 5377, 5435, 5617, 5747, 5911, 6065, 6117, 6537, 6711, 6722, 6782, 7087, 7157, 7327, 7538, 7661, 7813, 7827

Offset: 1

Jonathan Vos Post, Aug 06 2006

nonn

Semiprimes in A064826 Numbers which are the sum of two pentagonal numbers (A000326) in exactly two different ways.

A001358 intersect A064826.

Cf. A000326, A001358, A064826.

T = Range[10^4]*0; q = # (3 # - 1)/2 &@ Range[81]; Do[ s = q[[i]] + q[[j]]; If[s <= 10^4, T[[s]]++], {i, Length[q]}, {j, i}]; Select[ Flatten@ Position[T, 2], PrimeOmega[#] == 2 &] (* Giovanni Resta, Jun 13 2016 *)

a(14)-a(44) from Giovanni Resta, Jun 13 2016

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A117065 Primes that are not the sum of 3 pentagonal numbers.

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A117104 Sum of two positive heptagonal numbers A000566.

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A117105 Numbers that are the sum of three positive heptagonal numbers (A000566) in at least one way.

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A117111 Sum of four positive heptagonal numbers A000566.

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A120536 Semiprimes which are the sum of two pentagonal numbers (A000326) in exactly two different ways.

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