A333015 -id:A333015 - cp's OEIS frontend

A333011 Integers which can be written in only one way as a sum of two distinct nonzero pentagonal numbers.

6, 13, 17, 23, 27, 34, 36, 40, 47, 52, 56, 57, 63, 71, 73, 75, 82, 86, 92, 93, 97, 104, 105, 114, 118, 121, 122, 127, 129, 139, 143, 146, 150, 152, 157, 162, 167, 168, 177, 180, 181, 187, 188, 196, 198, 209, 222, 227, 232, 237, 245, 246, 248, 252, 259, 261, 262

Offset: 1

Olivier Gérard, Mar 05 2020

nonn

The first term of A332988 not in this sequence is 211.

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A000326, A332988, A333011, A333012, A333013, A333014, A333015.

is(k) = sum(i=1, sqrt(1+12*k)\6, sqrt(1+24*k+12*i-36*i*i)%6==5) == 1; \\ Jinyuan Wang, Mar 06 2020

A333012 Integers which can be written in exactly two ways as a sum of two distinct nonzero pentagonal numbers.

Original entry on oeis.org

211, 215, 381, 447, 602, 663, 766, 807, 853, 874, 1002, 1172, 1197, 1248, 1259, 1372, 1427, 1457, 1571, 1612, 1622, 1639, 1652, 1665, 1752, 1862, 1927, 1996, 2047, 2152, 2245, 2297, 2302, 2332, 2351, 2415, 2472, 2497, 2506, 2523, 2618, 2887, 2912, 2952

Offset: 1

Olivier Gérard, Mar 05 2020

nonn

			211 = P(5) + P(11) = P(1) + P(12) = 35 + 176 = 1 + 210, where P(n) is the n-th pentagonal number A000326.

Cf. A000326, A333011, A333013, A333014, A333015.

is(k) = sum(i=1, sqrt(1+12*k)\6, sqrt(1+24*k+12*i-36*i*i)%6==5) == 2; \\ Jinyuan Wang, Mar 06 2020

A333013 Integers which can be written in exactly three ways as sum of two distinct nonzero pentagonal numbers.

Original entry on oeis.org

2999, 6450, 6552, 7177, 8422, 9204, 9652, 10037, 10622, 11380, 11467, 16577, 17652, 17772, 17789, 17818, 19132, 19761, 20177, 21327, 21477, 22277, 22702, 22855, 23008, 23212, 23387, 23427, 23444, 24402, 24662, 25677, 25847, 26286, 26902, 27649, 27802, 27847, 28567, 29927

Offset: 1

Olivier Gérard, Mar 05 2020

nonn

			2999 = P(24) + P(38) = P(13) + P(43) = P(9) + P(44), where P(n) is the n-th pentagonal number A000326.

Cf. A000326, A333011, A333012, A333014, A333015.

dnpQ[n_]:=Count[IntegerPartitions[n,{2}],?(AllTrue[(1+Sqrt[1+24#])/6,IntegerQ]&)]==3; Parallelize[Select[Range[30000],dnpQ]] (* or *) Select[Tally[Total/@Subsets[ PolygonalNumber[ 5,Range[200]],{2}]],#[[2]]==3&][[;;,1]]//Union (* _Harvey P. Dale, Jul 20 2023 *)

is(k) = sum(i=1, sqrt(1+12*k)\6, sqrt(1+24*k+12*i-36*i*i)%6==5) == 3; \\ Jinyuan Wang, Mar 06 2020

cp's OEIS Frontend

A333011 Integers which can be written in only one way as a sum of two distinct nonzero pentagonal numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A333012 Integers which can be written in exactly two ways as a sum of two distinct nonzero pentagonal numbers.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

A333013 Integers which can be written in exactly three ways as sum of two distinct nonzero pentagonal numbers.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI