A054859 -id:A054859 - cp's OEIS frontend

A334010 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero hexagonal numbers in exactly n ways.

1, 703, 274550, 11132303325

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

			Let S(k, m) denote the sum of m hexagonal numbers starting from the k-th. We have
a(1) = S(1, 1);
a(2) = S(19, 1) = S(13, 2);
a(3) = S(62, 25) = S(184, 4) = S(25, 51);
a(4) = S(3065, 505) = S(22490, 11) = S(1215, 1430) = S(1938, 946).

Eric Weisstein's World of Mathematics, Hexagonal Number
Index to sequences related to polygonal numbers

Cf. A000384, A054859, A068314, A186337, A298467, A319185, A334007, A334008, A334011, A334012.

a(4) from Giovanni Resta, Apr 13 2020

A334011 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero heptagonal numbers in exactly n ways.

Original entry on oeis.org

1, 872, 8240232, 263346158075

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

			Let S(k, m) denote the sum of m heptagonal numbers starting from the k-th. We have
a(1) = S(1, 1);
a(2) = S(13, 2) = S(3, 8);
a(3) = S(133, 98) = S(479, 14) = S(168, 77);
a(4) = S(6773, 1785) = S(810, 6006) = S(7467, 1547) = S(38758, 70).

Eric Weisstein's World of Mathematics, Heptagonal Number
Index to sequences related to polygonal numbers

Cf. A000566, A054859, A068314, A186337, A298467, A322636, A334007, A334008, A334010, A334012.

a(4) from Giovanni Resta, Apr 14 2020

A055000 Integers that can be expressed as the sum of consecutive primes in exactly 5 ways.

Original entry on oeis.org

311, 863, 14369, 14699, 15329, 16277, 19717, 20272, 25416, 28500, 29033, 36467, 37607, 40433, 41074, 42463, 45101, 46660, 48731, 49253, 49499, 50560, 53424, 55813, 59068, 67141, 68787, 70104, 70429, 70692, 71548, 76423, 78756, 78791

Offset: 1

Jud McCranie, May 30 2000

nonn

R. K. Guy, Unsolved Problems in Number Theory, section C2.

Ray Chandler, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A054845, A054859, A054996, A054997, A054998, A054999, A055001.

A054845(a(n)) = 5. - Ray Chandler, Sep 20 2023

A067376 Smallest integer expressible as the sum of (at least two) consecutive primes in n ways.

Original entry on oeis.org

5, 36, 240, 311, 16277, 130638, 218918, 9186778, 274452156, 4611108324, 12941709050

Offset: 1

Patrick De Geest, Feb 04 2002

a(10)-a(11) found by Wilfred Whiteside in 2007 (see Rivera link). - Michael S. Branicky, Jul 27 2022

			In n=7 ways: 218918 = (#12,18199) (#16,13619) (#22,9851) (#28,7691) (#38,5623) (#46,4561) (#62,3301).

Patrick De Geest, WONplate 122
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A050936, A067372-A067381, A054859.

Offset corrected and a(8)-a(9) from Donovan Johnson, Mar 14 2010

a(10) confirmed and a(10)-a(11) entered by Michael S. Branicky, Jul 27 2022

A272041 Smallest integer that can be expressed as the sum of n primes in at least n distinct ways.

Original entry on oeis.org

2, 10, 15, 18, 19, 22, 25, 27, 29, 32, 34, 36, 39, 42, 44, 46, 49, 51, 53, 55, 58, 60, 63, 65, 67, 69, 72, 74, 76, 78, 80, 83, 85, 87, 90, 92, 94, 96, 98, 100, 102, 105, 107, 109, 111, 113, 115, 117, 120, 122, 124, 126, 128, 131, 133, 135, 137, 139, 141, 143

Offset: 1

Matthew Ryan, Apr 21 2016

nonn

Initial terms found by exhaustive search in Excel.

			The sequence is defined here as starting at n=1 to avoid the term a(0). Even though there cannot be exactly zero ways to add zero primes, there is always at least one way to add 0 primes to get any n (i.e., the sum of itself for any nonprime or (1+..+1) for any prime), and zero would be the lowest such number.
Sum of 1 prime in 1 way: 2.
Sum of 2 primes in 2 ways: 3+7 = 5+5 = 10.
Sum of 3 primes in 3 ways: 3+5+7 = 5+5+5 = 2+2+11 = 15.
Sum of 4 primes in 4 ways: 2+2+3+11 = 2+2+7+7 = 3+3+5+7 = 3+5+5+5 = 18.
Sum of 60 primes in 61 ways, e.g.: 57*2 + 3 + 7 + 19 = 37*2 + 23*3 = 143. - _Lars Blomberg_, Jul 18 2017

Lars Blomberg and Giovanni Resta, Table of n, a(n) for n = 1..5000 (first 99 terms from Lars Blomberg)

Cf. A054845, A054859.

a[n_] := Block[{k = 1}, While[Length@ Quiet@ IntegerPartitions[k,{n}, Prime@ Range@ PrimePi@ k, n] < n, k++]; k]; Array[a, 50]

a(36)-a(60) from Lars Blomberg, Jul 18 2017

A329236 a(n) is the least integer that can be expressed as the sum of one or more consecutive centered triangular numbers in exactly n ways.

Original entry on oeis.org

1, 64, 1789760

Offset: 1

Ilya Gutkovskiy, Apr 13 2020

If it exists, a(4) > 10^18. - Bert Dobbelaere, Apr 17 2020

Eric Weisstein's World of Mathematics, Centered Triangular Number

Cf. A005448, A054859, A068314, A298467, A322610, A334007.

A359386 a(n) is the least positive integer that can be expressed as the sum of one or more consecutive prime powers (not including 1) in exactly n ways.

Original entry on oeis.org

1, 2, 5, 9, 29, 1027, 6659, 13560, 2149512, 38239583

Offset: 0

Ilya Gutkovskiy, Mar 13 2023

			For n = 3: 9 = 9 = 4 + 5 = 2 + 3 + 4.

Cf. A054859, A246655.

A360837 a(n) is the least positive integer that can be expressed as the sum of one or more consecutive prime-indexed primes in exactly n ways.

Original entry on oeis.org

1, 3, 59, 10079, 744666, 163710521

Offset: 0

Ilya Gutkovskiy, Feb 23 2023

			For n = 2: 59 = prime(prime(7)) = prime(prime(3)) + prime(prime(4)) + prime(prime(5)).

Cf. A006450, A054859.

a(5) from Michael S. Branicky, Feb 23 2023

A361435 a(n) is the least positive integer that can be expressed as the sum of one or more consecutive squarefree numbers in exactly n ways.

Original entry on oeis.org

1, 3, 11, 34, 144, 165, 229, 517, 790, 6870, 12757, 21134, 54155, 226470, 193225, 431900, 948949, 3960994, 6674779, 7594013, 14204939, 32720909, 20369309, 176923605, 335119938

Offset: 1

Ilya Gutkovskiy, Mar 11 2023

			For n = 3: 11 = 11 = 5 + 6 = 1 + 2 + 3 + 5.

Cf. A005117, A054859.

a(21)-a(25) from Michael S. Branicky, Mar 12 2023

A361473 a(n) is the least positive integer that can be expressed as the sum of one or more consecutive nonprime numbers in exactly n ways.

Original entry on oeis.org

1, 10, 27, 45, 143, 306, 903, 465, 1215, 3037, 2418, 4809, 17193, 8349, 32055, 75847, 117705, 306075, 379395, 467955, 1269075, 2517687, 1809295, 4720023, 6375915, 12961575, 21540987, 35647010, 16615305, 192717405, 268822806, 186269391, 247067415

Offset: 1

Ilya Gutkovskiy, Mar 13 2023

nonn

			For n = 2: 10 = 10 = 4 + 6.

Cf. A018252, A054859.

a(25)-a(33) from Michael S. Branicky, Mar 13 2023

cp's OEIS Frontend

A334010 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero hexagonal numbers in exactly n ways.

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A334011 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero heptagonal numbers in exactly n ways.

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A055000 Integers that can be expressed as the sum of consecutive primes in exactly 5 ways.

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A067376 Smallest integer expressible as the sum of (at least two) consecutive primes in n ways.

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A272041 Smallest integer that can be expressed as the sum of n primes in at least n distinct ways.

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A329236 a(n) is the least integer that can be expressed as the sum of one or more consecutive centered triangular numbers in exactly n ways.

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A359386 a(n) is the least positive integer that can be expressed as the sum of one or more consecutive prime powers (not including 1) in exactly n ways.

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A360837 a(n) is the least positive integer that can be expressed as the sum of one or more consecutive prime-indexed primes in exactly n ways.

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A361435 a(n) is the least positive integer that can be expressed as the sum of one or more consecutive squarefree numbers in exactly n ways.

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A361473 a(n) is the least positive integer that can be expressed as the sum of one or more consecutive nonprime numbers in exactly n ways.

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