A186337 -id:A186337 - cp's OEIS frontend

A334007 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero triangular numbers in exactly n ways.

1, 10, 2180, 10053736, 13291443468940

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

			Let S(k, m) denote the sum of m triangular numbers starting from k(k+1)/2. We have
a(1) = S(1, 1);
a(2) = S(4, 1) = S(1, 3);
a(3) = S(31, 4) = S(27, 5) = S(9, 15);
a(4) = S(945, 22) = S(571, 56) = S(968, 21) = S(131, 266);
a(5) = S(4109, 38947) = S(25213, 20540) = S(10296, 32943) = S(32801, 15834) = S(31654, 16472).

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

Cf. A000217, A054859, A068314, A034706, A050943, A186337, A298467, A307666, A309783, A334008, A334010, A334011, A334012.

a(5) from Giovanni Resta, Apr 13 2020

A334012 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero octagonal numbers in exactly n ways.

Original entry on oeis.org

1, 1045, 5985

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

			From _Seiichi Manyama_, May 16 2021: (Start)
Let S(k, m) denote the sum of m octagonal numbers starting from k*(3*k-2). We have
a(1) = S(1, 1);
a(2) = S(19, 1) = S(1, 10);
a(3) = S(45, 1) = S(11, 9) = S(1, 18). (End)

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

Cf. A000567, A054859, A068314, A186337, A298467, A322637, A334007, A334008, A334010, A334011, A344376.

A334008 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero pentagonal numbers in exactly n ways.

Original entry on oeis.org

1, 287, 472320, 89051435880

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

			Let S(k, m) denote the sum of m pentagonal numbers starting from the k-th. We have
a(1) = S(1, 1);
a(2) = S(14, 1) = S(2, 7);
a(3) = S(103, 24) = S(19, 80) = S(67, 41);
a(4) = S(10833, 484) = S(4542, 1936) = S(9153, 660) = S(2817, 3036);

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

Cf. A000326, A054859, A068314, A186337, A298467, A319184, A334007, A334010, A334011, A334012.

a(4) from Giovanni Resta, Apr 13 2020

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.

Original entry on oeis.org

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

A186336 Number of ways of representing n as the sum of one or more consecutive semiprimes.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 0, 0, 2, 0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 0, 0, 0, 2, 1, 1, 1, 0, 1, 3, 0, 0, 0, 2, 0, 0, 1, 1, 1, 1, 1, 2, 0, 0, 1, 1, 0, 1, 3, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 3, 0, 0, 1, 2, 1, 1, 0, 2, 0, 1, 0, 0, 2, 1, 1, 2, 1, 1, 0, 0, 0, 2, 0, 2, 2, 2, 0, 2, 0, 0, 1, 1, 1, 0, 0, 0, 3, 2, 0, 1, 0, 1, 2, 0, 0, 2, 1, 0, 2, 1, 1

Offset: 0

Alois P. Heinz, Feb 18 2011

nonn

			a(4)  = 1:  4 = A001358(1) is the first semiprime.
a(10) = 2: 10 = A001358(1)+A001358(2) = 4+6 = A001358(4) = 10.
a(39) = 3: 39 = 6+9+10+14 = 10+14+15 = 39.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A001358, A112020, A186337.

a186336 n = f $ takeWhile (<= n) a001358_list where
   f []       = 0
   f (sp:sps) = g sp sps + f sps
   g spSum []                    = fromEnum (spSum == n)
   g spSum (sp:sps) | spSum < n  = g (sp + spSum) sps
                    | spSum == n = 1
                    | otherwise  = 0
-- Reinhard Zumkeller, Feb 28 2011

b:= proc(n) option remember; local k;
      if n=0 then 0
    else for k from b(n-1)+1
           while isprime(k) or 2<>add(i[2], i=ifactors(k)[2])
         do od; k
      fi
    end:
pis:= proc(n) option remember; local k;
        if n<4 then 0
      elif n=4 then 1
      else k:= pis(n-1);
           k +`if`(b(k+1)=n, 1 ,0)
        fi
      end:
ssp:= proc(i,j) option remember;
        b(j) + `if`(i=j, 0, ssp(i, j-1))
      end:
a:= proc(n) option remember; local i, j, cnt, s;
      cnt:= 0;
      j:= pis(n);
      i:= j;
      while i>0 do
        s:= ssp(i,j);
        if sn then j:= j-1
      else cnt:= cnt+1;
           i, j:= i-1, j-1
        fi
      od; cnt
    end:
seq(a(n), n=0..200);

nmax = 120;
sp = Select[Range[nmax], PrimeOmega[#] == 2&];
lsp = Length[sp]; Clear[a]; a[_] = 0;
Do[n = Total[sp[[i ;; j]]]; a[n] = a[n]+1, {i, 1, lsp}, {j, i, lsp}];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Mar 13 2019 *)

A365507 a(n) is the least positive integer that can be expressed as the sum of one or more consecutive n-almost primes in exactly n ways, or -1 if no such integer exists.

Original entry on oeis.org

2, 10, 105, 2410, 45010, 708408

Offset: 1

Ilya Gutkovskiy, Sep 07 2023

			For n = 3: 105 = 3*5*7 = 2*2*3 + 2*3*3 + 2*2*5 + 3*3*3 + 2*2*7 = 2*2*5 + 3*3*3 + 2*2*7 + 2*3*5.

Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A054859, A091538, A186337.

cp's OEIS Frontend

A334007 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero triangular numbers in exactly n ways.

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A334012 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero octagonal numbers in exactly n ways.

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A334008 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero pentagonal numbers in exactly n ways.

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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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A186336 Number of ways of representing n as the sum of one or more consecutive semiprimes.

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A365507 a(n) is the least positive integer that can be expressed as the sum of one or more consecutive n-almost primes in exactly n ways, or -1 if no such integer exists.

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