A326461 -id:A326461

A326455

Sum of all the parts in the partitions of n into 8 primes.

+10
9

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 17, 18, 38, 40, 63, 88, 92, 120, 150, 182, 216, 280, 261, 360, 434, 512, 528, 714, 665, 936, 962, 1178, 1170, 1560, 1394, 1932, 1849, 2332, 2160, 2990, 2632, 3696, 3234, 4250, 3927, 5408, 4452, 6372, 5445

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,17

LINKS

Table of n, a(n) for n=0..55.

Index entries for sequences related to partitions

FORMULA

a(n) = n * Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p), where c = A010051.

a(n) = n * A259198(n).

a(n) = A326456(n) + A326457(n) + A326458(n) + A326459(n) + A326460(n) + A326461(n) + A326462(n) + A326463(n).

MATHEMATICA

a[n_] := n*Length[IntegerPartitions[n, {8}, Prime[Range[PrimePi[n]]]]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 07 2019 *)

CROSSREFS

Cf. A010051, A259198, A326456, A326457, A326458, A326459, A326460, A326461, A326462, A326463.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 06 2019

STATUS

approved

A326456

Sum of the smallest parts of the partitions of n into 8 primes.

+10
8

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 4, 6, 8, 8, 11, 12, 15, 16, 22, 18, 26, 28, 36, 32, 47, 38, 59, 52, 71, 60, 93, 68, 109, 86, 128, 96, 157, 112, 190, 132, 210, 154, 262, 168, 300, 198, 344, 224, 399, 246, 464, 286, 515, 324, 605

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,17

LINKS

Table of n, a(n) for n=0..64.

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p) * p, where c = A010051.

a(n) = A326455(n) - A326457(n) - A326458(n) - A326459(n) - A326460(n) - A326461(n) - A326462(n) - A326463(n).

MATHEMATICA

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[p * (PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[p] - PrimePi[p - 1]) (PrimePi[n - i - j - k - l - m - o - p] - PrimePi[n - i - j - k - l - m - o - p - 1]), {i, j, Floor[(n - j - k - l - m - o - p)/2]}], {j, k, Floor[(n - k - l - m - o - p)/3]}], {k, l, Floor[(n - l - m - o - p)/4]}], {l, m, Floor[(n - m - o - p)/5]}], {m, o, Floor[(n - o - p)/6]}], {o, p, Floor[(n - p)/7]}], {p, Floor[n/8]}], {n, 0, 50}]

CROSSREFS

Cf. A010051, A259198, A326455, A326457, A326458, A326459, A326460, A326461, A326462, A326463.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 06 2019

STATUS

approved

A326457

Sum of the seventh largest parts in the partitions of n into 8 primes.

+10
8

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 4, 6, 8, 9, 11, 13, 15, 18, 22, 20, 26, 32, 36, 37, 47, 45, 59, 63, 73, 74, 95, 84, 111, 109, 132, 123, 163, 145, 196, 173, 220, 209, 278, 225, 316, 272, 366, 309, 427, 343, 494, 405, 557, 466, 659

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,17

LINKS

Table of n, a(n) for n=0..64.

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p) * o, where c = A010051.

a(n) = A326455(n) - A326456(n) - A326458(n) - A326459(n) - A326460(n) - A326461(n) - A326462(n) - A326463(n).

MATHEMATICA

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[o * (PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[p] - PrimePi[p - 1]) (PrimePi[n - i - j - k - l - m - o - p] - PrimePi[n - i - j - k - l - m - o - p - 1]), {i, j, Floor[(n - j - k - l - m - o - p)/2]}], {j, k, Floor[(n - k - l - m - o - p)/3]}], {k, l, Floor[(n - l - m - o - p)/4]}], {l, m, Floor[(n - m - o - p)/5]}], {m, o, Floor[(n - o - p)/6]}], {o, p, Floor[(n - p)/7]}], {p, Floor[n/8]}], {n, 0, 50}]

CROSSREFS

Cf. A010051, A259198, A326455, A326456, A326458, A326459, A326460, A326461, A326462, A326463.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 06 2019

STATUS

approved

A326458

Sum of the sixth largest parts of the partitions of n into 8 primes.

+10
8

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 4, 6, 9, 9, 12, 13, 17, 18, 24, 20, 30, 32, 41, 37, 56, 47, 71, 65, 88, 76, 114, 88, 138, 115, 162, 129, 205, 157, 246, 187, 282, 225, 352, 247, 404, 298, 469, 339, 553, 385, 641, 453, 731, 522, 855

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,17

LINKS

Table of n, a(n) for n=0..64.

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p) * m, where c = A010051.

a(n) = A326455(n) - A326456(n) - A326457(n) - A326459(n) - A326460(n) - A326461(n) - A326462(n) - A326463(n).

MATHEMATICA

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[m * (PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[p] - PrimePi[p - 1]) (PrimePi[n - i - j - k - l - m - o - p] - PrimePi[n - i - j - k - l - m - o - p - 1]), {i, j, Floor[(n - j - k - l - m - o - p)/2]}], {j, k, Floor[(n - k - l - m - o - p)/3]}], {k, l, Floor[(n - l - m - o - p)/4]}], {l, m, Floor[(n - m - o - p)/5]}], {m, o, Floor[(n - o - p)/6]}], {o, p, Floor[(n - p)/7]}], {p, Floor[n/8]}], {n, 0, 50}]

CROSSREFS

Cf. A010051, A259198, A326455, A326456, A326457, A326459, A326460, A326461, A326462, A326463.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 06 2019

STATUS

approved

A326459

Sum of the fifth largest parts of the partitions of n into 8 primes.

+10
8

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 4, 7, 9, 10, 12, 15, 17, 20, 24, 24, 30, 39, 43, 47, 60, 58, 75, 81, 94, 97, 124, 116, 152, 146, 180, 172, 229, 207, 278, 246, 320, 298, 400, 332, 464, 395, 539, 463, 643, 521, 749, 612, 855, 720

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,17

LINKS

Table of n, a(n) for n=0..63.

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p) * l, where c = A010051.

a(n) = A326455(n) - A326456(n) - A326457(n) - A326458(n) - A326460(n) - A326461(n) - A326462(n) - A326463(n).

MATHEMATICA

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[l * (PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[p] - PrimePi[p - 1]) (PrimePi[n - i - j - k - l - m - o - p] - PrimePi[n - i - j - k - l - m - o - p - 1]), {i, j, Floor[(n - j - k - l - m - o - p)/2]}], {j, k, Floor[(n - k - l - m - o - p)/3]}], {k, l, Floor[(n - l - m - o - p)/4]}], {l, m, Floor[(n - m - o - p)/5]}], {m, o, Floor[(n - o - p)/6]}], {o, p, Floor[(n - p)/7]}], {p, Floor[n/8]}], {n, 0, 50}]

CROSSREFS

Cf. A010051, A259198, A326455, A326456, A326457, A326458, A326460, A326461, A326462, A326463.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 06 2019

STATUS

approved

A326460

Sum of the fourth largest parts of the partitions of n into 8 primes.

+10
8

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 5, 7, 10, 10, 14, 15, 19, 20, 30, 26, 38, 43, 54, 51, 74, 64, 97, 93, 118, 111, 159, 132, 193, 172, 231, 202, 293, 243, 357, 296, 407, 352, 517, 402, 600, 495, 706, 577, 851, 661, 1004, 800, 1150

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,17

LINKS

Table of n, a(n) for n=0..62.

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p) * k, where c = A010051.

a(n) = A326455(n) - A326456(n) - A326457(n) - A326458(n) - A326459(n) - A326461(n) - A326462(n) - A326463(n).

MATHEMATICA

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[k * (PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[p] - PrimePi[p - 1]) (PrimePi[n - i - j - k - l - m - o - p] - PrimePi[n - i - j - k - l - m - o - p - 1]), {i, j, Floor[(n - j - k - l - m - o - p)/2]}], {j, k, Floor[(n - k - l - m - o - p)/3]}], {k, l, Floor[(n - l - m - o - p)/4]}], {l, m, Floor[(n - m - o - p)/5]}], {m, o, Floor[(n - o - p)/6]}], {o, p, Floor[(n - p)/7]}], {p, Floor[n/8]}], {n, 0, 50}]

CROSSREFS

Cf. A010051, A259198, A326455, A326456, A326457, A326458, A326459, A326461, A326462, A326463.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 06 2019

STATUS

approved

A326462

Sum of the second largest parts in the partitions of n into 8 primes.

+10
8

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 5, 6, 8, 14, 14, 19, 23, 31, 35, 46, 45, 62, 71, 86, 85, 119, 115, 162, 163, 205, 212, 275, 260, 356, 344, 435, 411, 559, 516, 709, 640, 829, 786, 1060, 914, 1272, 1112, 1485, 1299, 1795, 1501, 2133

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,17

LINKS

Table of n, a(n) for n=0..60.

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p) * i, where c = A010051.

a(n) = A326455(n) - A326456(n) - A326457(n) - A326458(n) - A326459(n) - A326460(n) - A326461(n) - A326463(n).

MATHEMATICA

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[i * (PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[p] - PrimePi[p - 1]) (PrimePi[n - i - j - k - l - m - o - p] - PrimePi[n - i - j - k - l - m - o - p - 1]), {i, j, Floor[(n - j - k - l - m - o - p)/2]}], {j, k, Floor[(n - k - l - m - o - p)/3]}], {k, l, Floor[(n - l - m - o - p)/4]}], {l, m, Floor[(n - m - o - p)/5]}], {m, o, Floor[(n - o - p)/6]}], {o, p, Floor[(n - p)/7]}], {p, Floor[n/8]}], {n, 0, 50}]

CROSSREFS

Cf. A010051, A259198, A326455, A326456, A326457, A326458, A326459, A326460, A326461, A326463.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 06 2019

STATUS

approved

A326463

Sum of the largest parts of the partitions of n into 8 primes.

+10
8

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 3, 8, 8, 15, 20, 20, 27, 40, 47, 62, 78, 73, 104, 132, 152, 172, 223, 211, 298, 324, 387, 394, 509, 470, 640, 645, 775, 756, 1015, 916, 1265, 1146, 1445, 1403, 1852, 1576, 2200, 1953, 2565, 2330, 3143

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,17

LINKS

Table of n, a(n) for n=0..58.

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p) * (n-i-j-k-l-m-o-p), where c = A010051.

a(n) = A326455(n) - A326456(n) - A326457(n) - A326458(n) - A326459(n) - A326460(n) - A326461(n) - A326462(n).

MATHEMATICA

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[(n-i-j-k-l-m-o-p) * (PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[p] - PrimePi[p - 1]) (PrimePi[n - i - j - k - l - m - o - p] - PrimePi[n - i - j - k - l - m - o - p - 1]), {i, j, Floor[(n - j - k - l - m - o - p)/2]}], {j, k, Floor[(n - k - l - m - o - p)/3]}], {k, l, Floor[(n - l - m - o - p)/4]}], {l, m, Floor[(n - m - o - p)/5]}], {m, o, Floor[(n - o - p)/6]}], {o, p, Floor[(n - p)/7]}], {p, Floor[n/8]}], {n, 0, 50}]

CROSSREFS

Cf. A010051, A259198, A326455, A326456, A326457, A326458, A326459, A326460, A326461, A326462.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 06 2019

STATUS

approved