cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A326540 Sum of all the parts in the partitions of n into 9 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 19, 20, 42, 44, 69, 96, 100, 130, 189, 196, 261, 300, 341, 384, 528, 544, 700, 756, 888, 988, 1287, 1240, 1599, 1638, 2021, 2024, 2655, 2438, 3243, 3120, 3920, 3850, 4998, 4420, 6042, 5616, 7205, 6608
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

Cf. A010051, A259200, A326541, A326542, A326543, A326544, A326545, A326546, A326547, A326548, A326549.

Programs

  • Mathematica
    Table[n * Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[(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[q] - PrimePi[q - 1]) (PrimePi[n - i - j - k - l - m - o - p - q] - PrimePi[n - i - j - k - l - m - o - p - q - 1]), {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 50}]

Formula

a(n) = n * Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} c(q) * c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p-q), where c = A010051.
a(n) = n * A259200(n).
a(n) = A326541(n) + A326542(n) + A326543(n) + A326544(n) + A326545(n) + A326546(n) + A326547(n) + A326548(n) + A326549(n).

A326541 Sum of the smallest parts of the partitions of n into 9 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 4, 6, 8, 8, 10, 15, 14, 19, 20, 24, 24, 34, 32, 44, 42, 53, 52, 73, 62, 87, 78, 107, 92, 136, 106, 161, 130, 186, 154, 232, 170, 271, 208, 315, 236, 373, 266, 439, 310, 493, 354, 589, 392, 668
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

Cf. A010051, A259200, A326540, A326542, A326543, A326544, A326545, A326546, A326547, A326548, A326549.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[q * (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[q] - PrimePi[q - 1]) (PrimePi[n - i - j - k - l - m - o - p - q] - PrimePi[n - i - j - k - l - m - o - p - q - 1]), {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 50}]

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} c(q) * c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p-q) * q, where c = A010051.
a(n) = A326540(n) - A326542(n) - A326543(n) - A326544(n) - A326545(n) - A326546(n) - A326547(n) - A326548(n) - A326549(n).

A326542 Sum of the eighth largest parts of the partitions of n into 9 primes.

Original entry on oeis.org

0, 0, 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, 15, 15, 19, 22, 24, 26, 34, 36, 44, 47, 53, 59, 73, 71, 87, 93, 109, 109, 138, 128, 163, 157, 190, 190, 238, 210, 277, 262, 325, 300, 387, 344, 457, 399, 515, 464, 617, 515
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

Cf. A010051, A259200, A326540, A326541, A326543, A326544, A326545, A326546, A326547, A326548, A326549.

Programs

  • Mathematica
    Table[Total[Select[IntegerPartitions[n,{9}],AllTrue[#,PrimeQ]&][[All,8]]],{n,0,70}] (* Harvey P. Dale, May 03 2022 *)

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} c(q) * c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p-q) * p, where c = A010051.
a(n) = A326540(n) - A326541(n) - A326543(n) - A326544(n) - A326545(n) - A326546(n) - A326547(n) - A326548(n) - A326549(n).

A326543 Sum of the seventh largest parts of the partitions of n into 9 primes.

Original entry on oeis.org

0, 0, 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, 16, 15, 21, 22, 26, 26, 38, 36, 49, 47, 60, 59, 84, 73, 103, 95, 127, 111, 163, 132, 194, 163, 229, 196, 285, 220, 342, 278, 398, 316, 477, 366, 566, 427, 640, 494, 768, 557
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

Cf. A010051, A259200, A326540, A326541, A326542, A326544, A326545, A326546, A326547, A326548, A326549.

Programs

  • Mathematica
    Table[Sum[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[q] - PrimePi[q - 1]) (PrimePi[n - i - j - k - l - m - o - p - q] - PrimePi[n - i - j - k - l - m - o - p - q - 1]), {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 50}]

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} c(q) * c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p-q) * o, where c = A010051.
a(n) = A326540(n) - A326541(n) - A326542(n) - A326544(n) - A326545(n) - A326546(n) - A326547(n) - A326548(n) - A326549(n).

A326544 Sum of the sixth largest parts of the partitions of n into 9 primes.

Original entry on oeis.org

0, 0, 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, 16, 17, 21, 24, 26, 30, 38, 41, 49, 56, 62, 71, 88, 88, 107, 114, 133, 138, 173, 162, 206, 205, 247, 246, 311, 282, 372, 352, 438, 404, 527, 469, 624, 553, 716, 641, 862, 731
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

Cf. A010051, A259200, A326540, A326541, A326542, A326543, A326545, A326546, A326547, A326548, A326549.

Programs

  • Mathematica
    Table[Total[Select[IntegerPartitions[n,{9}],AllTrue[#,PrimeQ]&][[All,6]]],{n,0,70}] (* Harvey P. Dale, Jan 01 2023 *)

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} c(q) * c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p-q) * m, where c = A010051.
a(n) = A326540(n) - A326541(n) - A326542(n) - A326543(n) - A326545(n) - A326546(n) - A326547(n) - A326548(n) - A326549(n).

A326546 Sum of the fourth largest parts in the partitions of n into 9 primes.

Original entry on oeis.org

0, 0, 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, 18, 19, 23, 30, 32, 38, 49, 54, 65, 74, 83, 97, 120, 118, 148, 159, 189, 193, 242, 231, 299, 293, 359, 357, 454, 407, 543, 517, 649, 600, 788, 706, 952, 851, 1102, 1004, 1351
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

Cf. A010051, A259200, A326540, A326541, A326542, A326543, A326544, A326545, A326547, A326548, A326549.

Programs

  • Mathematica
    Table[Total[Select[IntegerPartitions[n,{9}],AllTrue[#,PrimeQ]&][[;;,4]]],{n,0,70}] (* Harvey P. Dale, Mar 09 2023 *)

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} c(q) * c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p-q) * k, where c is the prime characteristic (A010051).
a(n) = A326540(n) - A326541(n) - A326542(n) - A326543(n) - A326544(n) - A326545(n) - A326547(n) - A326548(n) - A326549(n).

A326547 Sum of the third largest parts in the partitions of n into 9 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 5, 5, 8, 10, 12, 14, 22, 21, 30, 34, 41, 44, 65, 64, 83, 88, 108, 115, 154, 142, 189, 191, 241, 233, 314, 289, 384, 369, 472, 455, 604, 537, 729, 687, 889, 816, 1089, 974, 1318, 1179, 1548, 1392
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

Cf. A010051, A259200, A326540, A326541, A326542, A326543, A326544, A326545, A326546, A326548, A326549.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[j * (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[q] - PrimePi[q - 1]) (PrimePi[n - i - j - k - l - m - o - p - q] - PrimePi[n - i - j - k - l - m - o - p - q - 1]), {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 50}]

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} c(q) * c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p-q) * j, where c = A010051.
a(n) = A326540(n) - A326541(n) - A326542(n) - A326543(n) - A326544(n) - A326545(n) - A326546(n) - A326548(n) - A326549(n).

A326548 Sum of the second largest parts of the partitions of n into 9 primes.

Original entry on oeis.org

0, 0, 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, 26, 31, 38, 46, 53, 62, 81, 86, 105, 119, 140, 162, 204, 205, 263, 275, 341, 356, 452, 435, 562, 559, 700, 709, 904, 829, 1101, 1060, 1343, 1272, 1643, 1485, 1982, 1795, 2318
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

Cf. A010051, A259200, A326540, A326541, A326542, A326543, A326544, A326545, A326546, A326547, A326549.

Programs

  • Mathematica
    Table[Sum[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[q] - PrimePi[q - 1]) (PrimePi[n - i - j - k - l - m - o - p - q] - PrimePi[n - i - j - k - l - m - o - p - q - 1]), {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 50}]

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} c(q) * c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p-q) * i, where c = A010051.
a(n) = A326540(n) - A326541(n) - A326542(n) - A326543(n) - A326544(n) - A326545(n) - A326546(n) - A326547(n) - A326549(n).

A326549 Sum of the largest parts of the partitions of n into 9 primes.

Original entry on oeis.org

0, 0, 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, 43, 47, 67, 78, 85, 104, 144, 152, 202, 223, 254, 298, 385, 387, 485, 509, 609, 640, 827, 775, 1017, 1015, 1230, 1265, 1584, 1445, 1944, 1852, 2301, 2200, 2840, 2565, 3439
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

Cf. A010051, A259200, A326540, A326541, A326542, A326543, A326544, A326545, A326546, A326547, A326548.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[(n-i-j-k-l-m-o-p-q) * (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[q] - PrimePi[q - 1]) (PrimePi[n - i - j - k - l - m - o - p - q] - PrimePi[n - i - j - k - l - m - o - p - q - 1]), {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 50}]

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} c(q) * c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p-q) * (n-i-j-k-l-m-o-p-q), where c = A010051.
a(n) = A326540(n) - A326541(n) - A326542(n) - A326543(n) - A326544(n) - A326545(n) - A326546(n) - A326547(n) - A326548(n).
Showing 1-9 of 9 results.

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