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Search: a283762 -id:a283762
Displaying 1-9 of 9 results found. page 1
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A341982 Number of ways to write n as an ordered sum of 2 primes (counting 1 as a prime). +10
9
1, 2, 3, 2, 3, 2, 4, 2, 3, 0, 4, 2, 5, 2, 4, 0, 6, 2, 6, 2, 5, 0, 8, 2, 5, 0, 4, 0, 8, 2, 6, 2, 7, 0, 8, 0, 5, 2, 6, 0, 10, 2, 8, 2, 7, 0, 12, 2, 8, 0, 6, 0, 12, 2, 6, 0, 7, 0, 14, 2, 7, 2, 10, 0, 12, 0, 6, 2, 10, 0, 14, 2, 11, 2, 10, 0, 14, 0, 10, 2, 9, 0, 18, 2, 9, 0, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 2..10000
FORMULA
G.f.: ( x + Sum_{k>=1} x^prime(k) )^2.
MAPLE
b:= proc(n) option remember; series(`if`(n=0, 1, x*add(
`if`(j=1 or isprime(j), b(n-j), 0), j=1..n)), x, 3)
end:
a:= n-> coeff(b(n), x, 2):
seq(a(n), n=2..88); # Alois P. Heinz, Feb 24 2021
MATHEMATICA
nmax = 88; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &
CROSSREFS
Cf. A008578, A073610, A096139, A280917, A283762, A341945, A341973, A341983, A341984.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 24 2021
STATUS
approved
A341983 Number of ways to write n as an ordered sum of 4 primes (counting 1 as a prime). +10
9
1, 4, 10, 16, 23, 28, 38, 44, 55, 52, 62, 60, 84, 80, 106, 88, 123, 108, 160, 128, 184, 136, 214, 168, 261, 172, 270, 168, 304, 204, 352, 200, 382, 232, 442, 264, 470, 232, 502, 268, 557, 300, 608, 292, 672, 340, 722, 372, 789, 356, 856, 396, 900, 432, 968, 380, 1024, 432 (list; graph; refs; listen; history; text; internal format)
OFFSET
4,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 4..10000
FORMULA
G.f.: ( x + Sum_{k>=1} x^prime(k) )^4.
MAPLE
b:= proc(n) option remember; series(`if`(n=0, 1, x*add(
`if`(j=1 or isprime(j), b(n-j), 0), j=1..n)), x, 5)
end:
a:= n-> coeff(b(n), x, 4):
seq(a(n), n=4..61); # Alois P. Heinz, Feb 24 2021
MATHEMATICA
nmax = 61; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^4, {x, 0, nmax}], x] // Drop[#, 4] &
CROSSREFS
Cf. A008578, A280917, A283762, A340960, A341947, A341975, A341982, A341984.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 24 2021
STATUS
approved
A341984 Number of ways to write n as an ordered sum of 5 primes (counting 1 as a prime). +10
8
1, 5, 15, 30, 50, 71, 100, 130, 170, 195, 231, 250, 310, 340, 420, 430, 525, 535, 685, 680, 851, 800, 1025, 970, 1280, 1145, 1470, 1250, 1685, 1440, 1991, 1600, 2230, 1790, 2615, 2070, 2985, 2190, 3250, 2410, 3700, 2665, 4125, 2840, 4560, 3200, 5135, 3470, 5670, 3705, 6226, 4120 (list; graph; refs; listen; history; text; internal format)
OFFSET
5,2
LINKS
Table of n, a(n) for n=5..56.
FORMULA
G.f.: ( x + Sum_{k>=1} x^prime(k) )^5.
MAPLE
b:= proc(n) option remember; series(`if`(n=0, 1, x*add(
`if`(j=1 or isprime(j), b(n-j), 0), j=1..n)), x, 6)
end:
a:= n-> coeff(b(n), x, 5):
seq(a(n), n=5..56); # Alois P. Heinz, Feb 24 2021
MATHEMATICA
nmax = 56; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^5, {x, 0, nmax}], x] // Drop[#, 5] &
CROSSREFS
Cf. A008578, A280917, A283762, A340961, A341948, A341976, A341982, A341983.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 24 2021
STATUS
approved
A341985 Number of ways to write n as an ordered sum of 6 primes (counting 1 as a prime). +10
5
1, 6, 21, 50, 96, 156, 237, 336, 465, 596, 747, 882, 1077, 1260, 1536, 1736, 2067, 2286, 2761, 3030, 3627, 3842, 4578, 4806, 5826, 6000, 7167, 7116, 8562, 8430, 10318, 9906, 12093, 11396, 14286, 13386, 16868, 15270, 19242, 17180, 22218, 19536, 25393, 21750, 28680, 24456 (list; graph; refs; listen; history; text; internal format)
OFFSET
6,2
LINKS
Table of n, a(n) for n=6..51.
FORMULA
G.f.: ( x + Sum_{k>=1} x^prime(k) )^6.
MATHEMATICA
nmax = 51; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^6, {x, 0, nmax}], x] // Drop[#, 6] &
CROSSREFS
Cf. A008578, A280917, A283762, A340962, A341949, A341977, A341982, A341983, A341984, A341986, A341987, A341988, A341989.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 24 2021
STATUS
approved
A341986 Number of ways to write n as an ordered sum of 7 primes (counting 1 as a prime). +10
5
1, 7, 28, 77, 168, 308, 511, 785, 1155, 1603, 2142, 2723, 3430, 4207, 5202, 6216, 7497, 8729, 10451, 12061, 14350, 16205, 19033, 21182, 24934, 27482, 32109, 34587, 40139, 42714, 49791, 52290, 60718, 62699, 73297, 75278, 88571, 89488, 104993, 104482, 123760, 122066 (list; graph; refs; listen; history; text; internal format)
OFFSET
7,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 7..10000
FORMULA
G.f.: ( x + Sum_{k>=1} x^prime(k) )^7.
MAPLE
b:= proc(n) option remember; series(`if`(n=0, 1, x*add(
`if`(j=1 or isprime(j), b(n-j), 0), j=1..n)), x, 8)
end:
a:= n-> coeff(b(n), x, 7):
seq(a(n), n=7..48); # Alois P. Heinz, Feb 25 2021
MATHEMATICA
nmax = 48; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^7, {x, 0, nmax}], x] // Drop[#, 7] &
CROSSREFS
Cf. A008578, A280917, A283762, A340963, A341950, A341978, A341982, A341983, A341984, A341985, A341987, A341988, A341989.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 24 2021
STATUS
approved
A341987 Number of ways to write n as an ordered sum of 8 primes (counting 1 as a prime). +10
4
1, 8, 36, 112, 274, 560, 1016, 1688, 2647, 3928, 5580, 7568, 9990, 12832, 16332, 20336, 25167, 30472, 37004, 44136, 53054, 62272, 73788, 85240, 100276, 114752, 134072, 151144, 174834, 194616, 224304, 247240, 283467, 308448, 352668, 381032, 436368, 467272, 533520 (list; graph; refs; listen; history; text; internal format)
OFFSET
8,2
LINKS
Table of n, a(n) for n=8..46.
FORMULA
G.f.: ( x + Sum_{k>=1} x^prime(k) )^8.
MATHEMATICA
nmax = 46; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^8, {x, 0, nmax}], x] // Drop[#, 8] &
CROSSREFS
Cf. A008578, A280917, A283762, A340964, A341951, A341979, A341982, A341983, A341984, A341985, A341986, A341988, A341989.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 24 2021
STATUS
approved
A341988 Number of ways to write n as an ordered sum of 9 primes (counting 1 as a prime). +10
4
1, 9, 45, 156, 423, 954, 1887, 3384, 5661, 8935, 13446, 19332, 26838, 36126, 47691, 61668, 78696, 98631, 122665, 150516, 184230, 222438, 268146, 318564, 379383, 445572, 525942, 610344, 712872, 817290, 947166, 1075680, 1238148, 1391475, 1591236, 1773684, 2022241 (list; graph; refs; listen; history; text; internal format)
OFFSET
9,2
LINKS
Table of n, a(n) for n=9..45.
FORMULA
G.f.: ( x + Sum_{k>=1} x^prime(k) )^9.
MATHEMATICA
nmax = 45; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^9, {x, 0, nmax}], x] // Drop[#, 9] &
CROSSREFS
Cf. A008578, A280917, A283762, A340965, A341719, A341980, A341982, A341983, A341984, A341985, A341986, A341987, A341989.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 24 2021
STATUS
approved
A341989 Number of ways to write n as an ordered sum of 10 primes (counting 1 as a prime). +10
4
1, 10, 55, 210, 625, 1542, 3310, 6390, 11400, 19090, 30353, 46060, 67210, 94780, 130230, 174862, 230650, 298800, 382115, 482090, 603373, 746860, 918770, 1118100, 1355110, 1626742, 1949190, 2312380, 2740220, 3212640, 3769784, 4375900, 5092485, 5854680, 6758935, 7703112 (list; graph; refs; listen; history; text; internal format)
OFFSET
10,2
LINKS
Table of n, a(n) for n=10..45.
FORMULA
G.f.: ( x + Sum_{k>=1} x^prime(k) )^10.
MAPLE
b:= proc(n) option remember; series(`if`(n=0, 1, x*add(
`if`(j=1 or isprime(j), b(n-j), 0), j=1..n)), x, 11)
end:
a:= n-> coeff(b(n), x, 10):
seq(a(n), n=10..45); # Alois P. Heinz, Feb 25 2021
MATHEMATICA
nmax = 45; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^10, {x, 0, nmax}], x] // Drop[#, 10] &
CROSSREFS
Cf. A008578, A280917, A283762, A340966, A341972, A341981, A341982, A341983, A341984, A341985, A341986, A341987, A341988.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 24 2021
STATUS
approved
A347745 Number of compositions (ordered partitions) of n into at most 3 noncomposite parts. +10
4
1, 1, 2, 4, 6, 9, 10, 12, 13, 15, 15, 16, 13, 18, 17, 24, 19, 25, 18, 30, 24, 36, 23, 37, 23, 44, 29, 45, 19, 43, 20, 54, 30, 54, 25, 60, 29, 67, 29, 60, 21, 70, 28, 78, 38, 77, 31, 88, 33, 95, 44, 91, 30, 100, 30, 110, 42, 97, 25, 109, 35, 129, 49, 113, 31, 135 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
"Noncomposite" means "1 or a prime".
LINKS
Table of n, a(n) for n=0..65.
MATHEMATICA
Table[Length@Flatten[Permutations/@IntegerPartitions[n, 3, Join[{1}, Prime@Range@PrimePi@n]], 1], {n, 0, 65}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)
CROSSREFS
Cf. A008578, A280917, A283762, A347740, A347744.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 11 2021
STATUS
approved
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Last modified August 30 05:37 EDT 2024. Contains 375526 sequences. (Running on oeis4.)