Search: a283762 -id:a283762
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A341982
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Number of ways to write n as an ordered sum of 2 primes (counting 1 as a prime).
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+0
9
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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
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OFFSET
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2,2
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LINKS
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FORMULA
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G.f.: ( x + Sum_{k>=1} x^prime(k) )^2.
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MAPLE
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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):
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MATHEMATICA
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nmax = 88; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A347745
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Number of compositions (ordered partitions) of n into at most 3 noncomposite parts.
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+0
4
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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
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OFFSET
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0,3
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COMMENTS
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"Noncomposite" means "1 or a prime".
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LINKS
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MATHEMATICA
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Table[Length@Flatten[Permutations/@IntegerPartitions[n, 3, Join[{1}, Prime@Range@PrimePi@n]], 1], {n, 0, 65}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A341983
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Number of ways to write n as an ordered sum of 4 primes (counting 1 as a prime).
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+0
9
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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
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OFFSET
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4,2
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LINKS
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FORMULA
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G.f.: ( x + Sum_{k>=1} x^prime(k) )^4.
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MAPLE
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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):
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MATHEMATICA
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nmax = 61; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^4, {x, 0, nmax}], x] // Drop[#, 4] &
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A341989
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Number of ways to write n as an ordered sum of 10 primes (counting 1 as a prime).
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+0
4
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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
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OFFSET
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10,2
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LINKS
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FORMULA
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G.f.: ( x + Sum_{k>=1} x^prime(k) )^10.
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MAPLE
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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):
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MATHEMATICA
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nmax = 45; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^10, {x, 0, nmax}], x] // Drop[#, 10] &
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CROSSREFS
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Cf. A008578, A280917, A283762, A340966, A341972, A341981, A341982, A341983, A341984, A341985, A341986, A341987, A341988.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A341986
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Number of ways to write n as an ordered sum of 7 primes (counting 1 as a prime).
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+0
5
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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
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OFFSET
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7,2
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LINKS
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FORMULA
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G.f.: ( x + Sum_{k>=1} x^prime(k) )^7.
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MAPLE
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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):
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MATHEMATICA
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nmax = 48; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^7, {x, 0, nmax}], x] // Drop[#, 7] &
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CROSSREFS
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Cf. A008578, A280917, A283762, A340963, A341950, A341978, A341982, A341983, A341984, A341985, A341987, A341988, A341989.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A341988
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Number of ways to write n as an ordered sum of 9 primes (counting 1 as a prime).
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+0
4
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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
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OFFSET
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9,2
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LINKS
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FORMULA
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G.f.: ( x + Sum_{k>=1} x^prime(k) )^9.
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MATHEMATICA
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nmax = 45; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^9, {x, 0, nmax}], x] // Drop[#, 9] &
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CROSSREFS
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Cf. A008578, A280917, A283762, A340965, A341719, A341980, A341982, A341983, A341984, A341985, A341986, A341987, A341989.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A341987
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Number of ways to write n as an ordered sum of 8 primes (counting 1 as a prime).
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+0
4
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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
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OFFSET
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8,2
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LINKS
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FORMULA
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G.f.: ( x + Sum_{k>=1} x^prime(k) )^8.
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MATHEMATICA
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nmax = 46; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^8, {x, 0, nmax}], x] // Drop[#, 8] &
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CROSSREFS
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Cf. A008578, A280917, A283762, A340964, A341951, A341979, A341982, A341983, A341984, A341985, A341986, A341988, A341989.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A341985
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Number of ways to write n as an ordered sum of 6 primes (counting 1 as a prime).
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+0
5
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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
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OFFSET
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6,2
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LINKS
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FORMULA
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G.f.: ( x + Sum_{k>=1} x^prime(k) )^6.
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MATHEMATICA
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nmax = 51; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^6, {x, 0, nmax}], x] // Drop[#, 6] &
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CROSSREFS
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Cf. A008578, A280917, A283762, A340962, A341949, A341977, A341982, A341983, A341984, A341986, A341987, A341988, A341989.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A341984
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Number of ways to write n as an ordered sum of 5 primes (counting 1 as a prime).
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+0
8
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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
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OFFSET
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5,2
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LINKS
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FORMULA
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G.f.: ( x + Sum_{k>=1} x^prime(k) )^5.
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MAPLE
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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):
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MATHEMATICA
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nmax = 56; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^5, {x, 0, nmax}], x] // Drop[#, 5] &
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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