A372887
Number of integer partitions of n whose distinct parts are the binary indices of some prime number.
Original entry on oeis.org
0, 0, 1, 1, 3, 3, 6, 8, 12, 14, 21, 29, 36, 48, 56, 74, 94, 123, 144, 195, 235, 301, 356, 456, 538, 679, 803, 997, 1189, 1467, 1716, 2103, 2488, 2968, 3517, 4185, 4907, 5834, 6850, 8032, 9459, 11073, 12933, 15130, 17652, 20480, 24011, 27851, 32344, 37520
Offset: 0
The partition y = (4,3,1,1) has distinct parts {1,3,4}, which are the binary indices of 13, which is prime, so y is counted under a(9).
The a(2) = 1 through a(9) = 14 partitions:
(2) (21) (22) (221) (51) (331) (431) (3321)
(31) (311) (222) (421) (521) (4221)
(211) (2111) (321) (511) (2222) (4311)
(2211) (2221) (3221) (5211)
(3111) (3211) (3311) (22221)
(21111) (22111) (4211) (32211)
(31111) (5111) (33111)
(211111) (22211) (42111)
(32111) (51111)
(221111) (222111)
(311111) (321111)
(2111111) (2211111)
(3111111)
(21111111)
These partitions have Heinz numbers
A372850.
A014499 lists binary indices of prime numbers.
A372689 lists numbers whose binary indices sum to a prime.
A372885 lists primes whose binary indices sum to a prime, indices
A372886.
Binary indices:
-
Table[Length[Select[IntegerPartitions[n], PrimeQ[Total[2^(Union[#]-1)]]&]],{n,0,30}]
A316154
Number of integer partitions of prime(n) into a prime number of prime parts.
Original entry on oeis.org
0, 0, 1, 2, 3, 5, 9, 12, 19, 39, 50, 93, 136, 166, 239, 409, 682, 814, 1314, 1774, 2081, 3231, 4272, 6475, 11077, 14270, 16265, 20810, 23621, 30031, 68251, 85326, 118917, 132815, 226097, 251301, 342448, 463940, 565844, 759873, 1015302, 1117708, 1787452, 1961624
Offset: 1
The a(7) = 9 partitions of 17 into a prime number of prime parts: (13,2,2), (11,3,3), (7,7,3), (7,5,5), (7,3,3,2,2), (5,5,3,2,2), (5,3,3,3,3), (5,2,2,2,2,2,2), (3,3,3,2,2,2,2).
Cf.
A000040,
A000586,
A000607,
A038499,
A056768,
A064688,
A070215,
A085755,
A302590,
A316092,
A316153,
A316185,
A344782.
-
b:= proc(n, p, c) option remember; `if`(n=0 or p=2,
`if`(n::even and isprime(c+n/2), 1, 0),
`if`(p>n, 0, b(n-p, p, c+1))+b(n, prevprime(p), c))
end:
a:= n-> b(ithprime(n)$2, 0):
seq(a(n), n=1..50); # Alois P. Heinz, Jun 26 2018
-
Table[Length[Select[IntegerPartitions[Prime[n]],And[PrimeQ[Length[#]],And@@PrimeQ/@#]&]],{n,20}]
(* Second program: *)
b[n_, p_, c_] := b[n, p, c] = If[n == 0 || p == 2, If[EvenQ[n] && PrimeQ[c + n/2], 1, 0], If[p>n, 0, b[n - p, p, c + 1]] + b[n, NextPrime[p, -1], c]];
a[n_] := b[Prime[n], Prime[n], 0];
Array[a, 50] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)
-
seq(n)={my(p=vector(n,k,prime(k))); my(v=Vec(1/prod(k=1, n, 1 - x^p[k]*y + O(x*x^p[n])))); vector(n, k, sum(i=1, k, polcoeff(v[1+p[k]], p[i])))} \\ Andrew Howroyd, Jun 26 2018
A316185
Number of strict integer partitions of the n-th prime into a prime number of prime parts.
Original entry on oeis.org
0, 0, 1, 1, 0, 1, 0, 2, 2, 3, 5, 5, 6, 8, 10, 13, 18, 20, 26, 32, 34, 45, 54, 66, 90, 106, 117, 135, 142, 165, 269, 311, 375, 398, 546, 579, 689, 823, 938, 1107, 1301, 1352, 1790, 1850, 2078, 2153, 2878, 3811, 4241, 4338, 4828, 5495, 5637, 7076, 8000, 9032
Offset: 1
The a(14) = 8 partitions of 43 into a prime number of distinct prime parts: (41,2), (31,7,5), (29,11,3), (23,17,3), (23,13,7), (19,17,7), (19,13,11), (17,11,7,5,3).
Cf.
A000586,
A000607,
A038499,
A045450,
A056768,
A064688,
A070215,
A085755,
A302590,
A316092,
A316153,
A316154.
-
h:= proc(n) option remember; `if`(n=0, 0,
`if`(isprime(n), n, h(n-1)))
end:
b:= proc(n, i, c) option remember; `if`(n=0,
`if`(isprime(c), 1, 0), `if`(i<2, 0, b(n, h(i-1), c)+
`if`(i>n, 0, b(n-i, h(min(n-i, i-1)), c+1))))
end:
a:= n-> b(ithprime(n)$2, 0):
seq(a(n), n=1..56); # Alois P. Heinz, May 26 2021
-
Table[Length[Select[IntegerPartitions[Prime[n]],And[UnsameQ@@#,PrimeQ[Length[#]],And@@PrimeQ/@#]&]],{n,10}]
(* Second program: *)
h[n_] := h[n] = If[n == 0, 0, If[PrimeQ[n], n, h[n - 1]]];
b[n_, i_, c_] := b[n, i, c] = If[n == 0,
If[PrimeQ[c], 1, 0], If[i < 2, 0, b[n, h[i - 1], c] +
If[i > n, 0, b[n - i, h[Min[n - i, i - 1]], c + 1]]]];
a[n_] := b[Prime[n], Prime[n], 0];
Array[a, 56] (* Jean-François Alcover, Jun 11 2021, after Alois P. Heinz *)
-
seq(n)={my(p=vector(n, k, prime(k))); my(v=Vec(prod(k=1, n, 1 + x^p[k]*y + O(x*x^p[n])))); vector(n, k, sum(i=1, k, polcoeff(v[1+p[k]], p[i])))} \\ Andrew Howroyd, Jun 26 2018
A316153
Heinz numbers of integer partitions of prime numbers into a prime number of prime parts.
Original entry on oeis.org
15, 33, 45, 93, 153, 177, 275, 327, 369, 405, 425, 537, 603, 605, 775, 831, 891, 1025, 1059, 1125, 1413, 1445, 1475, 1641, 1705, 1719, 1761, 2057, 2075, 2319, 2511, 2577, 2979, 3175, 3179, 3189, 3459, 3485, 3603, 3609, 3825, 3925, 4299, 4475, 4497, 4565, 4581
Offset: 1
Sequence of integer partitions of prime numbers into a prime number of prime parts, preceded by their Heinz numbers, begins:
15: (3,2)
33: (5,2)
45: (3,2,2)
93: (11,2)
153: (7,2,2)
177: (17,2)
275: (5,3,3)
327: (29,2)
369: (13,2,2)
405: (3,2,2,2,2)
425: (7,3,3)
537: (41,2)
603: (19,2,2)
605: (5,5,3)
775: (11,3,3)
831: (59,2)
891: (5,2,2,2,2)
Cf.
A000586,
A000607,
A038499,
A056239,
A056768,
A064688,
A070215,
A085755,
A302590,
A316092,
A316151.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],And[PrimeQ[PrimeOmega[#]],PrimeQ[Total[primeMS[#]]],And@@PrimeQ/@primeMS[#]]&]
A316151
Heinz numbers of strict integer partitions of prime numbers into prime parts.
Original entry on oeis.org
3, 5, 11, 15, 17, 31, 33, 41, 59, 67, 83, 93, 109, 127, 157, 177, 179, 191, 211, 241, 277, 283, 327, 331, 353, 367, 401, 431, 461, 509, 537, 547, 563, 587, 599, 617, 709, 739, 773, 797, 831, 859, 877, 919, 967, 991, 1031, 1059, 1063, 1087, 1153, 1171, 1201
Offset: 1
Sequence of strict integer partitions of prime numbers into prime parts, preceded by their Heinz numbers, begins:
3: (2)
5: (3)
11: (5)
15: (3,2)
17: (7)
31: (11)
33: (5,2)
41: (13)
59: (17)
67: (19)
83: (23)
93: (11,2)
-
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And[SquareFreeQ[#],PrimeQ[Total[primeMS[#]]],And@@PrimeQ/@primeMS[#]]&]
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