A102378 -id:A102378 - cp's OEIS frontend

A325094 Write n as a sum of distinct powers of 2, then take the primes of those powers of 2 and multiply them together.

1, 2, 3, 6, 7, 14, 21, 42, 19, 38, 57, 114, 133, 266, 399, 798, 53, 106, 159, 318, 371, 742, 1113, 2226, 1007, 2014, 3021, 6042, 7049, 14098, 21147, 42294, 131, 262, 393, 786, 917, 1834, 2751, 5502, 2489, 4978, 7467, 14934, 17423, 34846, 52269, 104538, 6943

Offset: 0

Gus Wiseman, Mar 27 2019

The sorted sequence is A325093.

For example, 11 = 1 + 2 + 8, so a(11) = prime(1) * prime(2) * prime(8) = 114.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    6: {1,2}
    7: {4}
   14: {1,4}
   21: {2,4}
   42: {1,2,4}
   19: {8}
   38: {1,8}
   57: {2,8}
  114: {1,2,8}
  133: {4,8}
  266: {1,4,8}
  399: {2,4,8}
  798: {1,2,4,8}
   53: {16}
  106: {1,16}
  159: {2,16}
  318: {1,2,16}
  371: {4,16}

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000720, A001222, A005117, A018819, A019565, A033844, A056239, A102378, A112798, A247935, A318400.

Cf. A325091, A325092, A325093, A325106.

P:= [seq(ithprime(2^i),i=0..10)]:
f:= proc(n) local L,i;
  L:= convert(n,base,2);
  mul(P[i]^L[i],i=1..nops(L))
end proc:
map(f, [$0..100]); # Robert Israel, Mar 28 2019

Table[Times@@MapIndexed[If[#1==0,1,Prime[2^(#2[[1]]-1)]]&,Reverse[IntegerDigits[n,2]]],{n,0,100}]

A324927 Matula-Goebel numbers of rooted trees of depth 2. Numbers that are not powers of 2 but whose prime indices are all powers of 2.

Original entry on oeis.org

3, 6, 7, 9, 12, 14, 18, 19, 21, 24, 27, 28, 36, 38, 42, 48, 49, 53, 54, 56, 57, 63, 72, 76, 81, 84, 96, 98, 106, 108, 112, 114, 126, 131, 133, 144, 147, 152, 159, 162, 168, 171, 189, 192, 196, 212, 216, 224, 228, 243, 252, 262, 266, 288, 294, 304, 311, 318

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

Numbers n such that A109082(n) = 2.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Also Heinz numbers of integer partitions into powers of 2 with at least one part > 1 (counted by A102378).

			The sequence of terms together with their prime indices begins:
   3: {2}
   6: {1,2}
   7: {4}
   9: {2,2}
  12: {1,1,2}
  14: {1,4}
  18: {1,2,2}
  19: {8}
  21: {2,4}
  24: {1,1,1,2}
  27: {2,2,2}
  28: {1,1,4}
  36: {1,1,2,2}
  38: {1,8}
  42: {1,2,4}
  48: {1,1,1,1,2}
  49: {4,4}
  53: {16}
  54: {1,2,2,2}
  56: {1,1,1,4}

Cf. A000081, A000720, A003963, A007097, A018819, A033844, A056239, A102378, A112798, A302242, A318400, A324928, A324929.

Select[Range[100],And[!IntegerQ[Log[2,#]],And@@Cases[FactorInteger[#],{p_,_}:>IntegerQ[Log[2,PrimePi[p]]]]]&]

A324928 Matula-Goebel numbers of rooted trees of depth 3.

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 23, 25, 26, 30, 34, 35, 37, 39, 40, 43, 45, 46, 50, 51, 52, 60, 61, 65, 67, 68, 69, 70, 73, 74, 75, 78, 80, 85, 86, 89, 90, 91, 92, 95, 100, 102, 103, 104, 105, 107, 111, 115, 117, 119, 120, 122, 125, 129, 130, 134, 135, 136, 138, 140

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

Numbers n such that A109082(n) = 3.

			The sequence of all rooted trees of depth 3 together with their Matula-Goebel numbers begins:
   5: (((o)))
  10: (o((o)))
  13: ((o(o)))
  15: ((o)((o)))
  17: (((oo)))
  20: (oo((o)))
  23: (((o)(o)))
  25: (((o))((o)))
  26: (o(o(o)))
  30: (o(o)((o)))
  34: (o((oo)))
  35: (((o))(oo))
  37: ((oo(o)))
  39: ((o)(o(o)))
  40: (ooo((o)))
  43: ((o(oo)))
  45: ((o)(o)((o)))
  46: (o((o)(o)))
  50: (o((o))((o)))
  51: ((o)((oo)))
  52: (oo(o(o)))
  60: (oo(o)((o)))

Cf. A000081, A003963, A007097, A102378, A318400, A324927, A324930.

Select[Range[100],Length[NestWhileList[Times@@PrimePi/@FactorInteger[#][[All,1]]&,#,#>1&]]-1==3&]

A325093 Heinz numbers of integer partitions into distinct powers of 2.

Original entry on oeis.org

1, 2, 3, 6, 7, 14, 19, 21, 38, 42, 53, 57, 106, 114, 131, 133, 159, 262, 266, 311, 318, 371, 393, 399, 622, 719, 742, 786, 798, 917, 933, 1007, 1113, 1438, 1619, 1834, 1866, 2014, 2157, 2177, 2226, 2489, 2751, 3021, 3238, 3671, 4314, 4354, 4857, 4978, 5033

Offset: 1

Gus Wiseman, Mar 27 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are squarefree numbers whose prime indices are powers of 2. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    6: {1,2}
    7: {4}
   14: {1,4}
   19: {8}
   21: {2,4}
   38: {1,8}
   42: {1,2,4}
   53: {16}
   57: {2,8}
  106: {1,16}
  114: {1,2,8}
  131: {32}
  133: {4,8}
  159: {2,16}
  262: {1,32}
  266: {1,4,8}
  311: {64}

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000720, A001222, A018819, A033844, A056239, A102378, A112798, A318400, A325091, A325092.

P:= [seq(ithprime(2^i),i=0..20)]:f:= proc(S,N) option remember;
  if S = [] or S[1]>N then return {1} fi;
  procname(S[2..-1],N) union
    map(t -> S[1]*t, procname(S[2..-1], floor(N/S[1])))end proc:
sort(convert(f(P, P[20]),list));  # Robert Israel, Mar 28 2019

Select[Range[1000],SquareFreeQ[#]&&And@@IntegerQ/@Log[2,Cases[If[#==1,{},FactorInteger[#]],{p_,_}:>PrimePi[p]]]&]

isp2(q) = (q == 1) || (q == 2) || (ispower(q,,&p) && (p==2));
isok(n) = {if (issquarefree(n), my(f=factor(n)[,1]); for (k=1, #f, if (! isp2(primepi(f[k])), return (0));); return (1);); return (0);} \\ Michel Marcus, Mar 28 2019

A325091 Heinz numbers of integer partitions of powers of 2.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 10, 12, 16, 19, 34, 39, 49, 52, 53, 55, 63, 66, 70, 75, 81, 84, 88, 90, 94, 100, 108, 112, 120, 129, 131, 144, 160, 172, 192, 205, 246, 254, 256, 259, 311, 328, 333, 339, 341, 361, 370, 377, 391, 434, 444, 452, 465, 545, 558, 592, 598, 609, 614

Offset: 1

Gus Wiseman, Mar 27 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose sum of prime indices is a power of 2. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

1 is in the sequence because it has prime indices {} with sum 0 = 2^(-infinity).

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   7: {4}
   9: {2,2}
  10: {1,3}
  12: {1,1,2}
  16: {1,1,1,1}
  19: {8}
  34: {1,7}
  39: {2,6}
  49: {4,4}
  52: {1,1,6}
  53: {16}
  55: {3,5}
  63: {2,2,4}
  66: {1,2,5}
  70: {1,3,4}
  75: {2,3,3}
  81: {2,2,2,2}

Cf. A000720, A001222, A018819, A033844, A056239, A102378, A112798, A131577, A318400, A325092, A325093.

q:= n-> (t-> t=2^ilog2(t))(add(numtheory[pi](i[1])*i[2], i=ifactors(n)[2])):
select(q, [$1..1000])[];  # Alois P. Heinz, Mar 28 2019

Select[Range[100],#==1||IntegerQ[Log[2,Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]]]&]

A325092 Heinz numbers of integer partitions of powers of 2 into powers of 2.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 12, 16, 19, 49, 53, 63, 81, 84, 108, 112, 131, 144, 192, 256, 311, 361, 719, 931, 1197, 1539, 1596, 1619, 2052, 2128, 2401, 2736, 2809, 3087, 3648, 3671, 3969, 4116, 4864, 5103, 5292, 5488, 6561, 6804, 7056, 8161, 8748, 9072, 9408, 11664, 12096

Offset: 1

Gus Wiseman, Mar 27 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose prime indices are powers of 2 and whose sum of prime indices is also a power of 2. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

1 is in the sequence because it has prime indices {} with sum 0 = 2^(-infinity).

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    7: {4}
    9: {2,2}
   12: {1,1,2}
   16: {1,1,1,1}
   19: {8}
   49: {4,4}
   53: {16}
   63: {2,2,4}
   81: {2,2,2,2}
   84: {1,1,2,4}
  108: {1,1,2,2,2}
  112: {1,1,1,1,4}
  131: {32}
  144: {1,1,1,1,2,2}
  192: {1,1,1,1,1,1,2}
  256: {1,1,1,1,1,1,1,1}
  311: {64}

Cf. A000720, A001222, A018819, A033844, A056239, A102378, A112798, A318400, A325091, A325093.

q:= n-> andmap(t-> t=2^ilog2(t), (l-> [l[], add(i, i=l)])(
      map(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]))):
select(q, [$1..15000])[];  # Alois P. Heinz, Mar 28 2019

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pow2Q[n_]:=IntegerQ[Log[2,n]];
Select[Range[1000],#==1||pow2Q[Total[primeMS[#]]]&&And@@pow2Q/@primeMS[#]&]

A102379 a(n) is the minimal number of nodes in a binary tree of height n.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 17, 22, 29, 36, 46, 56, 69, 82, 100, 118, 141, 164, 194, 224, 261, 298, 345, 392, 449, 506, 576, 646, 729, 812, 913, 1014, 1133, 1252, 1394, 1536, 1701, 1866, 2061, 2256, 2481, 2706, 2968, 3230, 3529, 3828, 4174, 4520, 4913

Offset: 1

Mitch Harris, Jan 05 2005

nonn

Conjecture: Let b(n) be the number of fixed points of the set of binary partitions of n under Glaisher's function that proves Euler's odd-distinct theorem. Then b(1) = 1 and for n > 1, b(2*n) = b(2*n+1) = 2*a(n). - George Beck, Jul 23 2022

de Bruijn, N. G., On Mahler's partition problem. Nederl. Akad. Wetensch., Proc. 51, (1948) 659-669 = Indagationes Math. 10, 210-220 (1948).
Gonnet, Gaston H.; Olivie, Henk J.; and Wood, Derick, Height-ratio-balanced trees. Comput. J. 26 (1983), no. 2, 106-108.
Mahler, Kurt On a special functional equation. J. London Math. Soc. 15, (1940). 115-123.
Nievergelt, J.; Reingold, E. M., Binary search trees of bounded balance, SIAM J. Comput. 2 (1973), 33-43.

Cf. A000123, A033485, A102378.

Essentially partial sums of A040039.

from functools import cache
@cache
def a(n: int) -> int:
    return a(n - 1) + a(n // 2) + 1 if n > 1 else 0
print([a(n) for n in range(1, 51)])  # Peter Luschny, Jul 24 2022

a(n) = a(n-1) + a(floor(n/2)) + 1, a(1) = 0.
a(n) - a(n-1) = A018819(n+1).
G.f. A(x) satisfies (1-x)*A(x) = 2*(1 + x)*B(x^2), where B(x) is the g.f. of A033485.

A346912 a(0) = 1; a(n) = a(n-1) + a(floor(n/2)) + 1.

Original entry on oeis.org

1, 3, 7, 11, 19, 27, 39, 51, 71, 91, 119, 147, 187, 227, 279, 331, 403, 475, 567, 659, 779, 899, 1047, 1195, 1383, 1571, 1799, 2027, 2307, 2587, 2919, 3251, 3655, 4059, 4535, 5011, 5579, 6147, 6807, 7467, 8247, 9027, 9927, 10827, 11875, 12923, 14119, 15315

Offset: 0

Ilya Gutkovskiy, Aug 11 2021

nonn

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000123, A018819, A022907, A033485, A102378, A102379.

f:= proc(n) option remember; procname(n-1) + procname(floor(n/2)) + 1 end proc;
f(0):= 1:
map(f, [$1..50]); # Robert Israel, May 04 2025

a[0] = 1; a[n_] := a[n] = a[n - 1] + a[Floor[n/2]] + 1; Table[a[n], {n, 0, 47}]
nmax = 47; CoefficientList[Series[(1/(1 - x)) (-1 + 2 Product[1/(1 - x^(2^k)), {k, 0, Floor[Log[2, nmax]]}]), {x, 0, nmax}], x]

from itertools import islice
from collections import deque
def A346912_gen(): # generator of terms
    aqueue, f, b, a = deque([2]), True, 1, 2
    yield from (1, 3, 7)
    while True:
        a += b
        yield 4*a - 1
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A346912_list = list(islice(A346912_gen(),40)) # Chai Wah Wu, Jun 08 2022

G.f.: (1/(1 - x)) * (-1 + 2 * Product_{k>=0} 1/(1 - x^(2^k))).
a(n) = n + 1 + Sum_{k=1..n} a(floor(k/2)).
a(n) = 2 * A000123(n) - 1.
a(n) = 4 * A033485(n) - 1 for n > 0. - Hugo Pfoertner, Aug 12 2021
From Michael Tulskikh, Aug 12 2021: (Start)
2*a(2n) = a(2n-1) + a(2n+1).
a(2n) = a(2n-2) + a(n-1) + a(n) + 2.
a(2n) = 2*(Sum_{i=0..n} a(i)) - a(n) + 2n. (End)

A351621 a(1) = 1; a(n) = 1 + a(n-1) + Sum_{k=2..n} a(floor(n/k)).

Original entry on oeis.org

1, 3, 6, 12, 19, 32, 46, 69, 96, 133, 171, 234, 298, 379, 471, 595, 720, 891, 1063, 1288, 1531, 1815, 2100, 2496, 2900, 3371, 3873, 4479, 5086, 5848, 6611, 7530, 8491, 9580, 10691, 12088, 13486, 15059, 16700, 18642, 20585, 22885, 25186, 27818, 30580, 33630, 36681, 40363, 44060, 48208

Offset: 1

Ilya Gutkovskiy, Feb 20 2022

nonn

Partial sums of A345139.

Cf. A001906, A022825, A025523, A078346, A102378, A345139, A351620.

a[1] = 1; a[n_] := a[n] = 1 + a[n - 1] + Sum[a[Floor[n/k]], {k, 2, n}]; Table[a[n], {n, 1, 50}]

G.f. A(x) satisfies: A(x) = ( x + Sum_{k>=2} (1 - x^k) * A(x^k) ) / (1 - x)^2.

A179045 Triangle T(n,k), 1<=k<=n, read by rows, related to A033485.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 2, 1, 1, 7, 3, 1, 1, 1, 10, 4, 2, 1, 1, 1, 13, 5, 3, 1, 1, 1, 1, 18, 7, 4, 2, 1, 1, 1, 1, 23, 9, 5, 3, 1, 1, 1, 1, 1, 30, 12, 6, 4, 2, 1, 1, 1, 1, 1, 37, 15, 7, 5, 3, 1, 1, 1, 1, 1, 1, 47, 19, 9, 6, 4, 2, 1, 1, 1, 1, 1, 1, 57, 23, 11, 7, 5, 3, 1, 1, 1, 1, 1, 1, 1, 70, 28, 14, 8, 6, 4

Offset: 1

Philippe Deléham, Jun 26 2010

Column k=1 : A033485 ; column k=2 : A062188 ; row sums : A102378.

			Triangle begins : 1 ; 2,1 ; 3,1,1 ; 5,2,1,1 ; 7,3,1,1,1 ; ...

Cf. A033485, A062188, A102378

T(n,k)=T(n-1,k)+T([n/2],k), T(n,n)=1, T(n,k)=0 if k>n.

cp's OEIS Frontend

A325094 Write n as a sum of distinct powers of 2, then take the primes of those powers of 2 and multiply them together.

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A324927 Matula-Goebel numbers of rooted trees of depth 2. Numbers that are not powers of 2 but whose prime indices are all powers of 2.

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A324928 Matula-Goebel numbers of rooted trees of depth 3.

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A325093 Heinz numbers of integer partitions into distinct powers of 2.

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A325091 Heinz numbers of integer partitions of powers of 2.

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A325092 Heinz numbers of integer partitions of powers of 2 into powers of 2.

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A102379 a(n) is the minimal number of nodes in a binary tree of height n.

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Formula

A346912 a(0) = 1; a(n) = a(n-1) + a(floor(n/2)) + 1.

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