A325093 -id:A325093 - cp's OEIS frontend

A050315 Main diagonal of A050314.

1, 1, 1, 2, 1, 2, 2, 5, 1, 2, 2, 5, 2, 5, 5, 15, 1, 2, 2, 5, 2, 5, 5, 15, 2, 5, 5, 15, 5, 15, 15, 52, 1, 2, 2, 5, 2, 5, 5, 15, 2, 5, 5, 15, 5, 15, 15, 52, 2, 5, 5, 15, 5, 15, 15, 52, 5, 15, 15, 52, 15, 52, 52, 203, 1, 2, 2, 5, 2, 5, 5, 15, 2, 5, 5, 15, 5, 15, 15, 52, 2, 5, 5, 15, 5, 15

Offset: 0

Christian G. Bower, Sep 15 1999

nonn

Also, a(n) is the number of odd multinomial coefficients n!/(k_1!...k_m!) with 1 <= k_1 <= ... <= k_m and k_1 + ... + k_m = n. - Pontus von Brömssen, Mar 23 2018
From Gus Wiseman, Mar 30 2019: (Start)
Also the number of strict integer partitions of n with no binary carries. The Heinz numbers of these partitions are given by A325100. A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. For example, the a(1) = 1 through a(15) = 15 strict integer partitions with no binary carries are:
(1) (2) (3)  (4) (5)  (6)  (7)   (8) (9)  (A)  (B)   (C)  (D)   (E)   (F)
        (21)     (41) (42) (43)      (81) (82) (83)  (84) (85)  (86)  (87)
                           (52)                (92)       (94)  (A4)  (96)
                           (61)                (A1)       (C1)  (C2)  (A5)
                           (421)               (821)      (841) (842) (B4)
                                                                      (C3)
                                                                      (D2)
                                                                      (E1)
                                                                      (843)
                                                                      (852)
                                                                      (861)
                                                                      (942)
                                                                      (A41)
                                                                      (C21)
                                                                      (8421)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..16383
Michael Gilleland, Some Self-Similar Integer Sequences

Cf. A000110, A000120, A050314.
Cf. A070939, A080572, A247935, A267610.
Cf. A325093, A325095, A325096, A325099, A325100, A325103, A325110, A325123.
Main diagonal of A307431 and of A307505.

a:= n-> combinat[bell](add(i,i=convert(n, base, 2))):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 08 2019

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&stableQ[#,Intersection[binpos[#1],binpos[#2]]!={}&]&]],{n,0,20}] (* Gus Wiseman, Mar 30 2019 *)
a[n_] := BellB[DigitCount[n, 2, 1]];
a /@ Range[0, 100] (* Jean-François Alcover, May 21 2021 *)

Bell number of number of 1's in binary: a(n) = A000110(A000120(n)).

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

Original entry on oeis.org

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}]

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[#]&]

cp's OEIS Frontend

A050315 Main diagonal of A050314.

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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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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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