A276078 -id:A276078 - cp's OEIS frontend

A277008 Numbers k such that in the binary expansion of k no run of 1-bits is longer than 1 + the total number of 0-bits anywhere to the right of that run.

0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 28, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 48, 49, 50, 52, 53, 54, 56, 57, 58, 64, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 80, 81, 82, 84, 85, 86, 88, 89, 90, 92, 96, 97, 98, 100, 101, 102, 104, 105, 106, 108, 109, 112, 113, 114, 116, 117, 118, 120, 128

Offset: 0

Antti Karttunen, Sep 25 2016

Numbers k for which A277007(k) = 0.

Indexing starts from zero as a(0) = 0 is a special case in this sequence.

Complement: A277009.
Positions of zeros in A277007.
Sequence A277012 sorted into ascending order.
Cf. A005940, A276077, A276078.
Subsequence of A277018 from which this differs for the first time at n=41, where a(41)=64, skipping the value 60 present in A277018.

;; With Antti Karttunen's IntSeq-library.
(define A277008 (ZERO-POS 0 0 A277007))

Other identitities:

A276077(A005940(1+a(n))) = 0 for all n.

A351979 Numbers whose prime factorization has all odd prime indices and all even prime exponents.

Original entry on oeis.org

1, 4, 16, 25, 64, 100, 121, 256, 289, 400, 484, 529, 625, 961, 1024, 1156, 1600, 1681, 1936, 2116, 2209, 2500, 3025, 3481, 3844, 4096, 4489, 4624, 5329, 6400, 6724, 6889, 7225, 7744, 8464, 8836, 9409, 10000, 10609, 11881, 12100, 13225, 13924, 14641, 15376

Offset: 1

Gus Wiseman, Mar 11 2022

nonn

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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
Also Heinz numbers of integer partitions with all odd parts and all even multiplicities, counted by A035457 (see Emeric Deutsch's comment there).

			The terms together with their prime indices begin:
     1: 1
     4: prime(1)^2
    16: prime(1)^4
    25: prime(3)^2
    64: prime(1)^6
   100: prime(1)^2 prime(3)^2
   121: prime(5)^2
   256: prime(1)^8
   289: prime(7)^2
   400: prime(1)^4 prime(3)^2
   484: prime(1)^2 prime(5)^2
   529: prime(9)^2
   625: prime(3)^4
   961: prime(11)^2
  1024: prime(1)^10
  1156: prime(1)^2 prime(7)^2
  1600: prime(1)^6 prime(3)^2
  1681: prime(13)^2
  1936: prime(1)^4 prime(5)^2

Amiram Eldar, Table of n, a(n) for n = 1..10000

The second condition alone (exponents all even) is A000290, counted by A035363.
The distinct prime factors of terms all come from A031368.
These partitions are counted by A035457 or A000009 aerated.
The first condition alone (indices all odd) is A066208, counted by A000009.
The squarefree square roots are A258116, even A258117.
A056166 = exponents all prime, counted by A055923.
A066207 = indices all even, counted by complement of A086543.
A076610 = indices all prime, counted by A000607.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162641 counts even exponents, odd A162642.
A257991 counts odd indices, even A257992.
A268335 = exponents all odd, counted by A055922.
A325131 = disjoint indices from exponents, counted by A114639.
A346068 = indices and exponents all prime, counted by A351982.
A352140 = even indices with odd exponents, counted by A055922 (aerated).
A352141 = even indices with even exponents, counted by A035444.
A352142 = odd indices and odd multiplicities, counted by A117958.
Cf. A000720, A028260, A045931, A055396, A061395, A106529, A181819, A195017, A276078, A324588, A325698, A325700.

Select[Range[1000],#==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]

from sympy import factorint, primepi
def ok(n):
    return all(primepi(p)%2==1 and e%2==0 for p, e in factorint(n).items())
print([k for k in range(15500) if ok(k)]) # Michael S. Branicky, Mar 12 2022

Squares of elements of A066208.
Intersection of A066208 and A000290.
A257991(a(n)) = A001222(a(n)).
A162641(a(n)) = A001221(a(n)).
A162642(a(n)) = A257992(a(n)) = 0.
Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k-1)^2) = 1.4135142... . - Amiram Eldar, Sep 19 2022

A387112 Numbers with (strictly) choosable initial intervals of prime indices.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95

Offset: 1

Gus Wiseman, Aug 23 2025

First differs from A371088 in having a(86) = 121.
The initial interval of a nonnegative integer x is the set {1,...,x}.
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.
We say that a set or sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1,2,3},{1},{1,3},{2}) is not.
This sequence lists all numbers k such that if the prime indices of k are (x1,x2,...,xz), then the sequence of sets (initial intervals) ({1,...,x1},{1,...,x2},...,{1,...,xz}) is choosable.

			The prime indices of 85 are {3,7}, with initial intervals {{1,2,3},{1,2,3,4,5,6,7}}, which are choosable, so 85 is in the sequence
The prime indices of 90 are {1,2,2,3}, with initial intervals {{1},{1,2},{1,2},{1,2,3}}, which are not choosable, so 90 is not in the sequence.

Partitions of this type are counted by A238873, complement A387118.
For partitions instead of initial intervals we have A276078, complement A276079.
For prime factors instead of initial intervals we have A368100, complement A355529.
For divisors instead of initial intervals we have A368110, complement A355740.
These are all the positions of nonzero terms in A387111, complement A387134.
The complement is A387113.
For strict partitions instead of initial intervals we have A387176, complement A387137.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A367902 counts choosable set-systems, complement A367903.
A370582 counts sets with choosable prime factors, complement A370583.
A370585 counts maximal subsets with choosable prime factors.
Cf. A003963, A005703, A052335, A239312, A335433, A335448, A355739, A387110, A383706.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Tuples[Range/@prix[#]],UnsameQ@@#&]!={}&]

A387133 Number of ways to choose a sequence of distinct integer partitions, one of each prime factor of n (with multiplicity).

Original entry on oeis.org

1, 2, 3, 2, 7, 6, 15, 0, 6, 14, 56, 6, 101, 30, 21, 0, 297, 12, 490, 14, 45, 112, 1255, 0, 42, 202, 6, 30, 4565, 42, 6842, 0, 168, 594, 105, 12, 21637, 980, 303, 0, 44583, 90, 63261, 112, 42, 2510, 124754, 0, 210, 84, 891, 202, 329931, 12, 392, 0, 1470, 9130

Offset: 1

Gus Wiseman, Aug 26 2025

			The prime factors of 9 are (3,3), and the a(9) = 6 choices are:
  ((3),(2,1))
  ((3),(1,1,1))
  ((2,1),(3))
  ((2,1),(1,1,1))
  ((1,1,1),(3))
  ((1,1,1),(2,1))

For prime factors instead of partitions we have A008966, see A355741.
Twice partitions of this type are counted by A296122.
For prime indices instead of factors we have A387110, see A387136.
For strict partitions and prime indices we have A387115.
For constant partitions and prime indices we have A387120.
Positions of zero are A387326, for indices apparently A276079 (complement A276078).
Allowing repeated partitions gives A387327, see A299200, A357977.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A063834, A261049, A299201, A335433, A335448, A355739, A383706, A387111, A387135.

Table[Length[Select[Tuples[IntegerPartitions/@Flatten[ConstantArray@@@FactorInteger[n]]],UnsameQ@@#&]],{n,30}]

A387134 Number of integer partitions of n whose parts do not have choosable sets of integer partitions.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 6, 8, 12, 17, 25, 34, 49, 65, 89, 118, 158, 206, 271, 349, 453, 578, 740, 935, 1186, 1486, 1865, 2322, 2890, 3572, 4415, 5423, 6659, 8134, 9927, 12062, 14643, 17706, 21387, 25746, 30957, 37109, 44433, 53054, 63273, 75276, 89444, 106044

Offset: 0

Gus Wiseman, Aug 29 2025

Number of integer partitions of n such that it is not possible to choose a sequence of distinct integer partitions, one of each part.

Also the number of integer partitions of n with at least one part k satisfying that the multiplicity of k exceeds the number of integer partitions of k.

			The a(2) = 1 through a(8) = 12 partitions:
  (11)  (111)  (211)   (311)    (222)     (511)      (611)
               (1111)  (2111)   (411)     (2221)     (2222)
                       (11111)  (2211)    (3211)     (3311)
                                (3111)    (4111)     (4211)
                                (21111)   (22111)    (5111)
                                (111111)  (31111)    (22211)
                                          (211111)   (32111)
                                          (1111111)  (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

These partitions are ranked by A276079.
For divisors instead of partitions we have A370320, complement A239312.
The complement for prime factors is A370592, ranks A368100.
For prime factors instead of partitions we have A370593, ranks A355529.
For initial intervals instead of partitions we have A387118, complement A238873.
For just choices of strict partitions we have A387137.
The complement is counted by A387328, ranks A276078.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
Cf. A335433, A355535, A367867, A367901, A367903, A367905, A367907, A370583, A370594.

Table[Length[Select[IntegerPartitions[n],Length[Select[Tuples[IntegerPartitions/@#],UnsameQ@@#&]]==0&]],{n,0,15}]

A387176 Numbers whose prime indices do not have choosable sets of strict integer partitions. Zeros of A387115.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 45, 48, 52, 54, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 162, 164, 168, 171, 172

Offset: 1

Gus Wiseman, Aug 27 2025

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.

We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.

The complement for all partitions appears to be A276078, counted by A052335.
For all partitions we appear to have A276079, counted by A387134.
For divisors instead of strict partitions we have A355740, counted by A370320.
Twice-partitions of this type (into distinct strict partitions) are counted by A358914.
The complement for divisors is A368110, counted by A239312.
The complement for initial intervals is A387112, counted by A238873, see A387111.
For initial intervals instead of strict partitions we have A387113, counted by A387118.
These are the positions of 0 in A387115.
Partitions of this type are counted by A387137, complement A387178.
The complement is A387177.
The version for constant partitions is A387180, counted by A387329.
The complement for constant partitions is A387181, counted by A387330.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together the prime indices of n.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A000720, A261049, A270995, A335433, A335448, A355744, A357978, A357980, A383706, A387110, A387120.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@prix[#]],UnsameQ@@#&]=={}&]

A276941 Square array A(row,col): A(1,col) = A276937(col), and for row > 1, A(row,col) = A003961(A(row-1,col)), read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

2, 6, 3, 9, 15, 5, 10, 25, 35, 7, 14, 21, 49, 77, 11, 18, 33, 55, 121, 143, 13, 22, 75, 65, 91, 169, 221, 17, 26, 39, 245, 119, 187, 289, 323, 19, 30, 51, 85, 847, 209, 247, 361, 437, 23, 34, 105, 95, 133, 1859, 299, 391, 529, 667, 29, 38, 57, 385, 161, 253, 3757, 493, 551, 841, 899, 31, 42, 69, 115, 1001, 319, 377, 6137, 589, 713, 961, 1147, 37

Offset: 2

Antti Karttunen, Sep 25 2016

The starting offset is 2 because 1 is not included in the array proper. With it the terms are a permutation of A276078.

			The top left corner of the array:
   2,   6,   9,   10,   14,    18,   22,   26,    30,   34,   38,    42
   3,  15,  25,   21,   33,    75,   39,   51,   105,   57,   69,   165
   5,  35,  49,   55,   65,   245,   85,   95,   385,  115,  145,   455
   7,  77, 121,   91,  119,   847,  133,  161,  1001,  203,  217,  1309
  11, 143, 169,  187,  209,  1859,  253,  319,  2431,  341,  407,  2717
  13, 221, 289,  247,  299,  3757,  377,  403,  4199,  481,  533,  5083
  17, 323, 361,  391,  493,  6137,  527,  629,  7429,  697,  731,  9367
  19, 437, 529,  551,  589, 10051,  703,  779, 12673,  817,  893, 13547
  23, 667, 841,  713,  851, 19343,  943,  989, 20677, 1081, 1219, 24679
  29, 899, 961, 1073, 1189, 27869, 1247, 1363, 33263, 1537, 1711, 36859

Antti Karttunen, Table of n, a(n) for n = 2..1276; the first 50 antidiagonals of array

Transpose: A276942.
Topmost row: A276937, second row: A276938. Leftmost column: A000040.
Cf. A003961.
Cf. A276078 (sorted into ascending order).
Cf. also A276075, A276953.

(define (A276941 n) (A276941bi (A002260 (- n 1)) (A004736 (- n 1))))
(define (A276941bi row col) (if (= 1 row) (A276937 col) (A003961 (A276941bi (- row 1) col))))

A(1,col) = A276937(col), and for row > 1, A(row,col) = A003961(A(row-1,col)).

A276942 Square array A(row,col): A(row,1) = A276937(row), and for col > 1, A(row,col) = A003961(A(row,col-1)), read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

2, 3, 6, 5, 15, 9, 7, 35, 25, 10, 11, 77, 49, 21, 14, 13, 143, 121, 55, 33, 18, 17, 221, 169, 91, 65, 75, 22, 19, 323, 289, 187, 119, 245, 39, 26, 23, 437, 361, 247, 209, 847, 85, 51, 30, 29, 667, 529, 391, 299, 1859, 133, 95, 105, 34, 31, 899, 841, 551, 493, 3757, 253, 161, 385, 57, 38, 37, 1147, 961, 713, 589, 6137, 377, 319, 1001, 115, 69, 42

Offset: 2

Antti Karttunen, Sep 25 2016

The starting offset is 2 because 1 is not included in the array proper. With it the terms are a permutation of A276078.

All terms on each row have the same prime signature.

			The top left corner of the array:
   2,  3,  5,   7,  11,  13,  17,  19,  23,   29,   31,   37,   41,   43
   6, 15, 35,  77, 143, 221, 323, 437, 667,  899, 1147, 1517, 1763, 2021
   9, 25, 49, 121, 169, 289, 361, 529, 841,  961, 1369, 1681, 1849, 2209
  10, 21, 55,  91, 187, 247, 391, 551, 713, 1073, 1271, 1591, 1927, 2279
  14, 33, 65, 119, 209, 299, 493, 589, 851, 1189, 1333, 1739, 2173, 2537

Antti Karttunen, Table of n, a(n) for n = 2..631; the first 35 antidiagonals of array

Transpose: A276941.
Leftmost column: A276937, second column: A276938.
Rows from the top: A000040, A006094, A001248 (from 9 onward), A090076, A090090.
Cf. A003961.
Cf. A276078 (sorted into ascending order).
Cf. also A276075, A276955.

(define (A276942 n) (A276941bi (A004736 (- n 1)) (A002260 (- n 1)))) ;; Code for A276941bi given in A276941.

A(row,1) = A276937(row); for col > 1, A(row,col) = A003961(A(row,col-1)).

A376399 a(0) = 1, and for n > 0, a(n) is the least k such that A276075(k) = a(n-1) + A276075(a(n-1)), where A276075 is the factorial base log-function.

Original entry on oeis.org

1, 2, 6, 30, 1050, 519090, 1466909163669353522118

Offset: 0

Antti Karttunen, Nov 02 2024

nonn

a(7) has 212 digits, a(8) has 10654 digits.
The lexicographically earliest infinite sequence x for which A276075(x(n)) gives the partial sums of x (shifted right once).
For any a(n), the next term a(n+1) <= a(n) * A276076(a(n)).
Conjecture: there are infinitely many variants b of this sequence, such that A276075(b(n)) = partial sums of b (shifted once right). One way to construct them: set i for some value >= 4, construct b first as here, but at point i, set b(i+1) = b(i) * A276076(b(i)), and after that, proceed as before, always finding a minimal k satisfying the condition. Unless b(i+1) = a(i+1), then b differs from this sequence but satisfies the same general condition, except that it is not the lexicographically earliest one. See also A376400.
The n-th term can be computed by applying A276076 to A376403(n), i.e., to the partial sums of the preceding terms a(0) .. a(n-1) (see the examples). This follows because all terms are in A276078 by the "least k" condition of the definition (see comment in A376417).

			Starting with a(0) = 1, we take partial sums of previous terms, and apply A276076 to get the next term as:
a(1) = A276076(1) = 2,
a(2) = A276076(1+2) = 6,
a(3) = A276076(1+2+6) = 30,
a(4) = A276076(1+2+6+30) = 1050,
a(5) = A276076(1+2+6+30+1050) = 519090,
a(6) = A276076(1+2+6+30+1050+519090) = 1466909163669353522118,
etc.

Antti Karttunen, Table of n, a(n) for n = 0..7
Index entries for sequences related to factorial base representation

Cf. A276075, A276076, A376400 (variant).
Cf. A376403 (= A276075(a(n)), also gives the partial sums from its second term onward).
Subsequence of A276078.
Cf. also analogous sequences A002110 (for A276085), A093502 (for A056239), A376406 (for A048675).

\\ Do it hard way, by searching:
up_to = 12;
A276075(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*(primepi(f[k, 1])!)); };
A376399list(up_to) = { my(v=vector(up_to), x); v[1]=1; for(n=2,up_to,x=v[n-1]+A276075(v[n-1]); for(k=1,oo,if(A276075(k)==x,v[n]=k;break)); print1(v[n], ", ")); (v); };
v376399 = A376399list(1+up_to);
A376399(n) = v376399[1+n];

\\ Compute, do not search, much faster:
up_to = 8;
A276076(n) = { my(m=1, p=2, i=2); while(n, m *= (p^(n%i)); n = n\i; p = nextprime(1+p); i++); (m); };
A376399list(up_to) = { my(v=vector(up_to), s=1); v[1]=1; for(n=2,up_to,v[n] = A276076(s); s += v[n]); (v); };
v376399 = A376399list(1+up_to);
A376399(n) = v376399[1+n];

a(n) = A276076(A376403(n)) = A276076(Sum_{i=0..n-1} a(i)).

A376400 a(0) = 1, and for n > 0, a(n) = a(n-1) * A276076(a(n-1)), where A276076 is the factorial base exp-function.

Original entry on oeis.org

1, 2, 6, 30, 1050, 70814493750, 7568077812763134673885891483463343434987134201379042046671543939118568739667281250

Offset: 0

Antti Karttunen, Nov 02 2024

nonn

a(7) has 2129 (decimal) digits.
Like A376399, this satisfies A276075(a(n)) = a(n-1) + A276075(a(n-1)), for all n >= 1, so also here, applying A276075 to the terms gives the partial sums shifted right once, A376401.
However, unlike A376399, this is not a subsequence of A276078: a(5) = 70814493750 is the first term that is in A276079.

Index entries for sequences related to factorial base representation

Cf. A276075, A276076, A276078, A276079, A376399, A376403.
Cf. A376401 (= A276075(a(n)), also gives the partial sums from its second term onward).
Cf. also analogous sequences A002110 (for A276086) and A376408 (for A019565).

A276076(n) = { my(m=1, p=2, i=2); while(n, m *= (p^(n%i)); n = n\i; p = nextprime(1+p); i++); (m); };
A376400(n) = if(!n,1,my(x=A376400(n-1)); x*A276076(x));

cp's OEIS Frontend

A277008 Numbers k such that in the binary expansion of k no run of 1-bits is longer than 1 + the total number of 0-bits anywhere to the right of that run.

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A351979 Numbers whose prime factorization has all odd prime indices and all even prime exponents.

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A387112 Numbers with (strictly) choosable initial intervals of prime indices.

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A387133 Number of ways to choose a sequence of distinct integer partitions, one of each prime factor of n (with multiplicity).

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A387134 Number of integer partitions of n whose parts do not have choosable sets of integer partitions.

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A387176 Numbers whose prime indices do not have choosable sets of strict integer partitions. Zeros of A387115.

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A276941 Square array A(row,col): A(1,col) = A276937(col), and for row > 1, A(row,col) = A003961(A(row-1,col)), read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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A276942 Square array A(row,col): A(row,1) = A276937(row), and for col > 1, A(row,col) = A003961(A(row,col-1)), read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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A376399 a(0) = 1, and for n > 0, a(n) is the least k such that A276075(k) = a(n-1) + A276075(a(n-1)), where A276075 is the factorial base log-function.