A260443 -id:A260443 - cp's OEIS frontend

A206296 Prime factorization representation of Fibonacci polynomials: a(0) = 1, a(1) = 2, and for n > 1, a(n) = A003961(a(n-1)) * a(n-2).

1, 2, 3, 10, 63, 2750, 842751, 85558343750, 2098355820117528699, 769999781728184386440152910156250, 2359414683424785920146467280333749864720543920418139851

Offset: 0

Clark Kimberling, Feb 05 2012

nonn

These are numbers matched to the Fibonacci polynomials according to the scheme explained in A206284 (see also A104244). In this case, the exponent of the k-th prime p_k in the prime factorization of a(n) indicates the coefficient of term x^(k-1) in the n-th Fibonacci polynomial. See the examples.

			n    a(n)   prime factorization    Fibonacci polynomial
------------------------------------------------------------
0       1   (empty)                F_0(x) = 0
1       2   p_1                    F_1(x) = 1
2       3   p_2                    F_2(x) = x
3      10   p_3 * p_1              F_3(x) = x^2 + 1
4      63   p_4 * p_2^2            F_4(x) = x^3 + 2x
5    2750   p_5 * p_3^3 * p_1      F_5(x) = x^4 + 3x^2 + 1
6  842751   p_6 * p_4^4 * p_2^3    F_6(x) = x^5 + 4x^3 + 3x

Eric Weisstein's World of Mathematics, Fibonacci polynomial
Wikipedia, Fibonacci polynomials

Cf. A000045, A000129, A001222, A003961, A006190, A049310, A048675, A090880, A104244, A206284.
Other such mappings:
  polynomial sequence      integer sequence
  -----------------------------------------
  x^n                      A000040
  (x+1)^n                  A007188
  n*x^(n-1)                A062457
  (1-x^n)/(1-x)            A002110
  n + (n-1)x + ... +x^n    A006939
  Stern polynomials        A260443

c[n_] := CoefficientList[Fibonacci[n, x], x]
f[n_] := Product[Prime[k]^c[n][[k]], {k, 1, Length[c[n]]}]
Table[f[n], {n, 1, 11}]  (* A206296 *)

from functools import reduce
from sympy import factorint, prime, primepi
from operator import mul
def a003961(n):
    F=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**F[i] for i in F])
l=[1, 2]
for n in range(2, 11):
    l.append(a003961(l[n - 1])*l[n - 2])
print(l) # Indranil Ghosh, Jun 21 2017

From Antti Karttunen, Jul 29 2015: (Start)
a(0) = 1, a(1) = 2, and for n >= 2, a(n) = A003961(a(n-1)) * a(n-2).
Other identities. For all n >= 0:
A001222(a(n)) = A000045(n). [When each polynomial is evaluated at x=1.]
A048675(a(n)) = A000129(n). [at x=2.]
A090880(a(n)) = A006190(n). [at x=3.]
(End)

a(0) = 1 prepended (to indicate 0-polynomial), Name changed, Comments and Example section rewritten by Antti Karttunen, Jul 29 2015

A286378 Restricted growth sequence computed for Stern-polynomial related filter-sequence A278243.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 8, 5, 9, 2, 10, 6, 11, 4, 12, 7, 13, 3, 13, 8, 14, 5, 15, 9, 16, 2, 17, 10, 18, 6, 19, 11, 20, 4, 21, 12, 22, 7, 23, 13, 24, 3, 24, 13, 25, 8, 26, 14, 27, 5, 28, 15, 29, 9, 30, 16, 31, 2, 32, 17, 33, 10, 34, 18, 35, 6, 36, 19, 37, 11, 38, 20, 39, 4, 40, 21, 41, 12, 42, 22, 43, 7, 44, 23, 45, 13, 46, 24, 47, 3, 47, 24, 48

Offset: 0

Antti Karttunen, May 09 2017

nonn

Construction: we start with a(0)=1 for A278243(0)=1, and then after, for n > 0, we use the least unused natural number k for a(n) if A278243(n) has not been encountered before, otherwise [whenever A278243(n) = A278243(m), for some m < n], we set a(n) = a(m).
When filtering sequences (by equivalence class partitioning), this sequence (with its modestly sized terms) can be used instead of A278243, because for all i, j it holds that: a(i) = a(j) <=> A278243(i) = A278243(j).
For example, for all i, j: a(i) = a(j) => A002487(i) = A002487(j).
For pairs of distinct primes p, q for which a(p) = a(q) see comments in A317945. - Antti Karttunen, Aug 12 2018

			For n=1, A278243(1) = 2, which has not been encountered before, thus we allot for a(1) the least so far unused number, which is 2, thus a(1) = 2.
For n=2, A278243(2) = 2, which was already encountered as A278243(1), thus we set a(2) = a(1) = 2.
For n=3, A278243(3) = 6, which has not been encountered before, thus we allot for a(3) the least so far unused number, which is 3, thus a(3) = 3.
For n=23, A278243(23) = 2520, which has not been encountered before, thus we allot for a(23) the least so far unused number, which is 13, thus a(23) = 3.
For n=25, A278243(25) = 2520, which was already encountered at n=23, thus we set a(25) = a(23) = 13.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to Stern's sequences

Cf. A002487, A125184, A260443, A278243, A286377.
Cf. also A101296, A286603, A286605, A286610, A286619, A286621, A286622, A286626 for similarly constructed sequences.
Cf. also A317942, A317943, A317944, A317945.
Differs from A103391(1+n) for the first time at n=25, where a(25)=13, while A103391(26) = 14.

a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; With[{nn = 100}, Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Table[Times @@ MapIndexed[Prime[First@#2]^#1 &, Sort[FactorInteger[#][[All, -1]], Greater]] - Boole[# == 1] &@ a@ n, {n, 0, nn}]] (* Michael De Vlieger, May 12 2017 *)

up_to = 65537;
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2))));
A278243(n) = A046523(A260443(n));
v286378 = rgs_transform(vector(up_to+1,n,A278243(n-1)));
A286378(n) = v286378[1+n];

A277013 a(n) = number of irreducible polynomial factors (counted with multiplicity) in the n-th Stern polynomial B(n,t).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 2, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 4, 1, 2, 3, 6, 1, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 1, 3, 1, 5, 2, 2, 1, 4, 2, 2, 2, 4, 1, 4, 1, 3, 2, 2, 1, 6, 1, 3, 3, 3, 1, 3, 1, 4, 3, 2, 1, 5, 1, 2, 2, 5, 1, 3, 1, 3, 2, 2, 2, 5

Offset: 1

Antti Karttunen, Oct 07 2016

nonn

			B(11,t) = t^2 + 3t + 1 which is irreducible, so a(11) = 1.
B(12,t) = t^3 + t^2 = t^2(t+1), so a(12) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..85085

Cf. A186891 (positions of 0 and 1's in this sequence), A277027 (terms squared).
Cf. A000079, A125184, A277322, A260443.
Differs from A001222 for the first time at n=25, where a(25)=1. A277190 gives the positions of differing terms.

A277013 = n -> vecsum(factor(ps(n))[,2]);
ps(n) = if(n<2, n, if(n%2, ps(n\2)+ps(n\2+1), 'x*ps(n\2))); \\ From Charles R Greathouse IV code in A186891.
for(n=1, 85085, write("b277013.txt", n, " ", A277013(n)));

a(n) = A277322(A260443(n)).

It seems that for all n >= 1, a(2^n) = n.

A277905 Irregular table: Each row n (n >= 0) lists in ascending order all A018819(n) numbers k for which A048675(k) = n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 9, 12, 16, 10, 18, 24, 32, 15, 20, 27, 36, 48, 64, 30, 40, 54, 72, 96, 128, 7, 25, 45, 60, 80, 81, 108, 144, 192, 256, 14, 50, 90, 120, 160, 162, 216, 288, 384, 512, 21, 28, 75, 100, 135, 180, 240, 243, 320, 324, 432, 576, 768, 1024, 42, 56, 150, 200, 270, 360, 480, 486, 640, 648, 864, 1152, 1536, 2048, 35, 63, 84, 112, 125, 225, 300, 400

Offset: 1

Antti Karttunen, Nov 14 2016

Each row beginning with an odd number (rows with even index) is followed by a row of the same length, with the same terms, but multiplied by 2. See also comments in the Formula section of A018819.
Note that although the indexing of rows start from zero, the indexing of this sequence starts from 1, with a(1) = 1.
Also Heinz numbers of integer partitions whose binary rank is n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1). For example, row n = 6 is 15, 20, 27, 36, 48, 64, corresponding to the partitions (3,2), (3,1,1), (2,2,2), (2,2,1,1), (2,1,1,1,1), (1,1,1,1,1,1). - Gus Wiseman, May 25 2024
Also, row n lists in ascending order all A018819(n) numbers k for which A097248(k) = A019565(n). - Flávio V. Fernandes, Jul 19 2025

			The irregular table begins as:
  row terms
   0   1;
   1   2;
   2   3,  4;
   3   6,  8;
   4   5,  9,  12,  16;
   5  10, 18,  24,  32;
   6  15, 20,  27,  36,  48,  64;
   7  30, 40,  54,  72,  96, 128;
   8   7, 25,  45,  60,  80,  81, 108, 144, 192, 256;
   9  14, 50,  90, 120, 160, 162, 216, 288, 384, 512;
  10  21, 28,  75, 100, 135, 180, 240, 243, 320, 324, 432,  576,  768, 1024;
  11  42, 56, 150, 200, 270, 360, 480, 486, 640, 648, 864, 1152, 1536, 2048;
...

Antti Karttunen, Table of n, a(n) for n = 1..166
Index entries for sequences that are permutations of the natural numbers

Cf. A019565 (the left edge, the only terms that are squarefree).
Cf. A000079 (the trailing edge).
Cf. A048675, A260443, A277886, A277896, A277903, A277904.
Row lengths are A018819 (number of partitions of binary rank n).
A000009 counts strict partitions, ranks A005117.
A029837 stc_sum or A070939 bin_len, opposite A070940 binexp_lastpos_1.
A048675 gives binary rank of prime indices, distinct A087207.
A048793 lists binary indices, product A096111, reverse A272020.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, cf. A001222, A003963, A056239, A296150.
A372890 adds up binary ranks of partitions, strict A372888.
Cf. A000040, A000041, A001511, A005940, A118462, A215366, A225620, A246868.
Cf. A097248, A048675.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Select[Range[0,2^k],Total[2^(prix[#]-1)]==k&],{k,0,10}] (* Gus Wiseman, May 25 2024 *)

(definec (A277905 n) (A277905bi (A277903 n) (A277904 n)))
(define (A277905bi row col) (let outloop ((k (A019565 row)) (col col)) (if (zero? col) k (let inloop ((j (+ 1 k))) (if (= (A048675 j) row) (outloop j (- col 1)) (inloop (+ 1 j))))))) ;; Very slow implementation.
;; Implementation based on a naive recurrence:
(definec (A277905 n) (if (= 1 n) n (let ((maybe_next (A277896 (A277905 (- n 1))))) (if (not (zero? maybe_next)) maybe_next (A019565 (A277903 n))))))

a(1) = 1; for n > 1, if A277896(a(n-1)) > 0, then a(n) = A277896(a(n-1)), otherwise a(n) = A019565(A277903(n)). [A naive recurrence for a one-dimensional version.]
Other identities. For all n >= 1:
A048675(a(n)) = A277903(n).

A090880 Suppose n=(p1^e1)(p2^e2)... where p1,p2,... are the prime numbers and e1,e2,... are nonnegative integers. Then a(n) = e1 + (e2)3 + (e3)9 + (e4)27 + ... + (ek)(3^(k-1)) + ...

Original entry on oeis.org

0, 1, 3, 2, 9, 4, 27, 3, 6, 10, 81, 5, 243, 28, 12, 4, 729, 7, 2187, 11, 30, 82, 6561, 6, 18, 244, 9, 29, 19683, 13, 59049, 5, 84, 730, 36, 8, 177147, 2188, 246, 12, 531441, 31, 1594323, 83, 15, 6562, 4782969, 7, 54, 19, 732, 245, 14348907, 10, 90, 30, 2190

Offset: 1

Sam Alexander, Dec 12 2003

Replace "3" with "x" and extend the definition of a to positive rationals and a becomes an isomorphism between positive rationals under multiplication and polynomials over Z under addition. This remark generalizes A001222, A048675 and A054841: evaluate said polynomial at x=1, x=2 and x=10, respectively.

For examples of such evaluations at x=3, see "Other identities" in the Formula section. - Antti Karttunen, Jul 31 2015

Joseph J. Rotman, The Theory of Groups: An Introduction, 2nd ed. Boston: Allyn and Bacon, Inc. 1973. Page 9, problem 1.26.

Antti Karttunen, Table of n, a(n) for n = 1..512
Sam Alexander, Post to sci.math.

Row 3 of A104244.

Cf. A001222, A006190, A048675, A054841, A090881, A090882, A090883, A090884, A178590, A206296, A260443.

(define (A090880 n) (if (= 1 n) (- n 1) (+ (A000244 (- (A055396 n) 1)) (A090880 (A032742 n))))) ;; Antti Karttunen, Jul 29 2015

a(1) = 0; for n > 1, a(n) = 3^(A055396(n)-1) + a(A032742(n)). [Where A055396(n) gives the index of the smallest prime dividing n and A032742(n) gives the largest proper divisor of n.] - Antti Karttunen, Jul 29 2015
Other identities. For all n >= 0:
a(A206296(n)) = A006190(n).
a(A260443(n)) = A178590(n).

More terms from Ray Chandler, Dec 20 2003

A178590 a(2n) = 3*a(n), a(2n+1) = a(n) + a(n+1).

Original entry on oeis.org

1, 3, 4, 9, 7, 12, 13, 27, 16, 21, 19, 36, 25, 39, 40, 81, 43, 48, 37, 63, 40, 57, 55, 108, 61, 75, 64, 117, 79, 120, 121, 243, 124, 129, 91, 144, 85, 111, 100, 189, 103, 120, 97, 171, 112, 165, 163, 324, 169, 183, 136, 225, 139, 192, 181, 351, 196, 237, 199, 360, 241

Offset: 1

Gary W. Adamson, May 29 2010

In groups of 1, 2, 4, 8, ... terms; sums of group terms appears to be A081625: (1, 7, 41, 223,...), for example: 41 = (9 + 7 + 12 + 13).
Equals row 3 in the array shown in A178568, an infinite family of sequences of the form a(2n) = r*a(n), a(2n+1) = a(n) + a(n+1).
Let M = an infinite lower triangular matrix with (1, 3, 1, 0, 0, 0,...) in each column, and with successive columns shifted down twice from the previous column. A178590 = Lim_{n->inf} M^n, the left-shifted vector considered as a sequence.
The Stern polynomial B(n,x) evaluated at x=3. See A125184. - T. D. Noe, Feb 28 2011

			In groups of 2^n terms (n=0,1,2,...):
1;
3, 4;
9, 7, 12, 13;
27, 16, 21, 19, 36, 25, 39, 40;
...
a(6) = 12 = 3*a(3) = 3*4
a(7) = 13 = a(3) + a(4) = 4 + 9

Antti Karttunen, Table of n, a(n) for n = 1..8191

Row 3 of A178568.

Cf. A081625, A090880, A260443.

a[0] = a[1] = 1; a[n_] := a[n] = If[ OddQ@n, a[(n - 1)/2] + a[(n + 1)/2], 3*a[n/2]]; Array[a, 61] (* Robert G. Wilson v, Jun 11 2010 *)

a(2n) = 3*a(n), a(2n+1) = a(n) + a(n+1).
a(n) = A090880(A260443(n)). - Antti Karttunen, Jul 29 2015
G.f.: x * Product_{k>=0} (1 + 3*x^(2^k) + x^(2^(k+1))). - Ilya Gutkovskiy, Jul 07 2019

a(19) onwards from Robert G. Wilson v, Jun 11 2010

A277330 a(0)=1, a(1)=2, a(2n) = A003961(a(n)), a(2n+1) = lcm(a(n),a(n+1))/gcd(a(n),a(n+1)).

Original entry on oeis.org

1, 2, 3, 6, 5, 2, 15, 30, 7, 10, 3, 30, 35, 2, 105, 210, 11, 70, 21, 30, 5, 10, 105, 42, 77, 70, 3, 210, 385, 2, 1155, 2310, 13, 770, 231, 30, 55, 70, 105, 6, 7, 2, 21, 42, 385, 10, 165, 66, 143, 110, 231, 210, 5, 70, 1155, 66, 1001, 770, 3, 2310, 5005, 2, 15015, 30030, 17, 10010, 3003, 30, 715, 770, 105, 66, 91, 154, 231, 6, 385, 70, 15, 42, 11, 14, 3, 42, 55, 2

Offset: 0

Antti Karttunen, Oct 27 2016

nonn

Each term is a squarefree number, A005117.

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A001511, A003961, A005117, A007913, A019565, A048675, A055396, A260443, A264977, A277707.
Cf. A023758 (positions where coincides with A260443).
Cf. A277701, A277712, A277713 for the positions of 2's, 3's and 6's in this sequence, which are also the first three rows of array A277710.
Cf. also A255483.

a(0) = 1, a(1) = 2, a(2n) = A003961(a(n)), a(2n+1) = lcm(a(n),a(n+1))/gcd(a(n),a(n+1)).
Other identities. For all n >= 0:
a(n) = A007913(A260443(n)).
a(n) = A019565(A264977(n)), A048675(a(n)) = A264977(n).
A055396(a(n)) = A277707(A260443(n)) = A001511(n).

A101624 Stern-Jacobsthal numbers.

Original entry on oeis.org

1, 1, 3, 1, 7, 5, 11, 1, 23, 21, 59, 17, 103, 69, 139, 1, 279, 277, 827, 273, 1895, 1349, 2955, 257, 5655, 5141, 14395, 4113, 24679, 16453, 32907, 1, 65815, 65813, 197435, 65809, 460647, 329029, 723851, 65793, 1512983, 1381397, 3881019, 1118225

Offset: 0

Paul Barry, Dec 10 2004

The Stern diatomic sequence A002487 could be called the Stern-Fibonacci sequence, since it is given by A002487(n) = Sum_{k=0..floor(n/2)} (binomial(n-k,k) mod 2), where F(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k,k). Now a(n) = Sum_{k=0..floor(n/2)} (binomial(n-k,k) mod 2)*2^k, where J(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*2^k, with J(n) = A001045(n), the Jacobsthal numbers. - Paul Barry, Sep 16 2015

These numbers seem to encode Stern (0, 1)-polynomials in their binary expansion. See Dilcher & Ericksen paper, especially Table 1 on page 79, page 5 in PDF. See A125184 (A260443) for another kind of Stern-polynomials, and also A177219 for a reference to maybe a third kind. - Antti Karttunen, Nov 01 2016

Ivan Panchenko, Table of n, a(n) for n = 0..1000
K. Dilcher and L. Ericksen, Reducibility and irreducibility of Stern (0, 1)-polynomials, Communications in Mathematics, Volume 22/2014 , pp. 77-102.

Cf. A002487, A011973, A000079, A006921.

Cf. A125184, A260443, A177219.

a101624 = sum . zipWith (*) a000079_list . map (flip mod 2) . a011973_row
-- Reinhard Zumkeller, Jul 14 2015

prpr = 1
prev = 1
print("1, 1", end=", ")
for i in range(99):
    current = (prev)^(prpr*2)
    print(current, end=", ")
    prpr = prev
    prev = current
# Alex Ratushnyak, Apr 14 2012

def A101624(n): return sum(int(not k & ~(n-k))*2**k for k in range(n//2+1)) # Chai Wah Wu, Jun 20 2022

a(n) = Sum_{k=0..floor(n/2)} (binomial(n-k, k) mod 2)*2^k.
a(2^n-1)=1, a(2*n) = 2*a(n-1) + a(n+1) = A099902(n); a(2*n+1) = A101625(n+1).
a(n) = Sum_{k=0..n} (binomial(k, n-k) mod 2)*2^(n-k). - Paul Barry, May 10 2005
a(n) = Sum_{k=0..n} A106344(n,k)*2^(n-k). - Philippe Deléham, Dec 18 2008
a(0)=1, a(1)=1, a(n) = a(n-1) XOR (a(n-2)*2), where XOR is the bitwise exclusive-OR operator. - Alex Ratushnyak, Apr 14 2012
A000120(a(n-1)) = A002487(n). - Karl-Heinz Hofmann, Jun 18 2025

A277701 Positions of ones in A264977; positions of twos in A277330.

Original entry on oeis.org

1, 5, 13, 29, 41, 61, 85, 125, 173, 209, 253, 281, 313, 349, 421, 509, 565, 629, 701, 845, 929, 1021, 1133, 1261, 1405, 1693, 1861, 2045, 2269, 2525, 2665, 2813, 3121, 3313, 3389, 3725, 3905, 4093, 4541, 4841, 5053, 5209, 5257, 5333, 5629, 5993, 6245, 6629, 6781, 7453, 7813, 8189, 8537, 9085, 9593, 9685, 9905, 10109, 10421, 10517, 10669, 10921

Offset: 1

Antti Karttunen, Oct 27 2016

nonn

Positions in A260443 of terms that are twice square (terms in A001105, although not all of them are present in A260443).

Antti Karttunen, Table of n, a(n) for n = 1..720

Row 1 of A277710.
Cf. A001105, A260443, A264977, A277330.
Cf. also A277712, A277713.

A277712(n) = 2*a(n) for all n >= 1.

A277892 a(n) = A001222(A048675(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 2, 1, 4, 2, 5, 2, 2, 2, 6, 1, 7, 2, 2, 1, 8, 1, 3, 2, 2, 2, 9, 1, 10, 1, 3, 2, 3, 2, 11, 2, 2, 1, 12, 1, 13, 3, 3, 1, 14, 2, 4, 2, 3, 2, 15, 1, 3, 1, 3, 4, 16, 3, 17, 3, 3, 2, 4, 1, 18, 3, 3, 1, 19, 1, 20, 2, 2, 3, 4, 2, 21, 3, 3, 2, 22, 3, 3, 2, 2, 1, 23, 2, 4, 3, 5, 3, 4, 1, 24, 1, 3, 2, 25, 1, 26, 2, 2

Offset: 2

Antti Karttunen, Nov 08 2016

nonn

For n >= 3, a(n) = index of the row where n is located in array A277898.

Antti Karttunen (terms 2..3465) and Hans Havermann, Table of n, a(n) for n = 2..10000

Left inverse of A065091.
Cf. A001222, A048675.
Cf. A277319 (positions of ones).
Cf. A000040 (positions of records), A277900.
Cf. A277895 (ordinal transform from a(3) onward).
Cf. A277893, A277894, A277898.

A048675[n_] := If[n == 1, 0, Total[#[[2]]*2^(PrimePi[#[[1]]] - 1)& /@ FactorInteger[n]]];
a[n_] := PrimeOmega[A048675[n]];
Table[a[n], {n, 2, 105}] (* Jean-François Alcover, Jan 11 2022 *)

A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2;
A277892(n) = if(1==n,0,bigomega(A048675(n)));
for(n=1, 3465, write("b277892.txt", n, " ", A277892(n)));

from sympy import factorint, primepi, primefactors
def a001222(n): return 0 if n==1 else a001222(n//primefactors(n)[0]) + 1
def a048675(n):
    if n==1: return 0
    f=factorint(n)
    return sum(f[i]*2**(primepi(i) - 1) for i in f)
def a(n): return a001222(a048675(n))
print([a(n) for n in range(2, 101)]) # Indranil Ghosh, Jun 19 2017

(define (A277892 n) (if (= 1 n) 0 (A001222 (A048675 n))))

a(A019565(n)) = a(A260443(n)) = A001222(n).

For all n >= 2, a(A065091(n)) = n.

cp's OEIS Frontend

A206296 Prime factorization representation of Fibonacci polynomials: a(0) = 1, a(1) = 2, and for n > 1, a(n) = A003961(a(n-1)) * a(n-2).

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A277013 a(n) = number of irreducible polynomial factors (counted with multiplicity) in the n-th Stern polynomial B(n,t).

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A277905 Irregular table: Each row n (n >= 0) lists in ascending order all A018819(n) numbers k for which A048675(k) = n.

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A090880 Suppose n=(p1^e1)(p2^e2)... where p1,p2,... are the prime numbers and e1,e2,... are nonnegative integers. Then a(n) = e1 + (e2)*3 + (e3)*9 + (e4)*27 + ... + (ek)*(3^(k-1)) + ...

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A178590 a(2n) = 3*a(n), a(2n+1) = a(n) + a(n+1).

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A277330 a(0)=1, a(1)=2, a(2n) = A003961(a(n)), a(2n+1) = lcm(a(n),a(n+1))/gcd(a(n),a(n+1)).

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A101624 Stern-Jacobsthal numbers.

A090880 Suppose n=(p1^e1)(p2^e2)... where p1,p2,... are the prime numbers and e1,e2,... are nonnegative integers. Then a(n) = e1 + (e2)3 + (e3)9 + (e4)27 + ... + (ek)(3^(k-1)) + ...