A151821 -id:A151821 - cp's OEIS frontend

A280286 a(n) is the least k such that sopfr(k) - sopf(k) = n.

4, 9, 8, 25, 16, 49, 32, 81, 64, 121, 128, 169, 256, 625, 512, 289, 1024, 361, 2048, 1444, 1331, 529, 5324, 2116, 2197, 4232, 8788, 841, 17576, 961, 7569, 3844, 4913, 7688, 19652, 1369, 6859, 5476, 12321, 1681, 34225, 1849, 15129, 7396, 12167, 2209, 46225, 8836, 19881

Offset: 2

Michel Marcus, Dec 31 2016

nonn

Michel Marcus, Table of n, a(n) for n = 2..500

Cf. A001414 (sopfr), A008472 (sopf), A001248, A280163.
A multiplicative version is A064549 (sorted A001694), firsts of A003557.
For length instead of sum we have A151821.
These are the positions of first appearances in A280292 = A001414 - A008472.
For indices instead of factors we have A380956 (sorted A380957), firsts of A380955.
A multiplicative version for indices is A380987 (sorted A380988), firsts of A290106.
For prime exponents instead of factors we have A380989, firsts of A380958.
The sorted version is A381075.
For product instead of sum see A381076, sorted firsts of A066503.
A000040 lists the primes, differences A001223.
A005117 lists squarefree numbers, complement A013929.
A020639 gives least prime factor (index A055396), greatest A061395 (index A006530).
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A001222, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A364916, A366528.

prifacs[n_]:=If[n==1,{},Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]];
q=Table[Total[prifacs[n]]-Total[Union[prifacs[n]]],{n,1000}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[q,k][[1,1]],{k,2,mnrm[q/.(0->1)]}] (* Gus Wiseman, Feb 20 2025 *)

sopfr(n) = my(f=factor(n)); sum(j=1, #f~, f[j,1]*f[j,2]);
sopf(n) = my(f=factor(n)); sum(j=1, #f~, f[j,1]);
a(n) = {my(k = 2); while (sopfr(k) - sopf(k) != n, k++); k;}

For p prime, a(p) = p^2 (see A001248).

A257851 Triangle read by rows: row n contains the first n+1 numbers m such that A046660(m) = n.

Original entry on oeis.org

1, 4, 9, 8, 24, 27, 16, 48, 72, 80, 32, 96, 144, 160, 216, 64, 192, 288, 320, 432, 448, 128, 384, 576, 640, 864, 896, 1296, 256, 768, 1152, 1280, 1728, 1792, 2592, 2816, 512, 1536, 2304, 2560, 3456, 3584, 5184, 5632, 6400, 1024, 3072, 4608, 5120, 6912, 7168, 10368, 11264, 12800, 13312

Offset: 0

Reinhard Zumkeller, Nov 29 2015

At the suggestion of Michel Marcus's remark in Carlos Eduardo Olivieri's A261256.

			0:    1
1:    4     9
2:    8    24      27
3:   16    48      72    80
4:   32    96     144   160     216
5:   64   192     288   320     432   448
6:  128   384     576   640     864   896    1296
7:  256   768    1152  1280    1728  1792    2592   2816
8:  512  1536    2304  2560    3456  3584    5184   5632    6400
--  ------------------------------------------------------------
0:  1
1:  2^2   3^2
2:  2^3 2^3*3     3^3
3:  2^4 2^4*3 2^3*3^2 2^4*5
4:  2^5 2^5*3 2^4*3^2 2^5*5 2^3*3^3
5:  2^6 2^6*3 2^5*3^2 2^6*5 2^4*3^3 2^6*7
6:  2^7 2^7*3 2^6*3^2 2^7*5 2^5*3^3 2^7*7 2^4*3^4
7:  2^8 2^8*3 2^7*3^2 2^8*5 2^6*3^3 2^8*7 2^5*3^4 2^8*11
8:  2^9 2^9*3 2^8*3^2 2^9*5 2^7*3^3 2^9*7 2^6*3^4 2^9*11 2^8*5^2

Reinhard Zumkeller, Rows n = 0..20 of triangle, flattened

Cf. A046660, A151821, A261256.

Cf. A005117, A060687, A195086, A195087, A195088, A195089, A195090, A195091, A195092, A195093, A195069.

a257851 n k = a257851_tabl !! n !! k
a257851_row n = a257851_tabl !! n
a257851_tabl = map
   (\x -> take (x + 1) $ filter ((== x) . a046660) [1..]) [0..]

T[n_] := Reap[For[m = 1; k = 1, k <= n+1, If[PrimeOmega[m] - PrimeNu[m] == n, Sow[m]; k++]; m++]][[2, 1]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Sep 17 2021 *)

T(n,0) = A151821(n+1);
T(n,n-1) = A261256(n) for n > 0;
T(n,n) = A264959(n).
T(0,0) = A005117(1);
T(1,k) = A060687(k+1), k = 0..1;
T(2,k) = A195086(k+1), k = 0..2;
T(3,k) = A195087(k+1), k = 0..3;
T(4,k) = A195088(k+1), k = 0..4;
T(5,k) = A195089(k+1), k = 0..5;
T(6,k) = A195090(k+1), k = 0..6;
T(7,k) = A195091(k+1), k = 0..7;
T(8,k) = A195092(k+1), k = 0..8;
T(9,k) = A195093(k+1), k = 0..9;
T(10,k) = A195069(k+1), k = 0..10.

A381075 Sorted positions of first appearances in A280292 (sum of prime factors minus sum of distinct prime factors).

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 32, 49, 64, 81, 121, 128, 169, 256, 289, 361, 512, 529, 625, 841, 961, 1024, 1331, 1369, 1444, 1681, 1849, 2048, 2116, 2197, 2209, 2809, 3481, 3721, 3844, 4232, 4489, 4913, 5041, 5324, 5329, 5476, 6241, 6859, 6889, 7396, 7569, 7688, 7921

Offset: 1

Gus Wiseman, Feb 18 2025

nonn

			The initial terms of A280292 are (0,0,0,2,0,0,0,4,3,0,0,2,0,0,0,6,0,3,0,2,0,0,0,4,5,0,6,2,...), wherein a value appears for the first time at positions 1, 4, 8, 9, 16, 25, ...

Michel Marcus, Table of n, a(n) for n = 1..391

For length instead of sum we have A151821.
The unsorted version is A280286, firsts of A280292.
For indices instead of factors we have A380957 (unsorted A380956), firsts of A380955.
A multiplicative version is A380988 (unsorted A380987), firsts of A290106.
For prime multiplicities instead of factors see A380989, firsts of A380958.
For product instead of sum we have A381076, sorted firsts of A066503.
A000040 lists the primes, differences A001223.
A005117 lists squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A364916 counts partitions by (sum minus sum of distinct parts).
Cf. A000720, A001414, A008472, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A366528.

prifacs[n_]:=If[n==1,{},Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]];
q=Table[Total[prifacs[n]]-Total[Union[prifacs[n]]],{n,10000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

f(n) = my(f=factor(n)); sum(j=1, #f~, f[j, 1]*f[j, 2] - f[j, 1]); \\ A280292
lista(nn) = my(v=Set(vector(nn, i, f(i))), list=List()); for (i=1, #v, my(k=1); while(f(k) != v[i], k++); listput(list, k)); vecsort(Vec(list)); \\ Michel Marcus, Apr 15 2025

Sorted positions of first appearances in A001414 - A008472.

A380956 Position of first appearance of n in A380955 (sum of prime indices minus sum of distinct prime indices).

Original entry on oeis.org

1, 4, 8, 16, 27, 64, 81, 256, 243, 529, 729, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481

Offset: 0

Gus Wiseman, Feb 12 2025

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.

Also the position of first appearance of n in A374248.

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     8: {1,1,1}
    16: {1,1,1,1}
    27: {2,2,2}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
   256: {1,1,1,1,1,1,1,1}
   243: {2,2,2,2,2}
   529: {9,9}
   729: {2,2,2,2,2,2}
   961: {11,11}
  1369: {12,12}
  1681: {13,13}
  1849: {14,14}
  2209: {15,15}

For length instead of sum we have A151821.
For factors instead of indices we have A280286 (sorted A381075), firsts of A280292.
Counting partitions by this statistic gives A364916.
Positions of first appearances in A380955.
The sorted version is A380957.
For product instead of sum we have firsts of A380986.
A multiplicative version is A380987 (sorted A380988), firsts of A290106.
For prime multiplicities instead of prime indices we have A380989, firsts of A380958.
A000040 lists the primes, differences A001223.
A005117 lists squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000720, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A325033, A366528, A366749, A374248.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Table[Total[prix[n]]-Total[Union[prix[n]]],{n,1000}];
Table[Position[q,k][[1,1]],{k,0,mnrm[q+1]-1}]

After a(12) = 961, this appears to converge to prime(n)^2.

A380957 Sorted positions of first appearances in A380955 (sum of prime indices minus sum of distinct prime indices).

Original entry on oeis.org

1, 4, 8, 16, 27, 64, 81, 243, 256, 529, 729, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481

Offset: 1

Gus Wiseman, Feb 13 2025

nonn

Also appears to be sorted firsts of A374248.

For length instead of sum we have A151821.
Counting partitions by this statistic (sum minus sum of distinct parts) gives A364916.
Sorted positions of first appearances in A380955.
The unsorted version is A380956.
For product instead of sum we have sorted firsts of A380986.
The multiplicative version is A380988, unsorted A380987, firsts of A290106.
For prime multiplicities instead of prime indices we have A380989, firsts of A380958.
For factors instead of indices we have A381075, see A280286, A280292.
A000040 lists the primes, differences A001223.
A005117 lists squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Cf. A000720, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A325033, A366528, A366749, A374248.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[Total[prix[n]]-Total[Union[prix[n]]],{n,1000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

A134058 Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 2, 6, 6, 2, 2, 8, 12, 8, 2, 2, 10, 20, 20, 10, 2, 2, 12, 30, 40, 30, 12, 2, 2, 14, 42, 70, 70, 42, 14, 2, 2, 16, 56, 112, 140, 112, 56, 16, 2, 2, 18, 72, 168, 252, 252, 168, 72, 18, 2

Offset: 0

Gary W. Adamson, Oct 05 2007

Triangle T(n,k), 0 <= k <= n, read by rows, given by [2, -1, 0, 0, 0, 0, 0, ...] DELTA [2, -1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 07 2007
Equals A028326 for all but the first term. - R. J. Mathar, Jun 08 2008
Warning: the row sums do not give A046055. - N. J. A. Sloane, Jul 08 2009

			First few rows of the triangle:
  1
  2,  2;
  2,  4,  2;
  2,  6,  6,  2;
  2,  8, 12,  8,  2;
  2, 10, 20, 20, 10,  2;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318, A028326, A046055, A084938, A134059, A151821, A173048, A173049.

A134058:= func< n,k | n eq 0 select 1 else 2*Binomial(n,k) >;
[A134058(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 26 2021

T[n_, k_]:= SeriesCoefficient[(1+x+y)/(1-x-y), {x, 0, n-k}, {y, 0, k}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, Apr 09 2015, after Vladimir Kruchinin *)
Table[2*Binomial[n,k] -Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 26 2021 *)

def A134058(n,k): return 2*binomial(n,k) - bool(n==0)
flatten([[A134058(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 26 2021

Double Pascal's triangle and replace leftmost column with (1,2,2,2,...).
M*A007318, where M = an infinite lower triangular matrix with (1,2,2,2,...) in the main diagonal and the rest zeros.
Sum_{k=0..n} T(n,k) = A151821(n+1). - Philippe Deléham, Sep 17 2009
G.f.: (1+x+y)/(1-x-y). - Vladimir Kruchinin, Apr 09 2015
T(n, k) = 2*binomial(n, k) - [n=0]. - G. C. Greubel, Apr 26 2021
E.g.f.: 2*exp(x*(1+y)) - 1. - Stefano Spezia, Apr 03 2024

Title changed by G. C. Greubel, Apr 26 2021

A171476 a(n) = 6a(n-1) - 8a(n-2) for n > 1, a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 28, 120, 496, 2016, 8128, 32640, 130816, 523776, 2096128, 8386560, 33550336, 134209536, 536854528, 2147450880, 8589869056, 34359607296, 137438691328, 549755289600, 2199022206976, 8796090925056, 35184367894528

Offset: 0

Klaus Brockhaus, Dec 09 2009

Binomial transform of A048473; second binomial transform of A151821; third binomial transform of A010684; fourth binomial transform of A084633 without second term 0; fifth binomial transform of A168589.
Inverse binomial transform of A081625; second inverse binomial transform of A081626; third inverse binomial transform of A081627.
Partial sums of A010036.
Essentially first differences of A006095.
a(n) = A109241(n) converted from binary to decimal. - Robert Price, Jan 19 2016
a(n) is the area enclosed by a Hilbert curve with unit length segments after n iterations, when the start and end points are joined. - Jennifer Buckley, Apr 23 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jennifer Buckley, The Area Enclosed by a Hilbert Curve with Unit Length Segments
Gordon Hamilton, Cookie Monster Counts Cookies, Youtube video, 2010.
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A006516 (2^(n-1)*(2^n-1)), A020522 (4^n-2^n), A048473 (2*3^n-1), A151821 (powers of 2, omitting 2 itself), A010684 (repeat 1, 3), A084633 (inverse binomial transform of repeated odd numbers), A168589 ((2-3^n)*(-1)^n), A081625 (2*5^n-3^n), A081626 (2*6^n-4^n), A081627 (2*7^n-5^n), A010036 (sum of 2^n, ..., 2^(n+1)-1), A006095 (Gaussian binomial coefficient [n, 2] for q=2), A171472, A171473.

[2*4^n-2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011

LinearRecurrence[{6,-8},{1,6},30] (* Harvey P. Dale, Aug 02 2020 *)

m=23; v=concat([1, 6], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]); v

a(n) = Sum_{k=1..2^n-1} k.
a(n) = 2*4^n - 2^n.
G.f.: 1/((1-2*x)*(1-4*x)).
a(n) = A006516(n+1).
a(n) = 4*a(n-1) + 2^n for n > 0, a(0)=1. - Vincenzo Librandi, Jul 17 2011
a(n) = Sum_{k=0..n} 2^(n+k). - Bruno Berselli, Aug 07 2013
a(n) = A020522(n+1)/2. - Hussam al-Homsi, Jun 06 2021
E.g.f.: exp(2*x)*(2*exp(2*x) - 1). - Stefano Spezia, Dec 10 2021

A374252 Irregular triangle read by rows where T(n,k) is the number of permutations of the prime factors of n with k runs.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 0, 1, 0, 0, 2, 1, 0, 2, 1, 1, 0, 2, 0, 2, 1, 0, 0, 0, 1, 0, 2, 1, 1, 0, 2, 1, 0, 2, 0, 2, 1, 0, 2, 2, 0, 1, 0, 0, 2, 1, 0, 0, 0, 2, 1, 1, 0, 0, 6, 1, 1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 2, 2, 1, 0, 2, 0, 2, 0, 2, 2, 0, 1

Offset: 2

Gus Wiseman, Jul 07 2024

An alternative form of this sequence (with the same data) has offset 1 and begins with an empty row.

Note that the prime factors of n are separable (A335433) iff the last term of row n is positive.

			The T(36,3) = 2 permutations are (2,3,3,2) and (3,2,2,3).
Row n = 72 counts the following permutations:
  .  (2,2,2,3,3)  (2,2,3,3,2)  (2,2,3,2,3)  (2,3,2,3,2)
     (3,3,2,2,2)  (2,3,3,2,2)  (2,3,2,2,3)
                  (3,2,2,2,3)  (3,2,2,3,2)
                               (3,2,3,2,2)
Triangle begins:
   1:
   2: 1
   3: 1
   4: 1  0
   5: 1
   6: 0  2
   7: 1
   8: 1  0  0
   9: 1  0
  10: 0  2
  11: 1
  12: 0  2  1
  13: 1
  14: 0  2
  15: 0  2
  16: 1  0  0  0
  17: 1
  18: 0  2  1
  19: 1
  20: 0  2  1

Row-lengths are A001222.
Row-sums are A008480 (number of permutations of prime factors).
Column k = 1 is A069513.
For compositions instead of permutations of prime factors we have A238130.
Last column is A335452 (where k = A001222(n)), which counts separations.
Position of the last positive term in row n is A373957(n).
The number of zeros at the end of row n is A374246(n).
The number of nonzero terms in row n is A374247(n).
A001221 counts distinct prime factors.
A003242 counts run-compressed compositions, i.e., anti-runs.
A124767 counts runs in standard compositions, anti-runs A333381.
A333755 counts compositions by number of runs.
A335433 lists separable numbers, complement A335448.
A374250 maximizes sum of run-compression, for indices A373956.
Cf. A026549, A027748, A046660, A126706, A151821, A246655.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[Length[Select[Permutations[prifacs[n]], Length[Split[#]]==k&]],{n,100},{k,PrimeOmega[n]}]

A384781 Number of values of s, 0 < s <= n - 1, such that (-s)^s == s (mod n).

Original entry on oeis.org

0, 1, 0, 0, 1, 2, 1, 0, 1, 4, 0, 1, 1, 3, 3, 0, 0, 4, 0, 1, 2, 3, 1, 1, 3, 6, 1, 3, 1, 6, 1, 0, 3, 2, 2, 3, 3, 3, 2, 1, 1, 6, 0, 3, 5, 3, 1, 1, 3, 8, 2, 2, 2, 4, 3, 2, 1, 5, 0, 3, 3, 3, 7, 0, 5, 6, 0, 1, 3, 8, 1, 3, 3, 8, 5, 3, 4, 6, 1, 1, 4, 3, 0, 5, 2, 4, 6, 2, 4, 10, 5, 2, 3, 3, 2, 1, 4, 8, 5, 5

Offset: 1

Juri-Stepan Gerasimov, Jun 09 2025

nonn

Cf. A065295, A151821, A373901 (k such that a(k) = 0), A382752, A384854.

[#[s: s in [1..n-1] | Modexp((-s),s,n) eq s]: n in [1..100]];

a:= n-> add(`if`((-s)&^s-s mod n=0, 1, 0), s=1..n-1):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 09 2025

a[n_]:=Length[Select[Range[n-1],PowerMod[-#,#,n]==# &]]; Array[a,100] (* Stefano Spezia, Jun 11 2025 *)

a(n) = sum(s=1, n-1, Mod(-s, n)^s == s); \\ Michel Marcus, Jun 11 2025

A063759 Spherical growth series for modular group.

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072, 196608, 262144, 393216, 524288, 786432, 1048576, 1572864, 2097152

Offset: 0

N. J. A. Sloane, Aug 14 2001

Also number of sequences S of length n with entries in {1,..,q} where q = 3, satisfying the condition that adjacent terms differ in absolute value by exactly 1, see examples. - W. Edwin Clark, Oct 17 2008

			For n = 2 the a(2) = 4 sequences are (1,2),(2,1),(2,3),(3,2). - _W. Edwin Clark_, Oct 17 2008
From _Joerg Arndt_, Nov 23 2012: (Start)
There are a(6) = 16 such words of length 6:
[ 1]   [ 1 2 1 2 1 2 ]
[ 2]   [ 1 2 1 2 3 2 ]
[ 3]   [ 1 2 3 2 1 2 ]
[ 4]   [ 1 2 3 2 3 2 ]
[ 5]   [ 2 1 2 1 2 1 ]
[ 6]   [ 2 1 2 1 2 3 ]
[ 7]   [ 2 1 2 3 2 1 ]
[ 8]   [ 2 1 2 3 2 3 ]
[ 9]   [ 2 3 2 1 2 1 ]
[10]   [ 2 3 2 1 2 3 ]
[11]   [ 2 3 2 3 2 1 ]
[12]   [ 2 3 2 3 2 3 ]
[13]   [ 3 2 1 2 1 2 ]
[14]   [ 3 2 1 2 3 2 ]
[15]   [ 3 2 3 2 1 2 ]
[16]   [ 3 2 3 2 3 2 ]
(End)

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 156.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for sequences related to modular groups
Index entries for linear recurrences with constant coefficients, signature (0,2)

Cf. A054886, A029744.

The sequence (ternary strings) seems to be related to A029744 and A090989.

import Data.List (transpose)
a063759 n = a063759_list !! n
a063759_list = concat $ transpose [a151821_list, a007283_list]
-- Reinhard Zumkeller, Dec 16 2013

CoefficientList[Series[(1+3*x+2*x^2)/(1-2*x^2),{x,0,40}],x](* Jean-François Alcover, Mar 21 2011 *)
Join[{1},Transpose[NestList[{Last[#],2First[#]}&,{3,4},40]][[1]]] (* Harvey P. Dale, Oct 22 2011 *)

a(n)=([0,1; 2,0]^n*[1;3])[1,1] \\ Charles R Greathouse IV, Feb 09 2017

G.f.: (1+3*x+2*x^2)/(1-2*x^2).
a(n) = 2*a(n-2), n>2. - Harvey P. Dale, Oct 22 2011
a(2*n) = A151821(n+1); a(2*n+1) = A007283(n). - Reinhard Zumkeller, Dec 16 2013

Information from A145751 included by Joerg Arndt, Dec 03 2012

cp's OEIS Frontend

A280286 a(n) is the least k such that sopfr(k) - sopf(k) = n.

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A257851 Triangle read by rows: row n contains the first n+1 numbers m such that A046660(m) = n.

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A381075 Sorted positions of first appearances in A280292 (sum of prime factors minus sum of distinct prime factors).

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A380956 Position of first appearance of n in A380955 (sum of prime indices minus sum of distinct prime indices).

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A380957 Sorted positions of first appearances in A380955 (sum of prime indices minus sum of distinct prime indices).

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A134058 Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.

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A171476 a(n) = 6*a(n-1) - 8*a(n-2) for n > 1, a(0)=1, a(1)=6.

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A374252 Irregular triangle read by rows where T(n,k) is the number of permutations of the prime factors of n with k runs.

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A171476 a(n) = 6a(n-1) - 8a(n-2) for n > 1, a(0)=1, a(1)=6.