\n"} # _Ivan Neretin - cp's OEIS frontend

A283325 Lengths of runs of successive zeros in A283683.

1, 2, 3, 1, 2, 2, 2, 3, 2, 4, 2, 1, 3, 2, 2, 3, 2, 1, 1, 3, 2, 2, 1, 3, 2, 3, 1, 3, 2, 1, 2, 1, 3, 2, 2, 2, 1, 3, 2, 3, 2, 1, 3, 2, 1, 2, 2, 1, 3, 2, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 1, 3, 2, 3, 2, 2, 2, 1, 3, 2, 4, 2, 2, 2, 1, 3, 2, 1, 3, 2, 2, 2, 1, 3, 2, 2, 3

Offset: 1

N. J. A. Sloane, Mar 16 2017

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A283683.

Take[Length /@ Most@Split@ Nest[Flatten@ Table[#[[n - i]], {n, Length[#] + 1}, {i, n - 1}] &, {0, 1}, 4], {1, -1, 2}] (* Ivan Neretin, Mar 17 2017 *)

Initial 1 added by Ivan Neretin, Mar 17 2017

A268231 Indices of 1's in A047999.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 14, 15, 16, 19, 20, 21, 23, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 44, 45, 46, 53, 54, 55, 57, 63, 65, 66, 67, 68, 69, 74, 75, 76, 77, 78, 82, 86, 90, 91, 92, 95, 96, 99, 100, 103, 104, 105, 107, 109, 111, 113, 115, 117, 119, 120, 121, 122, 123, 124, 125, 126, 127

Offset: 1

N. J. A. Sloane, Feb 03 2016

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A047999, A268232.

Flatten@Position[Flatten@NestList[Mod[Append[#, 0] + Prepend[#, 0], 2] &, {1}, 15], 1] - 1 (* Ivan Neretin, Dec 19 2017 *)

A249036 a(1)=1, a(2)=2; thereafter a(n) = a(n-1-(number of even terms so far)) + a(n-1-(number of odd terms so far)).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 26 2014

nonn

Suggested by A006336 and A007604.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A006336, A007604.

A249037 and A249038 give numbers of even and odd terms so far.

M:=100;
v[1]:=1; v[2]:=2; w[1]:=0; w[2]:=1; x[1]:=1; x[2]:=1;
for n from 3 to M do
   v[n]:=v[n-1-w[n-1]]+v[n-1-x[n-1]];
if v[n] mod 2 = 0 then w[n]:=w[n-1]+1; x[n]:=x[n-1];
                  else w[n]:=w[n-1]; x[n]:=x[n-1]+1; fi;
od:
[seq(v[n], n=1..M)]; # A249036
[seq(w[n], n=1..M)]; # A249037
[seq(x[n], n=1..M)]; # A249038

Nest[Append[#, #[[Length@Select[#, OddQ]]] + #[[Length@Select[#, EvenQ]]]] &, {1, 2}, 75] (* Ivan Neretin, May 02 2016 *)

A231879 Numbers n such that bigomega(n)^2 (cf. A001222) does not divide n.

Original entry on oeis.org

1, 6, 8, 9, 10, 12, 14, 15, 20, 21, 22, 24, 25, 26, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115

Offset: 1

N. J. A. Sloane, Nov 17 2013

nonn

Contains all semiprimes (A001358) except 4. - Ivan Neretin, Apr 05 2016

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A001221, A075592, A185307, A001222, A074946, A134334, A231878 (complement).

Join[{1}, Select[Range[2, 115], ! Divisible[#, PrimeOmega[#]^2] &]] (* Ivan Neretin, Apr 05 2016 *)

lista(nn) = {print1(1, ", "); for(n=2, nn, if(n % bigomega(n)^2 != 0, print1(n, ", ")));} \\ Altug Alkan, Apr 05 2016

A231878 Numbers k such that bigomega(k)^2 (cf. A001222) divides k.

Original entry on oeis.org

2, 3, 4, 5, 7, 11, 13, 16, 17, 18, 19, 23, 27, 29, 31, 37, 41, 43, 45, 47, 53, 59, 61, 63, 67, 71, 73, 79, 83, 89, 97, 99, 101, 103, 107, 109, 113, 117, 127, 131, 137, 139, 144, 149, 151, 153, 157, 163, 167, 171, 173, 179, 181, 191, 193, 197, 199, 200, 207, 211, 216, 223, 227, 229, 233, 239, 241, 251, 256, 257, 261, 263

Offset: 1

N. J. A. Sloane, Nov 17 2013

nonn

Contains all primes. - Ivan Neretin, Apr 05 2016

Ivan Neretin, Table of n, a(n) for n = 1..13517 (all terms up to 10^5)

Cf. A001221, A075592, A185307, A001222, A074946, A134334, A231879 (complement).

Select[Range[2, 265], Divisible[#, PrimeOmega[#]^2] &] (* Ivan Neretin, Apr 05 2016 *)

isok(n) = !(n % bigomega(n)^2); \\ Michel Marcus, Apr 05 2016

A231500 a(n) = Sum_{i=0..n} wt(i)^2, where wt(i) = A000120(i).

Original entry on oeis.org

0, 1, 2, 6, 7, 11, 15, 24, 25, 29, 33, 42, 46, 55, 64, 80, 81, 85, 89, 98, 102, 111, 120, 136, 140, 149, 158, 174, 183, 199, 215, 240, 241, 245, 249, 258, 262, 271, 280, 296, 300, 309, 318, 334, 343, 359, 375, 400, 404, 413, 422, 438, 447, 463, 479, 504, 513, 529, 545, 570, 586, 611, 636, 672, 673, 677, 681, 690, 694

Offset: 0

N. J. A. Sloane, Nov 12 2013

Stolarsky (1977) has an extensive bibliography.

Amiram Eldar, Table of n, a(n) for n = 0..10000 (terms 0..1024 from Ivan Neretin)
Jean Coquet, Power sums of digital sums, J. Number Theory, Vol. 22, No. 2 (1986), pp. 161-176.
P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, On the moments of the sum-of-digits function, PDF, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993, pp. 263-271; alternative link.
J.-L. Mauclaire and Leo Murata, On q-additive functions. I, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 6 (1983), pp. 274-276.
J.-L. Mauclaire and Leo Murata, On q-additive functions. II, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 9 (1983), pp. 441-444.
Kenneth B. Stolarsky, Power and exponential sums of digital sums related to binomial coefficient parity, SIAM J. Appl. Math., Vol. 32, No. 4 (1977), pp. 717-730.
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag., Vol. 41, No. 1 (1968), pp. 21-25.

Cf. A000120, A000788, A231501, A231502, A360189.

digsum:=proc(n,B) local a; a := convert(n, base, B):
add(a[i], i=1..nops(a)): end;
f:=proc(n,k,B) global digsum; local i;
add( digsum(i,B)^k,i=0..n); end;
[seq(f(n,1,2),n=0..100)]; #A000788
[seq(f(n,2,2),n=0..100)]; #A231500
[seq(f(n,3,2),n=0..100)]; #A231501
[seq(f(n,4,2),n=0..100)]; #A231502

FoldList[#1 + DigitCount[#2, 2, 1]^2 &, 0, Range[1, 68]] (* Ivan Neretin, May 21 2015 *)

a(n) = sum(i=0, n, hammingweight(i)^2); \\ Michel Marcus, Sep 20 2017

Stolarsky (1977) studies the asymptotics.
a(n) ~ n * (log(n)/(2*log(2)))^2 + O(n*log(n)) (Stolarsky, 1977). - Amiram Eldar, Jan 20 2022
a(n) = Sum_{k=0..floor(log_2(n+1))} k^2 * A360189(n,k). - Alois P. Heinz, Mar 06 2023

A222313 A222311 sorted and duplicates removed (conjectured).

Original entry on oeis.org

1, 2, 3, 5, 6, 15, 17, 33, 41, 55, 57, 65, 70, 105, 129, 257, 273, 385, 561, 897, 969, 1001, 1105, 1353, 1430, 1785, 2049, 2145, 2337, 2665, 3553, 4097, 4305, 4745, 4845, 5633, 6105, 6545, 8193, 8385

Offset: 1

N. J. A. Sloane, Feb 16 2013

Obtained by sorting and removing duplicates from the first 500 terms of A222311. There is no proof as yet that this list is complete up to 105. Only the first three terms shown are certain. Is there a proof that 4 cannot appear?

Cristian Cobeli, Mihai Prunescu, Alexandru Zaharescu, A growth model based on the arithmetic Z-game, arXiv:1511.04315 [math.NT], 2015.
Cristian Cobeli, Alexandru Zaharescu, A game with divisors and absolute differences of exponents, arXiv:1411.1334 [math.NT], 2014 (see page 12).

Cf. A222310, A222311.

terms = 40; nmax0 = 5000;
seq[nmax_] := seq[nmax] = Union[Print[nmax]; Join[r = {1}, Table[Reverse[r = FoldList[#1*(#2/GCD[#1, #2]^2) & , n, r]], {n, 2, nmax}][[All, 1]]]][[1 ;; terms]];
seq[nmax = nmax0]; seq[nmax = 2 nmax]; While[seq[nmax] == seq[nmax/2], nmax = 2 nmax]; seq[nmax] (* Jean-François Alcover, Sep 04 2018, after Ivan Neretin in A222310 *)

Corrected and extended using data from Cobeli et al., 2015. - N. J. A. Sloane, Aug 27 2016

More terms (computed from a list of 10000) from Jean-François Alcover, Sep 04 2018

A222311 Leading diagonal of triangle in A222310.

Original entry on oeis.org

1, 2, 3, 6, 5, 15, 105, 70, 1, 5, 33, 55, 65, 273, 1001, 1430, 17, 17, 969, 4845, 1785, 6545, 37145, 81719, 17, 1105, 3553, 969969, 672945, 81345, 955049953, 66786710, 33, 561, 385, 6545, 6105, 657305, 15873, 8544965, 1353, 268345, 61705, 329681, 650793, 24173705985, 3065857, 250538768183, 561, 33, 21945

Offset: 1

N. J. A. Sloane, Feb 16 2013

nonn

See A222313 for the numbers that appear in this sequence.

N. J. A. Sloane, Table of n, a(n) for n = 1..500
C. Cobeli and A. Zaharescu, Promenade around Pascal Triangle-Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104) No. 1, 2013, 73-98.
Cristian Cobeli, Alexandru Zaharescu, A game with divisors and absolute differences of exponents, arXiv:1411.1334 [math.NT], 2014 (see page 12).
Cristian Cobeli, Mihai Prunescu, Alexandru Zaharescu, A growth model based on the arithmetic Z-game, arXiv:1511.04315 [math.NT], 2015.

Cf. A222310, A222313.

Join[r = {1}, Table[Reverse[r = FoldList[#1*#2/GCD[#1, #2]^2&, n, r]], {n, 2, 100}][[All, 1]]] (* Jean-François Alcover, Sep 04 2018, after Ivan Neretin in A222310 *)

A222310 Array read by antidiagonals: first row is 1, 2, 3, 4, ...; for subsequent rows, write i*j/gcd(i,j)^2 under ...i.j... in previous row.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 6, 2, 12, 4, 5, 30, 15, 20, 5, 15, 3, 10, 6, 30, 6, 105, 7, 21, 210, 35, 42, 7, 70, 6, 42, 2, 420, 12, 56, 8, 1, 70, 105, 10, 5, 84, 63, 72, 9, 5, 5, 14, 30, 3, 15, 1260, 20, 90, 10, 33, 165, 33, 462, 385, 1155, 77, 1980, 99, 110, 11, 55, 15, 11, 3, 154, 10, 462, 6, 330, 30, 132, 12, 65, 143, 2145, 195, 65, 10010, 1001, 78

Offset: 1

N. J. A. Sloane, Feb 16 2013

			Array begins:
1...2...3.....4......5......6.....7.....8.....9.....10
..2...6....12....20.....30....42.....56...72.....90
....3...2....15......6.....35....12....63....20
......6....30....10....210...420.....84..1260
........5.....3.....21......2.....5....15
...........15.....7.....42....10......3
.............105.....6.....105...30
........

Ivan Neretin, Table of n, a(n) for n = 1..8001
Cristian Cobeli, Mihai Prunescu, Alexandru Zaharescu, A growth model based on the arithmetic Z-game, arXiv:1511.04315 [math.NT], 2015.
C. Cobeli and A. Zaharescu, Promenade around Pascal Triangle-Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104) No. 1, 2013, 73-98.

Cf. A036262. Leading diagonal is A222311 (cf. A222313).

Similar array with primes in the starting row is A255483.

# To get first M rows of the array (s0 is A222311):
g:=(i,j)->i*j/gcd(i,j)^2;
M:=50;
s0:=[1]:
s1:=[seq(n,n=1..M)]:
for i1 from 1 to M-1 do
lprint(s1);
s2:=[seq(g(s1[i],s1[i+1]),i=1..nops(s1)-1)];
s0:=[op(s0),s2[1]];
s1:=[seq(s2[i],i=1..nops(s2))];
od:
# To produce A222310 (i.e., to read the array by antidiagonals):
g:=(i,j)->i*j/gcd(i,j)^2;
M:=15;
b1:=Array(1..M);
s0:=[1]:
s1:=[seq(n,n=1..M)]:
b1[1]:=s1;
for i1 from 1 to M-1 do
#lprint(s1);
s2:=[seq(g(s1[i],s1[i+1]),i=1..nops(s1)-1)];
b1[i1+1]:=s2;
s0:=[op(s0),s2[1]];
s1:=[seq(s2[i],i=1..nops(s2))];
od:
#[seq(s0[i],i=1..nops(s0))]; (that gives A222311)
lis:=[]:
for i from 1 to M do for j from 1 to i do
lis:=[op(lis),b1[i-j+1][j]];
od: od:
[seq(lis[k],k=1..nops(lis))];

a = r = {1}; Do[a = Join[a, Reverse[r = FoldList[#1*#2/GCD[#1, #2]^2 &, n, r]]], {n, 2, 13}]; a (* Ivan Neretin, May 14 2015 *)

A212438 Irregular triangle read by rows: T(n,k) is the number of polyhedra with n faces and k vertices (n >= 4, k=4..2n-4).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 2, 2, 0, 0, 2, 8, 11, 8, 5, 0, 0, 2, 11, 42, 74, 76, 38, 14, 0, 0, 0, 8, 74, 296, 633, 768, 558, 219, 50, 0, 0, 0, 5, 76, 633, 2635, 6134, 8822, 7916, 4442, 1404, 233, 0, 0, 0, 0, 38, 768, 6134, 25626, 64439, 104213, 112082, 79773, 36528, 9714, 1249

Offset: 4

N. J. A. Sloane, May 16 2012

Because of duality, T(n,k) = T(k,n). - Ivan Neretin, May 25 2016

The number of edges is n+k-2. - Andrew Howroyd, Mar 27 2021

			Triangle begins:
1
0 1 1
0 1 2  2  2
0 0 2  8 11   8    5
0 0 2 11 42  74   76   38   14
0 0 0  8 74 296  633  768  558  219   50
0 0 0  5 76 633 2635 6134 8822 7916 4442 1404 233
...

Andrew Howroyd, Table of n, a(n) for n = 4..199 (rows 4..17)
Gunnar Brinkmann and Brendan McKay, Fast generation of planar graphs (expanded edition), Table 9-11.
G. P. Michon, Counting Polyhedra

A049337, A058787, A212438 are all versions of the same triangle.
Row sums (the same as column sums) are A000944.
Main diagonal is A002856.
Cf. A002840 (by edges), A239893.

Terms a(53) and beyond from Andrew Howroyd, Mar 27 2021

cp's OEIS Frontend

User: \n"} # _Ivan Neretin

A283325 Lengths of runs of successive zeros in A283683.

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A268231 Indices of 1's in A047999.

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A249036 a(1)=1, a(2)=2; thereafter a(n) = a(n-1-(number of even terms so far)) + a(n-1-(number of odd terms so far)).

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A231879 Numbers n such that bigomega(n)^2 (cf. A001222) does not divide n.

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A231878 Numbers k such that bigomega(k)^2 (cf. A001222) divides k.

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A231500 a(n) = Sum_{i=0..n} wt(i)^2, where wt(i) = A000120(i).

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A222313 A222311 sorted and duplicates removed (conjectured).

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A222311 Leading diagonal of triangle in A222310.

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