A278784 -id:A278784 - cp's OEIS frontend

A278478 a(n) is the 2-adic valuation of A000041(n).

0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 3, 0, 0, 0, 4, 0, 0, 0, 1, 0, 3, 1, 0, 0, 1, 2, 1, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1, 2, 2, 0, 0, 2, 0, 1, 1, 6, 0, 0, 0, 5, 0, 0, 0, 2, 3, 0, 0, 2, 1, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 11, 1, 3, 0, 2, 1, 0, 1, 0, 0, 4, 0, 2, 7, 1, 0, 2, 2, 0, 0, 3, 2, 0

Offset: 0

Joerg Arndt, Nov 23 2016

nonn

Write A000041(n) = 2^k * s where s is odd, then a(n) = k.

Joerg Arndt, Table of n, a(n) for n = 0..10000

Cf. A000041, A007814, A069935, A278479.
Cf. A052002, A237280, A278779, A278780, A278781, A278782, A278783, A278784 (positions of terms 0, 1, 2, ..., 7 in this sequence).
Cf. also A278241.

a:= n-> padic[ordp](combinat[numbpart](n), 2):
seq(a(n), n=0..120);  # Alois P. Heinz, Nov 23 2016

a[n_] := IntegerExponent[PartitionsP[n], 2]; Array[a, 100, 0] (* Amiram Eldar, May 25 2024 *)

{ my( x='x+O('x^100), v=Vec(1/eta(x)) ); vector(#v,n,valuation(v[n],2)) }

From Amiram Eldar, May 25 2024: (Start)
a(n) = A007814(A000041(n)).
a(n) = log_2(A069935(n)). (End)

A278779 Numbers m such that A000041(m) is of the form 2^2 * k for odd k.

Original entry on oeis.org

26, 30, 55, 58, 59, 62, 74, 78, 80, 84, 100, 108, 112, 113, 117, 124, 126, 135, 153, 187, 191, 200, 205, 258, 265, 280, 291, 310, 317, 323, 336, 337, 342, 344, 351, 352, 353, 358, 359, 365, 374, 380, 384, 404, 409, 416, 444, 445, 458, 481, 485, 492, 501, 503

Offset: 1

Colin Barker, Nov 28 2016

nonn

Numbers m such that A278478(m) = 2.

Also numbers m such that A000041(m) has twice as many even divisors as odd divisors.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000041, A278196, A278780, A278781, A278782, A278783, A278784.

Positions of 2's in A278478.

Select[Range[3000], IntegerExponent[PartitionsP[#], 2] == 2 &] (* Amiram Eldar, May 25 2024 *)

isok(n) = valuation(numbpart(n), 2)==2;
select(n->isok(n), vector(1000, n, n))

A000041(a(n)) = A278196(n).

A278780 Numbers m such that A000041(m) is of the form 2^3 * k for odd k.

Original entry on oeis.org

11, 21, 75, 94, 98, 116, 120, 125, 128, 130, 133, 137, 141, 142, 180, 206, 231, 236, 243, 248, 255, 268, 292, 297, 303, 305, 322, 334, 340, 350, 364, 386, 397, 413, 415, 469, 471, 487, 494, 515, 550, 554, 605, 606, 609, 628, 631, 662, 676, 692, 699, 744, 745

Offset: 1

Colin Barker, Nov 28 2016

nonn

Numbers m such that A278478(m) = 3.

Also numbers m such that A000041(m) has three times as many even divisors as odd divisors.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000041, A278197, A278779, A278781, A278782, A278783, A278784.

Positions of 3's in A278478.

Select[Range[750], IntegerExponent[PartitionsP[#], 2] == 3 &] (* Amiram Eldar, May 25 2024 *)

isok(n) = valuation(numbpart(n),2)==3;
select(n->isok(n), vector(1000, n, n))

A000041(a(n)) = A278197(n).

A278781 Numbers m such that A000041(m) is of the form 2^4 * k for odd k.

Original entry on oeis.org

15, 106, 122, 131, 136, 253, 295, 327, 339, 383, 412, 449, 465, 517, 520, 551, 580, 581, 599, 602, 632, 648, 669, 677, 776, 806, 815, 838, 904, 927, 1071, 1137, 1166, 1174, 1199, 1263, 1275, 1298, 1325, 1375, 1399, 1404, 1425, 1554, 1564, 1641, 1684, 1688

Offset: 1

Colin Barker, Nov 28 2016

nonn

Numbers m such that A278478(m) = 4.

Also numbers m such that A000041(m) has four times as many even divisors as odd divisors.

Colin Barker, Table of n, a(n) for n = 1..600

Cf. A000041, A278198, A278779, A278780, A278782, A278783, A278784.

Positions of 4's in A278478.

Select[Range[1700], IntegerExponent[PartitionsP[#], 2] == 4 &] (* Amiram Eldar, May 25 2024 *)

isok(n) = valuation(numbpart(n), 2)==4;
select(n->isok(n), vector(2000, n, n))

A000041(a(n)) = A278198(n).

A278782 Numbers m such that A000041(m) is of the form 2^5 * k for odd k.

Original entry on oeis.org

70, 179, 262, 278, 419, 561, 682, 698, 767, 879, 1147, 1238, 1273, 1317, 1362, 1364, 1378, 1450, 1478, 1499, 1509, 1548, 1590, 1638, 1668, 1711, 1752, 1781, 1838, 1949, 2170, 2187, 2300, 2317, 2334, 2382, 2408, 2447, 2463, 2499, 2551, 2669, 2695, 2788, 2926

Offset: 1

Colin Barker, Nov 28 2016

nonn

Numbers m such that A278478(m) = 5.

Also numbers m such that A000041(m) has five times as many even divisors as odd divisors.

Colin Barker, Table of n, a(n) for n = 1..300

Cf. A000041, A278199, A278779, A278780, A278781, A278783, A278784.

Positions of 5's in A278478.

Select[Range[500], IntegerExponent[PartitionsP[#], 2] == 5 &] (* Amiram Eldar, May 25 2024 *)

isok(n) = valuation(numbpart(n), 2)==5;
select(n->isok(n), vector(3000, n, n))

A000041(a(n)) = A278199(n).

A278783 Numbers m such that A000041(m) is of the form 2^6 * k for odd k.

Original entry on oeis.org

66, 149, 298, 435, 450, 518, 615, 703, 751, 764, 765, 855, 982, 1389, 1398, 1411, 1555, 1896, 2113, 2124, 2286, 2400, 2575, 2618, 2816, 2890, 2989, 3032, 3113, 3202, 3351, 3430, 3454, 3485, 3509, 3562, 3652, 3786, 3994, 4061, 4202, 4690, 5042, 5055, 5067

Offset: 1

Colin Barker, Nov 28 2016

nonn

Also numbers m such that A000041(m) has six times as many even divisors as odd divisors.

Colin Barker, Table of n, a(n) for n = 1..150

Cf. A000041, A278200, A278779, A278780, A278781, A278782, A278784.

Positions of 6's in A278478.

Select[Range[5100], IntegerExponent[PartitionsP[#], 2] == 6 &] (* Amiram Eldar, May 25 2024 *)

isok(n) = valuation(numbpart(n), 2)==6;
select(n->isok(n), vector(6000, n, n))

A000041(a(n)) = A278200(n).

cp's OEIS Frontend

A278478 a(n) is the 2-adic valuation of A000041(n).

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A278779 Numbers m such that A000041(m) is of the form 2^2 * k for odd k.

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A278780 Numbers m such that A000041(m) is of the form 2^3 * k for odd k.

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A278781 Numbers m such that A000041(m) is of the form 2^4 * k for odd k.

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A278782 Numbers m such that A000041(m) is of the form 2^5 * k for odd k.

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A278783 Numbers m such that A000041(m) is of the form 2^6 * k for odd k.

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