A278478 -id:A278478 - cp's OEIS frontend

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

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).

A278784 Numbers m such that A000041(m) is of the form 2^7 * k for odd k.

Original entry on oeis.org

109, 230, 557, 675, 1094, 1679, 2070, 2314, 2594, 2781, 2873, 3133, 3498, 3548, 3595, 3601, 3629, 3911, 4059, 4171, 4264, 4274, 4501, 4568, 4878, 4889, 5112, 5136, 5507, 5675, 5907, 6059, 6152, 6260, 6490, 6667, 6669, 6938, 6961, 7667, 8013, 8640, 8729, 9171

Offset: 1

Colin Barker, Nov 28 2016

nonn

Numbers m such that A278478(m) = 7.

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

Colin Barker, Table of n, a(n) for n = 1..100

Cf. A000041, A278201, A278779, A278780, A278781, A278782, A278783.

Positions of 7's in A278478.

Select[Range[5000], IntegerExponent[PartitionsP[#], 2] == 7 &] (* Amiram Eldar, May 25 2024 *)

isok(n) = valuation(numbpart(n), 2)==7;
select(n->isok(n), vector(9000, n, n))

A000041(a(n)) = A278201(n).

A069935 Maximal power of 2 that divides the n-th partition number.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 8, 1, 1, 1, 16, 1, 1, 1, 2, 1, 8, 2, 1, 1, 2, 4, 2, 2, 1, 4, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 4, 1, 2, 4, 4, 1, 1, 4, 1, 2, 2, 64, 1, 1, 1, 32, 1, 1, 1, 4, 8, 1, 1, 4, 2, 4, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 8, 1, 2048, 2, 8, 1, 4, 2, 1, 2, 1, 1, 16, 1

Offset: 0

Sharon Sela (sharonsela(AT)hotmail.com), May 04 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000041, A006519, A278478.

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

for(n=0,10^3,my(p=numbpart(n));print1(2^valuation(p,2),", ")); \\ Joerg Arndt, May 09 2013

From Amiram Eldar, May 25 2024: (Start)
a(n) = A006519(A000041(n)).
a(n) = 2^A278478(n). (End)

More terms from John W. Layman, May 09 2002

A278479 a(n) is the smallest k such that the 2-adic valuation of A000041(k) equals n.

Original entry on oeis.org

0, 2, 26, 11, 15, 70, 66, 109, 154, 478, 441, 96, 3693, 15005, 66934, 99420, 18978, 43002, 55943, 972190, 1151214, 2799146, 26801784, 46632889, 15519397, 122101417, 210553237, 289585489, 473093534

Offset: 0

Joerg Arndt, Nov 23 2016

a(n) is the smallest k such that A278478(k) = n.

Next term > 10^9.

Joerg Arndt, C++ program for this sequence

Cf. A278478.

cp's OEIS Frontend

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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A278784 Numbers m such that A000041(m) is of the form 2^7 * k for odd k.

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A069935 Maximal power of 2 that divides the n-th partition number.

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A278479 a(n) is the smallest k such that the 2-adic valuation of A000041(k) equals n.

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