A324201 -id:A324201 - cp's OEIS frontend

A324816 Binary weight of A324815; number of 1-bits in common positions in 2*A156552(n) and A323243(n).

0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 1, 2, 0, 2, 0, 1, 2, 0, 0, 1, 2, 0, 2, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 1, 1, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 2, 1, 1, 0, 3, 2, 1, 1, 1, 0, 1, 0, 0, 2, 3, 1, 0, 0, 0, 1, 1, 0, 2, 0, 0, 1, 0, 2, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 0, 3, 1, 0, 0, 0, 2, 3, 0, 1, 1, 1, 0, 1, 0, 1, 1

Offset: 1

Antti Karttunen, Mar 17 2019

nonn

Numbers 0 .. 7 occur for the first time in 1, 4, 9, 54, 162, 972, 2816, 3456. These are also the positions of records so far.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000120, A156552, A323243, A324201, A324815.

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552 by David A. Corneth
A324816(n) = hammingweight(bitand(2*A156552(n),A323243(n))); \\ Needs code also from A323243.

a(n) = A000120(A324815(n)).
a(p) = 0 for all primes p.
a(A324201(n)) = A000043(n).

A324879 Numbers k such that A324863(k) is equal to A324874(k).

Original entry on oeis.org

1, 9, 15, 16, 21, 27, 35, 39, 55, 57, 64, 75, 77, 85, 87, 90, 91, 95, 99, 105, 111, 115, 119, 125, 129, 133, 143, 147, 155, 159, 161, 175, 183, 185, 189, 195, 201, 203, 205, 209, 213, 221, 235, 237, 243, 245, 253, 256, 259, 265, 267, 275, 285, 287, 295, 299, 301, 303, 319, 321, 323, 325, 335, 339, 341, 351, 355, 363, 365

Offset: 1

Antti Karttunen, Mar 27 2019

nonn

In range 1..10000, there are only three such numbers k for which A324868(k) == A000120(A324866(k)): 1, 9, 125. See A324201.

Antti Karttunen, Table of n, a(n) for n = 1..2240 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000120, A324863, A324866, A324868, A324874.

Subsequences: A324201, A324880 (even terms).

for(n=1,oo,if(A324863(n)==A324874(n), print1(n,", ")));

A329640 Numbers n for which A329639(n) is equal to gcd(A329638(n), A329639(n)).

Original entry on oeis.org

1, 9, 18, 27, 45, 54, 70, 75, 84, 125, 135, 144, 153, 198, 279, 366, 390, 392, 423, 459, 747, 837, 855, 858, 891, 927, 1269, 1341, 1494, 1503, 1690, 1899, 2097, 2241, 2493, 2604, 2679, 2763, 2781, 2888, 2979, 2988, 3177, 3411, 3507, 3879, 4023, 4041, 4050, 4482, 4491, 4509, 4707, 5067, 5283, 5463, 5679, 5697, 5817, 5877, 5982, 6093

Offset: 1

Antti Karttunen, Nov 21 2019

nonn

After the initial 1, numbers n such that A329638(n) is a multiple of A329639(n).

Antti Karttunen, Table of n, a(n) for n = 1..85 (based on Hans Havermann's factorization of A156552)

Cf. A156552, A329638, A329639, A329641.
Cf. A324201 (a subsequence).
Cf. also A326141.

isA329640(n) = { my(u=A329639(n)); (u==gcd(A329638(n), u)); };

A368989 Values of records in A368698.

Original entry on oeis.org

1, 2, 3, 4, 8, 24, 26, 48, 124, 342, 624, 1330, 2400, 3124, 14640, 16806, 28560, 29281, 161050, 371292, 483153, 1771560, 4826808, 19487170, 20049821, 24318700, 62748516, 214358880, 289608539, 410338672, 815730720, 2357947690, 3234870386, 6975757440, 10604499372, 25937424600

Offset: 1

Antti Karttunen, Jan 12 2024

nonn

Terms A324201(k) - 1: 8, 124, 161050, 410338672, ..., seem to form a subsequence. See comments in A368988 and in A344005.

Cf. A324201, A344005, A368988, A368698.

a(n) = A368698(A368988(n)).

A342652 a(n) = A331410(A156552(n)).

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 3, 2, 3, 0, 2, 0, 3, 2, 3, 0, 2, 1, 4, 1, 3, 0, 2, 0, 1, 3, 4, 2, 3, 0, 5, 3, 3, 0, 4, 0, 4, 2, 6, 0, 2, 1, 4, 4, 4, 0, 4, 2, 3, 4, 7, 0, 3, 0, 7, 3, 3, 3, 3, 0, 5, 5, 3, 0, 4, 0, 8, 2, 5, 2, 4, 0, 3, 3, 8, 0, 5, 3, 11, 6, 5, 0, 4, 2, 5, 7, 8, 4, 5, 0, 2, 3, 4, 0, 6, 0, 4, 2

Offset: 2

Antti Karttunen, Mar 18 2021

nonn

Positions of ones is given by a subsequence of A053810, i.e., prime powers whose exponent is one of the primes in A000043. See also A324201, A335431.

Index entries for sequences computed from indices in prime factorization

Cf. A000043, A003961, A053810, A156552, A324201, A331410, A335431, A342651.

A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res};
A331410(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A331410(f[k,1]+1)))); };
A342652(n) = A331410(A156552(n));

a(n) = A331410(A156552(n)).
a(p) = 0 for all primes p.
a(A003961(n)) = a(n).

A368988 Positions of records in A368698.

Original entry on oeis.org

0, 2, 3, 4, 6, 12, 14, 24, 28, 56, 60, 112, 120, 124, 240, 248, 480, 488, 496, 992, 1000, 1008, 2016, 2032, 4016, 4024, 4064, 4080, 8112, 8128, 8160, 8176, 16312, 16320, 16352, 16368

Offset: 1

Antti Karttunen, Jan 12 2024

nonn

The even perfect numbers seem to occur here, as via Doudna-mapping they encode terms of A324201, which in turn have large values in A344005.

Cf. A000396, A324201, A344005, A368698, A368989 (values of records).

A353308 Numbers k for which A046523(A332223(k)) is equal to A046523(k).

Original entry on oeis.org

1, 2, 9, 14, 15, 38, 39, 57, 68, 70, 92, 106, 110, 111, 125, 129, 130, 156, 170, 183, 190, 213, 230, 242, 245, 267, 275, 338, 350, 380, 393, 416, 441, 455, 494, 518, 522, 532, 572, 579, 585, 590, 595, 627, 638, 646, 650, 682, 686, 722, 740, 754, 782, 790, 850, 855, 879, 902, 946, 950, 957, 969, 994, 1090, 1118, 1227

Offset: 1

Antti Karttunen, Apr 17 2022

nonn

Numbers k such that A348717(A332223(k)) = A348717(k) form a subsequence of this sequence. As its subsequence, we further have sequences A005940(1+A336702(n)) and A005940(1+A027687(n)), computed for n >= 1, and sorted into ascending order.

Cf. A000203, A005940, A046523, A156552, A332223, A336702, A348717.
Cf. A324201 (subsequence), A353363.
Cf. also A019278, A323653.

cp's OEIS Frontend

A324816 Binary weight of A324815; number of 1-bits in common positions in 2*A156552(n) and A323243(n).

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A324879 Numbers k such that A324863(k) is equal to A324874(k).

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A329640 Numbers n for which A329639(n) is equal to gcd(A329638(n), A329639(n)).

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A368989 Values of records in A368698.

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A342652 a(n) = A331410(A156552(n)).

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A368988 Positions of records in A368698.

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A353308 Numbers k for which A046523(A332223(k)) is equal to A046523(k).

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