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Search: A061142
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A061142 Replace each prime factor of n by 2. +30
20
1, 2, 2, 4, 2, 4, 2, 8, 4, 4, 2, 8, 2, 4, 4, 16, 2, 8, 2, 8, 4, 4, 2, 16, 4, 4, 8, 8, 2, 8, 2, 32, 4, 4, 4, 16, 2, 4, 4, 16, 2, 8, 2, 8, 8, 4, 2, 32, 4, 8, 4, 8, 2, 16, 4, 16, 4, 4, 2, 16, 2, 4, 8, 64, 4, 8, 2, 8, 4, 8, 2, 32, 2, 4, 8, 8, 4, 8, 2, 32, 16, 4, 2, 16, 4, 4, 4, 16, 2, 16, 4, 8, 4, 4, 4 (list; graph; listen)
OFFSET

1,2
LINKS

R. Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

a(n)=sum( d divides n, 2^(bigomega(d)-omega(d)))=sum( d divides n, 2^(A001222(d)-A001221(d))) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 30 2002

a(n) = A000079(A001222(n)), i.e. a(n)=2^bigomega(n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 13 2005

Totally multiplicative with a(p) = 2. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Oct 04 2006

EXAMPLE

a(100)=16 since 100=2*2*5*5 and so a(100)=2*2*2*2.

MAPLE

with(numtheory): seq(2^bigomega(n), n=1..95);

CROSSREFS

Cf. A001222, A000079, A123667.

Cf. A034444, A124508.

Sequence in context: A055155 A085191 A165872 this_sequence A152858 A091248 A082991

Adjacent sequences: A061139 A061140 A061141 this_sequence A061143 A061144 A061145

KEYWORD

easy,nonn,mult

AUTHOR

Henry Bottomley (se16(AT)btinternet.com), May 29 2001

A124508 2^BigO(n) * 3^omega(n), where BigO=A001222 and omega=A001221, the numbers of prime factors of n with and without repetitions. +10
8
1, 6, 6, 12, 6, 36, 6, 24, 12, 36, 6, 72, 6, 36, 36, 48, 6, 72, 6, 72, 36, 36, 6, 144, 12, 36, 24, 72, 6, 216, 6, 96, 36, 36, 36, 144, 6, 36, 36, 144, 6, 216, 6, 72, 72, 36, 6, 288, 12, 72, 36, 72, 6, 144, 36, 144, 36, 36, 6, 432, 6, 36, 72, 192, 36, 216, 6, 72, 36, 216, 6, 288, 6 (list; graph; listen)
OFFSET

1,2
COMMENT

a(n) = A061142(n)*A074816(n) = A000079(A001222(n))*A000244(A001221(n));

A124509 gives the range: A124509(n) = a(A124510(n)) and a(m) <> a(A124510(n)) for m < A124510(n);

for primes p, q with p<>q: a(p) = 6; a(p*q) = 36; a(p^k) = 3*2^k, k>0;

for squarefree numbers m: a(m) = 6^omega(m);

A001222(a(n)) = A001222(n)+1; A001221(a(n)) = 2 for n > 1;

A124511(n) = a(a(n)); A124512(n) = a(a(a(n)));

LINKS

R. Zumkeller, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Prime Factorization

Eric Weisstein's World of Mathematics, Smooth number

FORMULA

Multiplicative with p^e -> 3*2^e, p prime and e>0.

CROSSREFS

Cf. A007283, A000400, A003586, A005117.

Sequence in context: A163757 A109538 A040031 this_sequence A028317 A055665 A127402

Adjacent sequences: A124505 A124506 A124507 this_sequence A124509 A124510 A124511

KEYWORD

nonn,mult

AUTHOR

Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 04 2006

A123667 a(n) = n * 2^bigomega(n). +10
3
1, 4, 6, 16, 10, 24, 14, 64, 36, 40, 22, 96, 26, 56, 60, 256, 34, 144, 38, 160, 84, 88, 46, 384, 100, 104, 216, 224, 58, 240, 62, 1024, 132, 136, 140, 576, 74, 152, 156, 640, 82, 336, 86, 352, 360, 184, 94, 1536, 196, 400, 204, 416, 106, 864, 220, 896, 228, 232, 118 (list; graph; listen)
OFFSET

1,2
COMMENT

Rearrangement of A123666.

FORMULA

Totally multiplicative with a(p) = 2p. a(n) = n * A061142(n). bigomega(n) = A001222(n).

CROSSREFS

Cf. A061142, A001222, A123666.

Sequence in context: A009463 A066260 A141042 this_sequence A129597 A088843 A154784

Adjacent sequences: A123664 A123665 A123666 this_sequence A123668 A123669 A123670

KEYWORD

mult,nonn

AUTHOR

Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Oct 04 2006

A079707 In prime factorization of n replace odd primes by their prime predecessor. +10
1
1, 2, 2, 4, 3, 4, 5, 8, 4, 6, 7, 8, 11, 10, 6, 16, 13, 8, 17, 12, 10, 14, 19, 16, 9, 22, 8, 20, 23, 12, 29, 32, 14, 26, 15, 16, 31, 34, 22, 24, 37, 20, 41, 28, 12, 38, 43, 32, 25, 18, 26, 44, 47, 16, 21, 40, 34, 46, 53, 24, 59, 58, 20, 64, 33, 28, 61, 52, 38, 30, 67, 32, 71, 62, 18 (list; graph; listen)
OFFSET

1,2
COMMENT

a(n)<=n; a(n)<n iff n>1 is odd; a(n)=n iff n=2^k;

A001222(a(n)) = A001222(n);

for 3-smooth numbers: a(2^i * 3^j) = 2^(i+j);

result after A061395(n)-1 iterations = A061142(n).

FORMULA

Multiplicative with 2->2 and prime(k)->prime(k-1), k>1.

CROSSREFS

Cf. A005408, A000079, A003586, A003961.

Sequence in context: A140472 A109168 A015134 this_sequence A131816 A128181 A125185

Adjacent sequences: A079704 A079705 A079706 this_sequence A079708 A079709 A079710

KEYWORD

nonn,mult

AUTHOR

Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 31 2003

A089693 Numbers n such that phi(n)=2^bigomega(n). +10
1
1, 3, 10, 20, 30, 40, 60, 80, 120, 160, 240, 320, 480, 640, 960, 1280, 1920, 2560, 3840, 5120, 7680, 10240, 15360, 20480, 30720, 40960, 61440, 81920 (list; graph; listen)
OFFSET

1,2
CROSSREFS

Cf. A000010, A061142.

Sequence in context: A048343 A056789 A098645 this_sequence A005997 A081205 A092305

Adjacent sequences: A089690 A089691 A089692 this_sequence A089694 A089695 A089696

KEYWORD

nonn

AUTHOR

Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 06 2004

A165872 Totally multiplicative sequence with a(p) = - 2. +10
1
1, -2, -2, 4, -2, 4, -2, -8, 4, 4, -2, -8, -2, 4, 4, 16, -2, -8, -2, -8, 4, 4, -2, 16, 4, 4, -8, -8, -2, -8, -2, -32, 4, 4, 4, 16, -2, 4, 4, 16, -2, -8, -2, -8, -8, 4, -2, -32, 4, -8 (list; graph; listen)
OFFSET

1,2
FORMULA

a(n) = A122803(A001222(n)) = (-2)^bigomega(n) = (-2)^A001222(n). Abs(a(n)) = A061142(n)

CROSSREFS

Sequence in context: A101113 A055155 A085191 this_sequence A061142 A152858 A091248

Adjacent sequences: A165869 A165870 A165871 this_sequence A165873 A165874 A165875

KEYWORD

sign

AUTHOR

Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Sep 28 2009

A166632 Totally multiplicative sequence with a(p) = 2*(p-1) for prime p. +10
1
1, 2, 4, 4, 8, 8, 12, 8, 16, 16, 20, 16, 24, 24, 32, 16, 32, 32, 36, 32, 48, 40, 44, 32, 64, 48, 64, 48, 56, 64, 60, 32, 80, 64, 96, 64, 72, 72, 96, 64, 80, 96, 84, 80, 128, 88, 92, 64, 144, 128 (list; graph; listen)
OFFSET

1,2
FORMULA

Multiplicative with a(p^e) = (2*(p-1))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(p(k)-1)^e(k). a(n) = A061142(n) * A003958(n) = 2^bigomega(n) * A003958(n) = 2^A001222(n) * A003958(n).

CROSSREFS

Sequence in context: A034583 A076347 A140513 this_sequence A116596 A048656 A107848

Adjacent sequences: A166629 A166630 A166631 this_sequence A166633 A166634 A166635

KEYWORD

nonn

AUTHOR

Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Oct 18 2009

A166642 Totally multiplicative sequence with a(p) = 2*(p+1) for prime p. +10
1
1, 6, 8, 36, 12, 48, 16, 216, 64, 72, 24, 288, 28, 96, 96, 1296, 36, 384, 40, 432, 128, 144, 48, 1728, 144, 168, 512, 576, 60, 576, 64, 7776, 192, 216, 192, 2304, 76, 240, 224, 2592, 84, 768, 88, 864, 768, 288, 96, 10368, 256, 864 (list; graph; listen)
OFFSET

1,2
FORMULA

Multiplicative with a(p^e) = (2*(p+1))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(p(k)+1)^e(k). a(n) = A061142(n) * A003959(n) = 2^bigomega(n) * A003959(n) = 2^A001222(n) * A003959(n).

CROSSREFS

Sequence in context: A076904 A025091 A028321 this_sequence A077709 A025627 A038262

Adjacent sequences: A166639 A166640 A166641 this_sequence A166643 A166644 A166645

KEYWORD

nonn

AUTHOR

Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Oct 18 2009

A167294 Totally multiplicative sequence with a(p) = 2*(p-2) for prime p. +10
1
1, 0, 2, 0, 6, 0, 10, 0, 4, 0, 18, 0, 22, 0, 12, 0, 30, 0, 34, 0, 20, 0, 42, 0, 36, 0, 8, 0, 54, 0, 58, 0, 36, 0, 60, 0, 70, 0, 44, 0, 78, 0, 82, 0, 24, 0, 90, 0, 100, 0 (list; graph; listen)
OFFSET

1,3
FORMULA

Multiplicative with a(p^e) = (2*(p-2))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(p(k)-2))^e(k). a(2k) = 0 for k >= 1. a(n) = A061142(n) * A166586(n) = 2^bigomega(n) * A166586(n) = 2^A001222(n) * A166586(n).

CROSSREFS

Sequence in context: A046520 A157195 A019781 this_sequence A081153 A126869 A094233

Adjacent sequences: A167291 A167292 A167293 this_sequence A167295 A167296 A167297

KEYWORD

nonn,new

AUTHOR

Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 01 2009

A167303 Totally multiplicative sequence with a(p) = 2*(p+2) for prime p. +10
1
1, 8, 10, 64, 14, 80, 18, 512, 100, 112, 26, 640, 30, 144, 140, 4096, 38, 800, 42, 896, 180, 208, 50, 5120, 196, 240, 1000, 1152, 62, 1120, 66, 32768, 260, 304, 252, 6400, 78, 336, 300, 7168, 86, 1440, 90, 1664, 1400, 400, 98, 40960, 324, 1568 (list; graph; listen)
OFFSET

1,2
FORMULA

Multiplicative with a(p^e) = (2*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(p(k)+2))^e(k). a(n) = A061142(n) * A166590(n) = 2^bigomega(n) * A166590(n) = 2^A001222(n) * A166590(n).

CROSSREFS

Sequence in context: A112547 A015657 A106131 this_sequence A025634 A038288 A070276

Adjacent sequences: A167300 A167301 A167302 this_sequence A167304 A167305 A167306

KEYWORD

nonn,new

AUTHOR

Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 01 2009

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Last modified November 21 21:21 EST 2009. Contains 167310 sequences.

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