A369666 -id:A369666

Search: a369666 -id:a369666

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page 1

A369663

Numbers k of the form 4m+3, whose arithmetic derivative k' is of the form 4u+2, and k' has an even number of prime factors.

+10
5

375, 459, 783, 819, 875, 1071, 1107, 1155, 1375, 1395, 1715, 1911, 1935, 1995, 2223, 2275, 2375, 2499, 2619, 2655, 2695, 2727, 2875, 2907, 2943, 3003, 3051, 3135, 3195, 3255, 3315, 3519, 3575, 3627, 3699, 3843, 3927, 3975, 4059, 4459, 4515, 4671, 4815, 4887, 4935, 4959, 5187, 5247, 5375, 5415, 5607, 5635, 5655

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

PROG

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

isA369663(n) = if(3!=(n%4), 0, my(d=A003415(n)); (2==(d%4) && !(bigomega(d)%2)));

CROSSREFS

Intersection of A004767 and A369661.

Subsequence of A369666.

Cf. A003415, A351228.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 06 2024

STATUS

approved

A369664

Numbers k of the form 4m+1, whose arithmetic derivative k' is of the form 4u+2, and k' has an odd number of prime factors.

+10
4

65, 77, 141, 161, 185, 201, 209, 221, 301, 305, 321, 341, 365, 377, 381, 413, 437, 453, 481, 485, 497, 501, 537, 545, 589, 649, 681, 689, 717, 721, 729, 737, 745, 749, 785, 789, 849, 893, 901, 905, 917, 921, 989, 1037, 1073, 1081, 1101, 1121, 1133, 1141, 1157, 1165, 1169, 1189, 1205, 1253, 1261, 1293, 1313, 1317

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

PROG

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

isA369664(n) = if(1!=(n%4), 0, my(d=A003415(n)); (2==(d%4) && (bigomega(d)%2)));

CROSSREFS

Intersection of A016813 and A369662.

Subsequence of A369666.

Cf. A003415, A351228.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 06 2024

STATUS

approved

A369665

a(n) = 1 if n > 1 and A276085(A003415(n)) == n (mod 4), otherwise 0, where A003415 is the arithmetic derivative, and A276085 is the primorial base log-function.

+10
2

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..100000

Index entries for characteristic functions

PROG

(PARI)

A002110(n) = prod(i=1, n, prime(i));

A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

A369665(n) = ((n>1) && ((A276085(A003415(n))%4)==(n%4)));

CROSSREFS