Displaying 1-10 of 474 results found.
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The terms of A329447 sorted into increasing order.
+0
0
0, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 44, 45, 46, 47, 48, 49, 50, 55, 56, 57, 58, 59, 60, 66, 67, 68, 69, 70, 77, 78, 79, 80, 88, 89, 90, 99, 100, 112, 113, 114, 115, 116, 117, 118, 119, 120, 123, 124, 125, 126, 127, 128, 129, 130, 134, 135, 136, 137, 138, 139, 140, 145, 146, 147, 148
COMMENTS
Is there an independent characterization of these numbers?
Numbers k for which A276085(k) is not a multiple of 4 and has at least one divisor of the form p^p, with p an odd prime, where A276085 is fully additive with a(p) = p#/p.
+0
0
174, 232, 282, 325, 376, 438, 462, 474, 539, 584, 606, 616, 632, 654, 678, 798, 808, 872, 904, 906, 931, 966, 978, 1002, 1064, 1074, 1075, 1105, 1127, 1182, 1208, 1288, 1302, 1304, 1336, 1398, 1432, 1506, 1519, 1576, 1626, 1662, 1736, 1755, 1842, 1864, 1866, 2008, 2168, 2216, 2226, 2340, 2425, 2442, 2456, 2488, 2514
EXAMPLE
A276085(55) = 216 = 2^3 * 3^3, which although it has a divisor of the form p^p, with p an odd prime, it is also a multiple of 4, and therefore 55 is NOT included in this sequence.
A276085(174) = 223092873 = 3^3 * 3 * 1063 * 2591, which has a divisor of the form p^p, with p an odd prime and therefore 174 is included in this sequence.
A276085(1104299) = 11231250 = 2 * 3 * 5^5 * 599, thus 1104299 is included.
PROG
(PARI)
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1, primepi(f[k, 1]-1), prime(i))); };
A377868(n) = if(isprime(n), 1, my(x= A276085(n), pp); forprime(p=2, , pp = p^p; if(!(x%pp), return(0)); if(pp > x, return(1))));
A variant of Golomb's sequence ( A001462): the n-th digit of the sequence gives the number of times n appears, with a(1) = 1 and a(2) = 2.
+0
0
1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 19, 19
COMMENTS
This sequence is a base-10 variant of A167500. Numbers corresponding to positions of zeros do not appear in the sequence.
This sequence first differ from A001462 for n = 169: a(169) = 31 whereas A001462(169) = 29.
PROG
(PARI) \\ See Links section.
30, 32, 34, 36, 38, 154, 156, 158, 160, 162, 164, 166, 168, 320, 340, 342, 370, 372, 408, 410, 412, 454, 456, 458, 460, 508, 510, 512, 514, 516, 570, 571, 573, 574, 576, 577, 579, 580, 582, 583, 585, 586, 588, 591, 594, 597, 600, 603, 606, 609, 612, 615, 618
COMMENTS
Also numbers k such that A377896(k) = 0. This sequence is infinite.
PROG
(PARI) \\ See Links section.
a(n) is the number of occurrences of n in A377862.
+0
0
1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 5, 1
COMMENTS
Equivalently, the list of successive digits of A377862.
PROG
(PARI) \\ See Links section.
Parity of A083345(n), where A083345(n) = n' / gcd(n,n') = numerator of Sum(e/p: n=Product(p^e)).
+0
0
0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0
COMMENTS
This is one's complement of A369001. See more comments and formulas in that sequence.
EXAMPLE
A083345(174) = 151, therefore a(174) = 1. Note also that A276085(174) = 223092873 is not a multiple of 4.
PROG
(PARI) A377874(n) = { my(f=factor(n)); (numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1])))%2); };
CROSSREFS
Characteristic function of A369003, whose complement A369002 gives the positions of 0's.
Numbers i for which A194627(i) is prime.
+0
0
2, 3, 9, 11, 29, 75, 77, 101, 105, 107, 221, 225, 235, 257, 315, 321, 323, 357, 363, 389, 411, 417, 431, 453, 455, 461, 501, 509, 515, 519, 557, 635, 645, 655, 689, 795, 799, 851, 885, 887, 911, 915, 921, 923, 933, 977, 989, 1029, 1033, 1037, 1071, 1073, 1145, 1167, 1175, 1187, 1197, 1201, 1241
FORMULA
A377791(n) = (n-1)^2 + (a(n)-n)^2 + 1.
MAPLE
v:= 1: p:= 0: q:= 0: np:= 0: R:= NULL:
for i from 1 while np < 100 do
if isprime(v) then p:= p+1; R:= R, i; np:= np+1 else q:= q+1 fi;
v:= p^2 + q^2 + 1;
od:
R;
2, 3, 41, 59, 593, 4787, 4937, 8699, 9281, 9491, 44201, 45491, 49429, 59219, 90197, 93251, 93893, 115211, 118661, 136523, 152501, 156467, 166949, 184571, 185477, 189851, 225353, 232091, 236981, 239963, 277577, 364571, 375569, 386731, 428873, 577307, 581941, 662339, 717161, 718931, 758501, 763811
FORMULA
a(n) = (n-1)^2 + ( A377882(n)-n)^2 + 1.
MAPLE
v:= 1: p:= 0: q:= 0: np:= 0: R:= NULL:
for i from 1 while np < 100 do
if isprime(v) then p:= p+1; R:= R, v; np:= np+1 else q:= q+1 fi;
v:= p^2 + q^2 + 1;
od:
R;
Cogrowth sequence of the 16-element quasihedral group SD16 = <S,T | S^8, T^2, STS^5T>.
+0
0
1, 1, 1, 4, 28, 136, 544, 2080, 8128, 32512, 130816, 524800, 2099200, 8390656, 33550336, 134201344, 536854528, 2147516416, 8590065664, 34359869440, 137438691328, 549754765312, 2199022206976, 8796095119360, 35184380477440, 140737496743936, 562949936644096
COMMENTS
Gives the even terms, all the odd terms are 0.
Also called QD16, Q8:C2. Gap identifier 16,8.
FORMULA
G.f.: (6*x^3-7*x^2+5*x-1) / ((4*x-1) * (4*x^2-2*x+1)).
G.f. A(x) = Sum_{n>=0} a(n)*x^n where a(n) = Sum_{k=0..n-1} [x^k] A(x)^(k*(n-k)) for n >= 1 with a(0) = 1.
+0
0
1, 1, 2, 8, 55, 525, 6202, 85842, 1350421, 23687392, 457238998, 9620344475, 219011293036, 5363006495793, 140567134618434, 3927060955253388, 116510112059820553, 3658928109471912657, 121273249515650581850, 4231012832296844451474, 155003839703746214942229, 5949765253601511005012122
COMMENTS
Compare to C(x) = Sum_{n>=0} c(n)*x^n where c(n) = Sum_{k=0..n-1} [x^k] C(x)^(n-k) for n >= 1 with c(0) = 1, holds when C(x) is the g.f. of the Catalan numbers ( A000108).
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) a(n) = Sum_{k=0..n-1} [x^k] A(x)^(k*(n-k)) for n >= 1, with a(0) = 1.
(2) A(x) = 1 + x*Sum_{n>=0} x^n/n! * ( d^n/dy^n A(y)^n/(1 - x*A(y)^n) ) evaluated at y = 0.
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 55*x^4 + 525*x^5 + 6202*x^6 + 85842*x^7 + 1350421*x^8 + 23687392*x^9 + 457238998*x^10 + 9620344475*x^11 + 219011293036*x^12 + ...
By definition, a(n) equals the sum of the coefficients of x^k in A(x)^(k*(n-k)), k = 0..n-1, for n >= 1, as illustrated below.
a(1) = [x^0] A(x)^0;
a(2) = 1 + [x^1] A(x)^1;
a(3) = 1 + [x^1] A(x)^2 + [x^2] A(x)^2;
a(4) = 1 + [x^1] A(x)^3 + [x^2] A(x)^4 + [x^3] A(x)^3;
a(5) = 1 + [x^1] A(x)^4 + [x^2] A(x)^6 + [x^3] A(x)^6 + [x^4] A(x)^4;
a(6) = 1 + [x^1] A(x)^5 + [x^2] A(x)^8 + [x^3] A(x)^9 + [x^4] A(x)^8 + [x^5] A(x)^5;
a(7) = 1 + [x^1] A(x)^6 + [x^2] A(x)^10 + [x^3] A(x)^12 + [x^4] A(x)^12 + [x^5] A(x)^10 + [x^6] A(x)^6;
a(8) = 1 + [x^1] A(x)^7 + [x^2] A(x)^12 + [x^3] A(x)^15 + [x^4] A(x)^16 + [x^5] A(x)^15 + [x^6] A(x)^12 + [x^7] A(x)^7;
...
Explicitly,
a(1) = 1 = 1;
a(2) = 1 + 1 = 2;
a(3) = 1 + 2 + 5 = 8;
a(4) = 1 + 3 + 14 + 37 = 55;
a(5) = 1 + 4 + 27 + 128 + 365 = 525;
a(6) = 1 + 5 + 44 + 300 + 1406 + 4446 = 6202;
a(7) = 1 + 6 + 65 + 580 + 3795 + 17892 + 63503 = 85842;
a(8) = 1 + 7 + 90 + 995 + 8460 + 53088 + 258212 + 1029568 = 1350421;
...
RELATED TABLES.
The table of coefficients of x^k in A(x)^n begins as follows.
n\k 0 1 2 3 4 5 6 7
A^1 = [1, 1, 2, 8, 55, 525, 6202, 85842, ...];
A^2 = [1, 2, 5, 20, 130, 1192, 13738, 187068, ...];
A^3 = [1, 3, 9, 37, 231, 2037, 22877, 306201, ...];
A^4 = [1, 4, 14, 60, 365, 3104, 33944, 446208, ...];
A^5 = [1, 5, 20, 90, 540, 4446, 47330, 610580, ...];
A^6 = [1, 6, 27, 128, 765, 6126, 63503, 803424, ...];
A^7 = [1, 7, 35, 175, 1050, 8218, 83020, 1029568, ...];
A^8 = [1, 8, 44, 232, 1406, 10808, 106540, ...];
A^9 = [1, 9, 54, 300, 1845, 13995, 134838, ...];
A^10 = [1, 10, 65, 380, 2380, 17892, 168820, ...];
A^11 = [1, 11, 77, 473, 3025, 22627, 209539, ...];
A^12 = [1, 12, 90, 580, 3795, 28344, 258212, ...];
A^13 = [1, 13, 104, 702, 4706, 35204, 316238, ...];
A^14 = [1, 14, 119, 840, 5775, 43386, 385217, ...];
A^15 = [1, 15, 135, 995, 7020, 53088, 466970, ...];
A^16 = [1, 16, 152, 1168, 8460, 64528, 563560, ...];
...
PROG
(PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^(k*(m-k)), k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
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