cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-30 of 94 results. Next

A250478 Number of times prime(n) occurs as the least prime factor among numbers 1 .. prime(n)^4: a(n) = A078898(A030514(n)).

Original entry on oeis.org

8, 14, 42, 92, 305, 455, 944, 1238, 2085, 3995, 4710, 7757, 10273, 11558, 14742, 20701, 28019, 30444, 39680, 46534, 49856, 62350, 71394, 86977, 111352, 124421, 130649, 145076, 151939, 167759, 236113, 257098, 291830, 302611, 370060, 382610, 427214, 475078
Offset: 1

Views

Author

Antti Karttunen, Dec 14 2014

Keywords

Links

Crossrefs

Column 8 of A249822.
Cf. also A250474 (column 4), A250477 (column 6).
Cf. A002110, A030514, A030078, A078898.

Programs

  • PARI
    allocatemem(234567890);
A002110(n) = prod(i=1, n, prime(i));
A250478(n) = { my(p3); p3 = (prime(n)^3); sumdiv(A002110(n-1), d, (moebius(d)*(p3\d))); };
for(n=1, 23, print1(A250478(n),", "));
  • Scheme
    (define (A250478 n) (A078898 (A030514 n)))

Formula

a(n) = A078898(A030514(n)).
a(1) = 1, a(n) = sum_{d | A002110(n-1)} moebius(d) * floor(prime(n)^3 / d). [Follows when A030514, prime(n)^4 is substituted to the similar formula given for A078898. Here A002110(n) gives the product of the first n primes. Because the latter is always squarefree, one could use here also Liouville's lambda (A008836) instead of Moebius mu (A008683).]

Extensions

More terms from Jon E. Schoenfield, Dec 14 2014

A302040 Numbers k such that A078898(k) is a power of 2; an analog for A000961 based on factorization-kind of process involving the sieve of Eratosthenes (A083221).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 29, 31, 32, 37, 41, 43, 45, 47, 49, 53, 55, 59, 61, 64, 67, 71, 73, 79, 83, 89, 91, 93, 97, 101, 103, 107, 109, 113, 115, 121, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 189, 191, 193, 197, 199, 203, 211, 223, 227, 229, 233, 235, 239, 241, 247, 251, 256, 257
Offset: 1

Views

Author

Antti Karttunen, Apr 02 2018

Keywords

Comments

Numbers k for which A302041(k) < 2, or equally, for which A302044(k) = 1.
Sequence A250245(A000961(k)) sorted into ascending order, or in other words, numbers k such that A250246(k) is a prime power (in A000961).
Numbers k such that all terms in iteration sequence k, A302042(k), A302042(A302042(k)), A302042(A302042(A302042(k))), ..., have an equal smallest prime factor (A020639) before the sequence settles to 1, in other words, that they all stay on the same row of A083221. This also forces the column position of each (A078898) to be a power of 2 (A000079).

Examples

			For k = 21 = 3*7, the smallest prime factor is 3. A302042(21) = 9, and A302042(9) = 3, both (9 and 3) which also have 3 as their smallest prime factor, and after that the sequence settles to 1, as A302042(3) = 1, thus 21 is included in this sequence.
For k = 27 = 3*3*3, the smallest prime factor is 3. However, A302042(27) = 7, thus 27 is not included in this sequence.

Links

Crossrefs

Cf. A000961, A078898, A302036, A302041, A302044, A302053.
Cf. A000040, A000079, A001248 (subsequences).

Programs

  • PARI
    for(n=1,257,if(2>A302041(n),print1(n,","))); \\ Other code as in A302041.

A317833 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A078898 (the ordinal transform of A020639, the smallest prime factor of n).

Original entry on oeis.org

1, 1, 1, 7, 1, 5, 1, 25, 7, 9, 1, 31, 1, 13, 5, 363, 1, 55, 1, 55, 7, 21, 1, 101, 7, 25, 33, 79, 1, 41, 1, 1335, 11, 33, 5, 305, 1, 37, 13, 177, 1, 59, 1, 127, 47, 45, 1, 1371, 7, 175, 17, 151, 1, 309, 7, 253, 19, 57, 1, 187, 1, 61, 67, 9923, 9, 95, 1, 199, 23, 113, 1, 927, 1, 73, 87, 223, 5, 113, 1, 2379, 715, 81, 1, 265, 11
Offset: 1

Views

Author

Antti Karttunen, Aug 10 2018

Keywords

Comments

The first negative term is a(840) = -445.

Links

Crossrefs

Cf. A046644, A078898.
Cf. also A305798, A305803, A305804, A317830, A317834.

Programs

  • Mathematica
    lpf[n_] := If[n == 1, 1, FactorInteger[n][[1, 1]]];
b[_] = 1;
A078898[n_] := A078898[n] = If[n == 0, 0, With[{t = lpf[n]}, b[t]++]];
f[n_] := f[n] = If[n == 1, 1, (1/2)(A078898[n] - Sum[If[1 < d < n, f[d]*f[n/d], 0], {d, Divisors[n]}])]
a[n_] := Numerator[f[n]];
Array[a, 100] (* Jean-François Alcover, Dec 19 2021 *)
  • PARI
    up_to = 16384;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A317833aux(n) = if(1==n,n,(A078898(n)-sumdiv(n,d,if((d>1)&&(dA317833aux(d)*A317833aux(n/d),0)))/2);
A317833(n) = numerator(A317833aux(n));

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A078898(n) - Sum_{d|n, d>1, d 1.

A280492 a(1) = 0; for n > 1, a(n) = A246277(n) - A078898(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 7, 0, 0, 0, 0, 0, -1, 0, 2, 0, 0, 0, 7, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, -6, 0, 0, 0, 5, 0, 8, 0, 0, 0, 1, 0, 13, 0, 6, 0, 0, 0, -3, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 12, 0, 0, 0, 9, 0, 2, 0, 2
Offset: 1

Views

Author

Antti Karttunen, Jan 09 2017

Keywords

Comments

For n > 1, a(n) gives the difference of column positions of n's location in arrays A246278 and A083221. Note that any n occurs on the same row in both arrays.

Links

Crossrefs

Cf. A078898, A083221, A246277, A246278, A249820.
Cf. also A278523, A279350, A280497, A280498.

Programs

Formula

a(1) = 0; for n > 1, a(n) = A246277(n) - A078898(n).

A300248 Filter sequence combining A046523(n) and A078898(n), the prime signature of n and the number of times the smallest prime factor of n is the smallest prime factor for numbers <= n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 6, 2, 7, 2, 8, 4, 9, 2, 10, 2, 11, 12, 13, 2, 14, 3, 15, 16, 17, 2, 18, 2, 19, 20, 21, 4, 22, 2, 23, 8, 24, 2, 25, 2, 26, 27, 28, 2, 29, 3, 30, 31, 32, 2, 33, 12, 34, 35, 36, 2, 37, 2, 38, 39, 40, 6, 41, 2, 42, 43, 44, 2, 45, 2, 46, 47, 48, 4, 49, 2, 50, 51, 52, 2, 53, 20, 54, 55, 56, 2, 57, 12, 58, 59, 60, 8, 61, 2, 62, 63, 64, 2
Offset: 1

Views

Author

Antti Karttunen, Mar 03 2018

Keywords

Comments

Restricted growth sequence transform of P(A046523(n), A078898(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

Examples

			a(10) = a(65) (= 6) because A078898(10) = A078898(65) = 5 (both numbers occur in column 5 of A083221) and because both have the same prime-signature (both are nonsquare semiprimes).

Links

Crossrefs

Cf. A046523, A078898.
Cf. also A300226, A300229, A300247.

Programs

  • PARI
    rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };
A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From A046523
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A078898(n) = { if(n<=1,n, my(spf=A020639(n),k=1,m=n/spf); while(m>1,if(A020639(m)>=spf,k++); m--); (k)); };
Aux300248(n) = if(1==n,0,(1/2)*(2 + ((A078898(n)+A046523(n))^2) - A078898(n) - 3*A046523(n)));
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300248(n))),"b300248.txt");

A305797 Dirichlet convolution of A078898 with A078899.

Original entry on oeis.org

1, 2, 2, 5, 2, 7, 2, 11, 6, 9, 2, 19, 2, 11, 8, 23, 2, 24, 2, 25, 9, 15, 2, 45, 8, 17, 17, 31, 2, 39, 2, 47, 11, 21, 10, 66, 2, 23, 12, 62, 2, 48, 2, 43, 27, 27, 2, 100, 10, 48, 14, 49, 2, 76, 11, 79, 15, 33, 2, 113, 2, 35, 32, 95, 12, 66, 2, 61, 17, 69, 2, 161, 2, 41, 37, 67, 12, 75, 2, 142, 44, 45, 2, 142, 13, 47, 20, 111, 2, 143, 13, 79, 21, 51, 14
Offset: 1

Views

Author

Antti Karttunen, Jun 13 2018

Keywords

Links

Crossrefs

Cf. A078898, A078899, A305798, A305799.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
v078899 = ordinal_transform(vector(up_to,n,A006530(n)));
A078899(n) = v078899[n];
A305797(n) = sumdiv(n,d,A078898(d)*A078899(n/d));

Formula

a(n) = Sum_{d|n} A078898(d)*A078899(n/d).

A305803 Dirichlet inverse of A078898.

Original entry on oeis.org

1, -1, -1, -1, -1, -1, -1, -1, -1, -3, -1, 1, -1, -5, -1, -1, -1, -2, -1, 1, -2, -9, -1, 3, -1, -11, -2, 1, -1, 1, -1, -1, -4, -15, -1, 5, -1, -17, -5, 5, -1, 1, -1, 1, -1, -21, -1, 5, -1, -14, -7, 1, -1, -4, -2, 7, -8, -27, -1, 17, -1, -29, -2, -1, -3, 1, -1, 1, -10, -11, -1, 8, -1, -35, -6, 1, -1, 1, -1, 9, -5, -39, -1, 26, -4, -41, -13, 11, -1, 6, -2, 1
Offset: 1

Views

Author

Antti Karttunen, Jun 13 2018

Keywords

Links

Crossrefs

Cf. A078898, A305798, A305804.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A305803(n) = if(1==n,1,-sumdiv(n,d,if(dA078898(n/d)*A305803(d),0)));

Formula

a(1) = 1; for n > 1, a(n) = -Sum_{d|n, dA078898(n/d)*a(d).

A305804 Möbius-transform of A078898.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 2, 1, 4, 0, 2, 0, 6, 2, 4, 0, 5, 0, 4, 3, 10, 0, 4, 1, 12, 3, 6, 0, 6, 0, 8, 5, 16, 2, 6, 0, 18, 6, 8, 0, 9, 0, 10, 4, 22, 0, 8, 1, 19, 8, 12, 0, 15, 3, 12, 9, 28, 0, 8, 0, 30, 6, 16, 4, 15, 0, 16, 11, 22, 0, 12, 0, 36, 9, 18, 2, 18, 0, 16, 9, 40, 0, 12, 5, 42, 14, 20, 0, 20, 3, 22, 15, 46, 6, 16, 0, 41, 10, 20, 0, 24, 0, 24, 10
Offset: 1

Views

Author

Antti Karttunen, Jun 13 2018

Keywords

Links

Crossrefs

Cf. A008683, A078898, A305798, A305803.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A305804(n) = sumdiv(n,d,moebius(n/d)*A078898(d));

Formula

a(n) = Sum_{d|n} A008683(n/d)*A078898(d).

A319689 Number of distinct values of A078898 that occur when map x -> A064989(x) is iterated, starting from x = n, until an even number is reached; a(1) = 0 by convention.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2
Offset: 1

Views

Author

Antti Karttunen, Nov 20 2018

Keywords

Comments

Number of distinct values that A078898 obtains when applied to n and all the terms above it in that column where it is located in array A246278.

Links

Crossrefs

Cf. A064989, A078898.
Cf. also A246278, A319679.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A319689(n) = if(1==n,0,my(m=Map(),s,k=0); while(1,if(!mapisdefined(m,s=A078898(n)), mapput(m,s,s); k++); if(!(n%2), return(k)); n = A064989(n)));

A286457 Compound filter: a(n) = P(A078898(n), A246277(n)), with a(1) = 0, where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 1, 1, 5, 1, 13, 1, 25, 5, 41, 1, 61, 1, 85, 13, 113, 1, 145, 1, 181, 32, 221, 1, 265, 5, 313, 33, 365, 1, 421, 1, 481, 72, 545, 13, 613, 1, 685, 143, 761, 1, 841, 1, 925, 86, 1013, 1, 1105, 5, 1201, 219, 1301, 1, 1405, 32, 1513, 335, 1625, 1, 1741, 1, 1861, 201, 1985, 60, 2113, 1, 2245, 447, 2381, 1, 2521, 1, 2665, 223, 2813, 13, 2965, 1, 3121, 224, 3281
Offset: 1

Views

Author

Antti Karttunen, May 14 2017

Keywords

Links

Crossrefs

Cf. A000027, A078898, A246277, A280492.

Programs

Formula

a(1) = 0 and for n > 1, a(n) = (1/2)*(2 + ((A078898(n)+A246277(n))^2) - A078898(n) - 3*A246277(n)).
Previous Showing 21-30 of 94 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.