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.

Showing 1-3 of 3 results.

A319693 Filter sequence combining sopfr(d) from all proper divisors d of n, where sopfr(d) is A001414(d) = sum of primes dividing d with repetition.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 40, 41, 42, 43, 2, 44, 2, 45, 46, 47, 48, 49, 2, 50, 51, 52, 2, 53, 2, 54, 55, 56, 57, 58, 2, 59, 60, 61, 2, 62, 63, 64, 65, 66, 2, 67, 68, 69, 70, 71, 72, 73, 2, 74, 75, 76, 2, 77, 2, 78, 79, 80, 2, 73, 2, 81, 82, 83, 2, 84, 85
Offset: 1

Views

Author

Antti Karttunen, Oct 02 2018

Keywords

Comments

Restricted growth sequence transform of A319692.
For all i, j: a(i) = a(j) => A305611(i) = A305611(j).

Examples

			The proper divisors of  96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, while
the proper divisors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54.
It happens that sopfr(8) = sopfr(9), sopfr(16) = sopfr(18), sopfr(24) = sopfr(27), sopfr(32) = sopfr(36) and sopfr(48) = sopfr(54), and the rest of proper divisors (1, 2, 3, 4, 6, 12) are shared by both numbers, from which follows that by taking product of sopfr over proper divisors gives an identical result for both, thus a(96) = a(108). Here sopfr = A001414.

Links

Crossrefs

Cf. A001414, A319692.
Cf. also A319353.
Differs from A305800, A296073 and A317943 for the first time at n=108, as here a(108) = 73.

Programs

  • PARI
    up_to = 65537;
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; };
A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
A319692(n) = { my(m=1); fordiv(n, d, if(dA001414(d)))); (m); };
v319693 = rgs_transform(vector(up_to,n,A319692(n)));
A319693(n) = v319693[n];

A319352 a(n) = Product_{d|n, dA056239(d)), where A056239(d) gives the weight of the partition whose Heinz-number is d.

Original entry on oeis.org

1, 2, 2, 6, 2, 30, 2, 30, 10, 42, 2, 1050, 2, 66, 70, 210, 2, 2310, 2, 2310, 110, 78, 2, 80850, 14, 102, 110, 4290, 2, 210210, 2, 2310, 130, 114, 154, 1651650, 2, 138, 170, 210210, 2, 510510, 2, 6630, 10010, 174, 2, 11561550, 22, 7854, 190, 9690, 2, 510510, 182, 510510, 230, 186, 2, 2555102550, 2, 222, 20570, 30030, 238, 881790, 2
Offset: 1

Views

Author

Antti Karttunen, Sep 17 2018

Keywords

Links

Crossrefs

Cf. A056239, A319353 (rgs-transform).

Programs

  • PARI
    A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A319352(n) = { my(m=1); fordiv(n, d, if(dA056239(d)))); (m); };

Formula

a(n) = Product_{d|n, dA056239(d)).
For all n >= 1:
A001221(a(n)) = A304793(n).
A001222(a(n)) = A032741(n).
1+A056169(a(n)) = A301855(n).

A319357 Filter sequence combining A003415(d) from all proper divisors d of n, where A003415(d) = arithmetic derivative of d.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 8, 2, 9, 4, 4, 2, 10, 3, 4, 11, 12, 2, 13, 2, 14, 4, 4, 4, 15, 2, 4, 4, 16, 2, 17, 2, 18, 19, 4, 2, 20, 3, 21, 4, 22, 2, 23, 4, 24, 4, 4, 2, 25, 2, 4, 26, 27, 4, 28, 2, 29, 4, 30, 2, 31, 2, 4, 32, 33, 4, 34, 2, 35, 36, 4, 2, 37, 4, 4, 4, 38, 2, 39, 4, 40, 4, 4, 4, 41, 2, 42, 43, 44, 2, 45, 2, 46, 47
Offset: 1

Views

Author

Antti Karttunen, Oct 02 2018

Keywords

Comments

Restricted growth sequence transform of A319356.
The only duplicates in range 1..65537 with a(n) > 4 are the following six pairs: a(1445) = a(2783), a(4205) = a(11849), a(5819) = a(8381), a(6727) = a(15523), a(8405) = a(31211) and a(28577) = a(44573). All these have prime signature p^2 * q^1. If all the other duplicates respect the prime signature as well, then also the last implication given below is valid.
For all i, j:
a(i) = a(j) => A000005(i) = A000005(j),
a(i) = a(j) => A319683(i) = A319683(j),
a(i) = a(j) => A319686(i) = A319686(j),
a(i) = a(j) => A101296(i) = A101296(j). [Conjectural, see notes above]

Examples

			Proper divisors of 1445 are [1, 5, 17, 85, 289], while the proper divisors of 2783 are [1, 11, 23, 121, 253]. 1 contributes 0 and primes contribute 1, so only the last two matter in each set. We have A003415(85) = 22 = A003415(121) and A003415(289) = 34 = A003415(253), thus the value of arithmetic derivative coincides for all proper divisors, thus a(1445) = a(2783).

Links

Crossrefs

Cf. A003415, A319356.
Cf. A000041 (positions of 2's), A001248 (positions of 3's), A006881 (positions of 4's),
Cf. also A300245, A300249, A319353, A319693.

Programs

  • PARI
    up_to = 65537;
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; };
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A319356(n) = { my(m=1); fordiv(n, d, if(dA003415(d)))); (m); };
v319357 = rgs_transform(vector(up_to,n,A319356(n)));
A319357(n) = v319357[n];
Showing 1-3 of 3 results.

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