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 1-8 of 8 results.

A086561 Duplicate of A083790.

Original entry on oeis.org

1, 1, 4, 6, 8, 23, 24, 1804, 1968, 8375, 53750, 243864, 639324, 41381620, 102621343
Offset: 1

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Author

Keywords

A034173 a(n) is minimal such that prime factorizations of a(n), ..., a(n)+n-1 have same exponents.

Original entry on oeis.org

1, 2, 33, 19940, 204323, 380480345, 440738966073
Offset: 1

Views

Author

Dean Hickerson, Oct 01 1998

Keywords

Comments

a(8) > 10^13. - Donovan Johnson, Oct 20 2009
Don Reble has shown that a(8) < 1.9*10^42, cf. link.
From David Wasserman, Jan 05 2019: (Start)
a(8) <= 108111092880293127811946663766147737122,
a(9) <= 6850672946809600696044301071559918192380244,
a(10) <= 96037988156124494415303285590850571857698741869620,
a(11) <= 9044737840075556371215937303485030235666252755947862558252154847122. (End)

Examples

			a(4) = 19940 because 19940, ..., 19943 all have the form p^2 q r.

Links

Crossrefs

Cf. A034174.
Cf. A006558, A083790.
Cf. A052213, A052214, A175590, A218448. This sequence is the first column of A083785 and first row of A113456. The latter generalizes to arithmetic progressions with step d>=1. - M. F. Hasler, Oct 28 2012

Programs

  • PARI
    A034173(n)={my(f);for(k=1,oo,f=0;for(i=1,n, f==(f=vecsort(factor(k+n-i)[,2])) || i==1 || [k+=n-i; next(2)]);return(k))} \\ M. F. Hasler, Oct 23 2012

Formula

a(n) = A034174(n) - n + 1. - Max Alekseyev, Nov 10 2009
a(n) = A083785(n,1) = A113456(1,n); a(2) = A052213(1), a(3) = A052214(1), a(4) = A175590(1), a(5) = A218448(1), a(6) = A218448(62) = A218448(63)-1. - M. F. Hasler, Oct 28 2012

Extensions

a(7) from Donovan Johnson, Oct 20 2009
Don Reble link repaired by N. J. A. Sloane, Oct 24 2024

A083788 Triangle read by rows, in which n-th row contains smallest set of n consecutive numbers with distinct prime signatures.

Original entry on oeis.org

1, 1, 2, 4, 5, 6, 6, 7, 8, 9, 8, 9, 10, 11, 12, 23, 24, 25, 26, 27, 28, 24, 25, 26, 27, 28, 29, 30, 1804, 1805, 1806, 1807, 1808, 1809, 1810, 1811, 1968, 1969, 1970, 1971, 1972, 1973, 1974, 1975, 1976, 8375, 8376, 8377, 8378, 8379, 8380, 8381, 8382, 8383, 8384
Offset: 1

Views

Author

Amarnath Murthy, May 07 2003

Keywords

Comments

Question: Given n with a prime signature say p^a*q^b*r^c... find a tight upper limit m such that there exists a number k, n < k < m with the same prime signature.

Examples

			1;
1,2;
4,5,6;
6,7,8,9;
8,9,10,11,12;
23,24,25,26,27,28;
24,25,26,27,28,29,30;
1804,1805,1806,1807,1808,1809,1810,1811;
1968,1969,1970,1971,1972,1973,1974,1975,1976;
8375,8376,8377,8378,8379,8380,8381,8382,8383,8384;
53750,53751,53752,53753,53754,53755,53756,53757,53758,53759,53760;
...
Row 4 contains 6,7,8 and 9 with prime signatures p*q,p,p^3,p^2 respectively.

Links

Crossrefs

Cf. A083789, A083790, A083791.

Extensions

Corrected and extended by Ray Chandler, Aug 17 2003

A218448 First of a run of 5 consecutive numbers with same prime signature.

Original entry on oeis.org

204323, 3252571, 5205074, 7201674, 20182921, 28387953, 36193650, 43216722, 51049537, 56155074, 57070850, 61961315, 62167075, 65425473, 76647074, 82507473, 92658049, 95943321, 100498849, 107236449, 109751473, 110899321, 112198075, 112477849, 116736323
Offset: 1

Views

Author

M. F. Hasler, Oct 28 2012

Keywords

Comments

A number n is in this sequence iff n and n+1 is in A175590; also: iff n and n+2 are in A052214 (in which case n+1 is in A052214, too); and also: iff {n,n+1,n+2,n+3} are in A052213.
A034173(6) = A218448(62) = A218448(63)-1 is the least term n such that n+1 is also in the sequence.

Links

Crossrefs

Cf. A034173, A034174, A006558, A083790, A052213, A052214, A175590.

Programs

  • PARI
    is_A218448(n)={my(f);!for(i=0,4,f!=(f=vecsort(factor(n+i)[,2])) & i & return)}
  • PARI
    f(k)=vecsort(factor(k)[,2]~,,4)
t=f(n=2);while(n<1e8, for(i=n+1, n+4, tt=f(i); if(tt!=t, n=i; t=tt; next(2))); print1(n", "); n++) \\ Charles R Greathouse IV, Oct 28 2012

Extensions

a(6)-a(8) from Charles R Greathouse IV, Oct 28 2012
a(9)-a(25) from Donovan Johnson, Oct 28 2012
Values up to a(140) computed using b175590.txt from Charles R Greathouse IV - M. F. Hasler, Oct 28 2012

A083789 Diagonal of A083788.

Original entry on oeis.org

1, 2, 6, 9, 12, 28, 30, 1811, 1976, 8384, 53760, 243875, 639336, 41381633, 102621357, 7802708751, 7802708752, 178874468632, 1133934949386
Offset: 1

Views

Author

Amarnath Murthy, May 07 2003

Keywords

Comments

a(16) > 750000000. - Ray Chandler, Aug 21 2003
a(18) > 10^11. [Donovan Johnson, Oct 24 2009]
a(20) > 10^13. - Giovanni Resta, Jul 13 2015

Crossrefs

Cf. A083788, A083790, A083791.

Extensions

Corrected and extended by Ray Chandler, Aug 17 2003
More terms from Ray Chandler, Aug 21 2003
a(16)-a(17) from Donovan Johnson, Oct 24 2009
a(18)-a(19) from Giovanni Resta, Jul 13 2015

A083791 Prime signatures pertaining to A083788.

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 6, 2, 8, 4, 8, 4, 6, 2, 12, 2, 24, 4, 6, 8, 12, 24, 4, 6, 8, 12, 2, 30, 60, 12, 210, 6, 48, 24, 30, 2, 240, 6, 30, 24, 60, 2, 210, 12, 120, 24, 120, 2, 30, 180, 60, 12, 210, 6, 192, 240, 210, 24, 12, 420, 30, 60, 120, 6, 2, 53760
Offset: 1

Views

Author

Amarnath Murthy, May 07 2003

Keywords

Examples

			a(n) = k where k is the smallest number with the same prime signature as that of A083788(n).
1;
1,2;
4,2,6;
6,2,8,4;
8,4,6,2,12;
2,24,4,6,8,12;
24,4,6,8,12,2,30;
60,12,210,6,48,24,30,2;
240,6,30,24,60,2,210,12,120;
24,120,2,30,180,60,12,210,6,192;
240,210,24,12,420,30,60,120,6,2,53760;

Links

Crossrefs

Cf. A083788, A083789, A083790.

Extensions

Corrected and extended by Ray Chandler, Aug 17 2003

A218455 First of a run of 6 consecutive numbers with same prime signature.

Original entry on oeis.org

380480345, 2713001274, 6282718946, 7209536449, 9809067073, 10684724346, 12008728850, 14824913049, 17231547073, 17552118546, 17659180314, 18036555273, 20473171322, 21507097001, 23676804346, 24742649321, 25401767522, 25694056449, 27656894273, 28259097818
Offset: 1

Views

Author

M. F. Hasler, Oct 29 2012

Keywords

Comments

A number n is in this sequence iff n and n+1 is in A218448; see the comment there for other characterizations in terms of membership in A175590 or A052214 or A052213.

Links

Crossrefs

Cf. A034173, A034174, A006558, A083790, A052213, A052214, A175590, A218448, A218865.

Programs

  • PARI
    is_A218455(n)={my(s(n)=vecsort(factor(n)[,2]),t=s(n));!for(m=n+1,n+5, t!=s(m) & return)}

Extensions

a(2)-a(20) from Donovan Johnson, Oct 29 2012

A124058 Smallest of the first occurrence of n consecutive integers with all different prime signatures.

Original entry on oeis.org

2, 3, 4, 6, 8, 23, 24, 1804, 1968, 8375, 53750, 243864, 639324, 41381620, 102621343, 7802708736, 7802708736, 178874468615, 1133934949368
Offset: 1

Views

Author

Joe McCauley (mccauleyj(AT)insightbb.com), Nov 03 2006

Keywords

Comments

Essentially the same as A083790. - T. D. Noe, Nov 10 2006
a(18) > 10^11. - Donovan Johnson, Oct 24 2009
a(20) > 10^13. - Giovanni Resta, Jul 13 2015

Examples

			a(6) = 23 because it begins the first occurrence of 6 consecutive integers that have all different prime signatures ({1}, {1,3}, {2}, {1,1}, {3}, {1,2} respectively for 23 through 28).

Links

Crossrefs

Cf. A034173, A025487, A083790, A083788.

Extensions

a(12)-a(15) added from A083790, Nov 10 2006
a(16)-a(17) from Donovan Johnson, Oct 24 2009
a(18)-a(19) from Giovanni Resta, Jul 13 2015
Showing 1-8 of 8 results.

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