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.

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A137795 Smallest positive m such that m*n is free of prime gaps in canonical factorization.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 1, 1, 1, 3, 5, 105, 1, 1, 1, 1155, 1, 15, 1, 1, 1, 1, 35, 15015, 1, 1, 1, 255255, 385, 3, 1, 5, 1, 105, 1, 4849845, 1, 1, 1, 3, 5005, 1155, 1, 1, 7, 15, 85085, 111546435, 1, 1, 1, 3234846615, 5, 1, 77, 35, 1, 15015, 1616615, 3, 1, 1
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 11 2008

Keywords

Examples

			n=42: A073490(42) = A073490([2*3]*[7]) = 1,
the gap is filled by a(42) = 5: A073490(42*5) = 0.

Links

Crossrefs

Cf. A020639, A034386, A073490, A073491, A073492, A083720, A137794.

Programs

  • PARI
    A137795(n) = if(1==n,1, my(f = factor(n), p = f[1, 1], gpf = f[#f~, 1], m = 1); while(pAntti Karttunen, Sep 06 2018

Formula

A073490(n*a(n)) = 0; A137794(n*a(n)) = 1.
For m < a(n), A073490(n*m) > 0 and A137794(n*m) = 0.
a(A073491(n)) = 1; a(A073492(n)) > 1.
a(n) = A083720(n) / A034386(A020639(n)-1). - Peter Munn, Feb 24 2024

A137722 Number of numbers not greater than n with exactly one prime gap in their factorization.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 9, 9, 9, 9, 10, 11, 12, 12, 13, 13, 14, 14, 15, 15, 15, 15, 16, 17, 18, 18, 18, 19, 20, 21, 22, 22, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 29, 29, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 09 2008

Keywords

Comments

a(n) > a(n-1) iff A073490(n) = 1;
A137721(n) > a(n) for n < 134;
A137721(n) < a(n) for n > 140.

Links

Crossrefs

Cf. A073493.

A137723 First occurrence of a set of n consecutive numbers having at least one prime gap in their factorization: a(n) = smallest number of this set.

Original entry on oeis.org

10, 33, 20, 55, 84, 114, 390, 513, 182, 200, 468, 2941, 774, 65522, 1832, 1261, 1130, 1332, 1638, 524289, 1952, 4298, 4524, 69960, 5120, 16385, 2972, 4832, 5352, 10801, 5592
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 09 2008

Keywords

Comments

A073490(a(n)+k)>0 for 0<=kA073490(a(n)-1)=A073490(a(n)+n)=0.
Continuation after the missing a(14): 1832, 1261, 1130, 1332, 1638, missing, 1952,4298, 4524, missing, 5120, 16385, 2972, 4832, 5352, 10801, 5592, missing, 8468, missing, 9552, missing, 39462, missing, 20810, missing, 38502, missing, 15684, ...
a(32) > 10^11. - Lucas A. Brown, Oct 07 2024

Examples

			a(5) = 84: #{84, 85, 86, 87, 88} = 5,
84=[7]*[3*2^2], 84+1=19*5, 84+2=43*2, 84+3=29*3, 84+4=11*2^3.

Links

Crossrefs

Cf. A073492.

Extensions

Discovered a(14) and some more terms from Sean A. Irvine, Sep 27 2009

A297173 Smallest difference between indices of prime divisors of n, or 0 if n is a prime power.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 3, 1, 0, 0, 1, 0, 2, 2, 4, 0, 1, 0, 5, 0, 3, 0, 1, 0, 0, 3, 6, 1, 1, 0, 7, 4, 2, 0, 1, 0, 4, 1, 8, 0, 1, 0, 2, 5, 5, 0, 1, 2, 3, 6, 9, 0, 1, 0, 10, 2, 0, 3, 1, 0, 6, 7, 1, 0, 1, 0, 11, 1, 7, 1, 1, 0, 2, 0, 12, 0, 1, 4, 13, 8, 4, 0, 1, 2, 8, 9, 14, 5, 1, 0, 3, 3, 2, 0, 1, 0, 5, 1
Offset: 1

Views

Author

Antti Karttunen, Mar 03 2018

Keywords

Examples

			For n = 130 = 2*5*13 = prime(1)*prime(3)*prime(6), the smallest difference between indices is 3-1 = 2, thus a(130) = 2.

Links

Crossrefs

Cf. A000720, A073490, A073491.
Cf. also A100573, A243055, A242411, A261079, A286470, A286471.

Programs

  • PARI
    A297173(n) = if(omega(n)<=1,0,my(ps=factor(n)[,1]); vecmin(vector((#ps)-1,i,primepi(ps[i+1])-primepi(ps[i]))));

Formula

a(A073491(n)) <= 1.

A072941 Least multiple of n having no prime gaps.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 30, 11, 12, 13, 210, 15, 16, 17, 18, 19, 60, 105, 2310, 23, 24, 25, 30030, 27, 420, 29, 30, 31, 32, 1155, 510510, 35, 36, 37, 9699690, 15015, 120, 41, 210, 43, 4620, 45, 223092870, 47, 48, 49, 150, 255255, 60060, 53, 54, 385
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 12 2002

Keywords

Comments

a(n) = smallest m such that m is a multiple of n and in the prime factorization of m every prime between the smallest prime factor of n and the largest appears at least once.
A073490(a(n))=0; a(n)=n iff A073490(A007947(n))=0; A006530(a(n)) = A006530(n); A020639(a(n)) = A020639(n); A001221(n) <= A001221(a(n)); A001221(a(n))=A049084(A006530(n))-A049084(A020639(n))+1; A001222(n) <= A001222(a(n)); A001222(a(n)) + A001221(n) = A001221(a(n)) + A001222(n).

Examples

			a(99)=a(3*3*11)=3*3*5*7*11=3465.

Links

Crossrefs

Cf. A073490, A074044, A067255.

Programs

  • Haskell
    a072941 n = product $ zipWith (^) ps $ map (max 1) es where
            (ps, es) = unzip $ dropWhile ((== 0) . snd) $
                       zip a000040_list $ a067255_row n
-- Reinhard Zumkeller, Dec 21 2013

Formula

a(n)=A002110(A049084(A006530(n)))*A003557(n)/A002110(A049084(A020639(n))-1).

Extensions

Example corrected by Nadia Heninger, Jul 06 2005

A074044 Largest divisor of n having no prime gaps.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 5, 11, 12, 13, 7, 15, 16, 17, 18, 19, 5, 7, 11, 23, 24, 25, 13, 27, 7, 29, 30, 31, 32, 11, 17, 35, 36, 37, 19, 13, 8, 41, 7, 43, 11, 45, 23, 47, 48, 49, 25, 17, 13, 53, 54, 11, 8, 19, 29, 59, 60, 61, 31, 9, 64, 13, 11, 67, 17, 23, 35, 71, 72
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 13 2002

Keywords

Links

Crossrefs

Cf. A072941, A073490, A073491.

Programs

  • Mathematica
    nogapsQ[n_] := Module[{p = FactorInteger[n][[;;, 1]]}, PrimePi[p[[-1]]] == PrimePi[p[[1]]] + Length[p] - 1]; a[n_] := SelectFirst[Reverse[Divisors[n]], nogapsQ]; Array[a, 100] (* Amiram Eldar, Apr 06 2025 *)
Previous Showing 11-16 of 16 results.

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