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 1-10 of 10 results.

A377782 First-differences of A031218(n) = greatest number <= n that is 1 or a prime-power.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 1, 0, 2, 0, 2, 0, 0, 3, 1, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 2, 0, 2, 1, 0, 0, 0, 0, 5, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 3, 0, 0, 3, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 6, 0, 2, 0, 2, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Nov 16 2024

Keywords

Comments

Note 1 is a power of a prime (A000961) but not a prime-power (A246655).

Crossrefs

Positions of 1 are A006549.
Positions of 0 are A080765 = A024619 - 1, complement A181062 = A000961 - 1.
Positions of 2 are A120432 (except initial terms).
Sorted positions of first appearances appear to include A167236 - 1.
Positions of terms > 1 are A373677.
The restriction to primes minus 1 is A377289.
Below, A (B) indicates that A is the first-differences of B:
- This sequence is A377782 (A031218), which has restriction to primes A065514 (A377781).
- The opposite is A377780 (A000015), restriction A377703 (A345531).
- For nonsquarefree we have A378036 (A378033), opposite A378039 (A120327).
- For squarefree we have A378085 (A112925), restriction A378038 (A070321).
A000040 lists the primes, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A361102 lists the non-powers of primes, differences A375708.
A378034 gives differences of A378032 (restriction of A378033).
Prime-powers between primes: A053607, A080101, A366833, A377057, A377286, A377287.
Cf. A053289, A343249, A375706, A377281, A377282.

Programs

  • Mathematica
    Differences[Table[NestWhile[#-1&,n,#>1&&!PrimePowerQ[#]&],{n,100}]]

A096022 Numbers that are congruent to {15, 27, 39, 51} mod 60.

Original entry on oeis.org

15, 27, 39, 51, 75, 87, 99, 111, 135, 147, 159, 171, 195, 207, 219, 231, 255, 267, 279, 291, 315, 327, 339, 351, 375, 387, 399, 411, 435, 447, 459, 471, 495, 507, 519, 531, 555, 567, 579, 591, 615, 627, 639, 651, 675, 687, 699, 711, 735, 747, 759, 771, 795
Offset: 1

Views

Author

Klaus Brockhaus, Jun 15 2004

Keywords

Comments

Numbers n such that (n+j) mod (2+j) = 1 for j from 0 to 2 and (n+3) mod 5 <> 1.
This is one of a family of sequences which are defined (or could be defined) according to the same scheme: Numbers n such that (n+j) mod (2+j) = 1 for j from 0 to k-1 and (n+k) mod (2+k) <> 1. We have A007310 for k = 1, A017629 for k = 2, this one (A096022) for k = 3, A096023 for k = 5, A096024 for k = 6, A096025 for k = 7, A096026 for k = 9, A096027 for k = 11. Remarkably these sequences are empty for k = 4, 8, 10, ... (i.e., if k+1 is a term of A080765).
Numbers n such that n mod 12 = 3 and n mod 60 <> 3.
Subsequence of A017557: 12n+3.

Examples

			51 mod 2 = 52 mod 3 = 53 mod 4 = 1 and 54 mod 5 = 4, hence 51 is in the sequence; 3 mod 2 = 4 mod 3 = 5 mod 4 = 6 mod 5 = 1, hence 3 is not in the sequence.

Links

Crossrefs

Cf. A007310, A017557, A017629, A080765, A096023, A096024, A096025, A096026, A096027.

Programs

  • Magma
    [ n : n in [1..1500] | n mod 60 in [15, 27, 39, 51] ] // Vincenzo Librandi, Mar 24 2011
  • Maple
    A096022:=n->3*(10*n-3-I^(2*n)-(1-I)*I^(-n)-(1+I)*I^n)/2: seq(A096022(n), n=1..80); # Wesley Ivan Hurt, Jun 04 2016
  • Mathematica
    Table[3*(10n-3-I^(2n)-(1-I)*I^(-n)-(1+I)*I^n)/2, {n, 80}] (* Wesley Ivan Hurt, Jun 04 2016 *)
  • PARI
    {k=3;m=800;for(n=1,m,j=0;b=1;while(b&&j

Formula

G.f.: 3*x*(5+4*x+4*x^2+4*x^3+3*x^4) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jun 04 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = 3*(10*n-3-i^(2*n)-(1-i)*i^(-n)-(1+i)*i^n)/2 where i=sqrt(-1). (End)
E.g.f.: 3*(3 + sin(x) - cos(x) + (5*x - 1)*sinh(x) - (2 - 5*x)*cosh(x)). - Ilya Gutkovskiy, Jun 05 2016

Extensions

New definition from Ralf Stephan, Dec 01 2004

A083390 m such that 2m + 1 divides lcm(1,3,5,...,2m - 1).

Original entry on oeis.org

7, 10, 16, 17, 19, 22, 25, 27, 28, 31, 32, 34, 37, 38, 42, 43, 45, 46, 47, 49, 52, 55, 57, 58, 59, 61, 64, 66, 67, 70, 71, 72, 73, 76, 77, 79, 80, 82, 85, 87, 88, 91, 92, 93, 94, 97, 100, 101, 102, 103, 104, 106, 107, 108, 109, 110, 112, 115, 117, 118, 122, 123, 124, 126
Offset: 1

Views

Author

Lekraj Beedassy, Jun 11 2003

Keywords

Comments

Also m for which A025547(m)=A025547(m+1). Query: a(n) seems to be equal to A030343(n+4) - 1. Is this true?
While any odd number>1 can be the leg of a primitive Pythagorean triangle, the m-th odd number 2m+1=A061346 forms leg common to more than one PPT. - Lekraj Beedassy, Jul 12 2006

Examples

			10 is in the sequence because we have 2*10 - 1 = 19 and lcm(1,3,5,...,19)=166966608033225=7950790858725*21 which is divisible by 2*10 + 1 = 21.

Links

Crossrefs

Cf. A080765.

Programs

  • Mathematica
    Select[Range[150],Divisible[LCM@@Range[1,2#-1,2],2#+1]&] (* Harvey P. Dale, Jan 22 2023 *)
  • PARI
    isok(n) = {lc = 1; for (i = 1, 2*n-1, lc = lcm(lc, i);); return (lc % (2*n+1) == 0);} \\ Michel Marcus, Jul 27 2013

Formula

a(n) = (A061346(n)-1)/2. - David Wasserman, Oct 26 2004

Extensions

More terms from David Wasserman, Oct 26 2004

A120109 Row sums of number triangle A120108.

Original entry on oeis.org

1, 3, 10, 21, 106, 107, 750, 1501, 4504, 4505, 49556, 49557, 644242, 644243, 644244, 1288489, 21904314, 21904315, 416181986, 416181987, 416181988, 416181989, 9572185748, 9572185749, 47860928746, 47860928747, 143582786242
Offset: 0

Views

Author

Paul Barry, Jun 09 2006

Keywords

Comments

It appears that the indices k such that a(k) = a(k-1) + 1 are A080765. - Michel Marcus, Mar 04 2019

Links

Crossrefs

Cf. A080765, A120108, A120110.

Programs

  • GAP
    List([0..30],n->Sum([0..n],k->Lcm(List([1..n+1],i->i))/Lcm(List([1..k+1],i->i)))); # Muniru A Asiru, Mar 03 2019
  • Magma
    A120108:= func< n,k | Lcm([1..n+1])/Lcm([1..k+1]) >;
[(&+[A120108(n,k): k in [0..n]]): n in [0..50]]; // G. C. Greubel, May 04 2023
  • Mathematica
    A120108[n_, k_]:= LCM@@Range[n+1]/(LCM@@Range[k+1]);
A120109[n_]:= Sum[A120108[n, k], {k,0,n}];
Table[A120109[n], {n,0,50}] (* G. C. Greubel, May 04 2023 *)
  • PARI
    a(n) = lcm([1..n+1])*sum(k=0, n, 1/lcm([1..k+1])); \\ Michel Marcus, Mar 04 2019
  • SageMath
    def f(n): return lcm(range(1,n+2))
def A120109(n):
    return sum(f(n)//f(k) for k in range(n+1))
[A120109(n) for n in range(51)] # G. C. Greubel, May 04 2023

Formula

a(n) = Sum_{k=0..n} lcm(1,...,n+1)/lcm(1,...,k+1).

A224503 Smallest nontrivial prime power congruent to 1 mod n.

Original entry on oeis.org

3, 4, 5, 11, 7, 8, 9, 19, 11, 23, 13, 27, 29, 16, 17, 103, 19, 191, 41, 43, 23, 47, 25, 101, 27, 109, 29, 59, 31, 32, 97, 67, 103, 71, 37, 149, 191, 79, 41, 83, 43, 173, 89, 181, 47, 283, 49, 197, 101, 103, 53, 107, 109, 331, 113, 229, 59
Offset: 2

Views

Author

R. J. Mathar, Apr 08 2013

Keywords

Links

Crossrefs

Cf. A000961, A024619, A080765.

Programs

  • Maple
    A224503 := proc(n)
    local i;
    for i from 2 do
        if (A000961(i) mod n) = 1 then
             return A000961(i);
        end if;
    end do:
end proc:
seq(A224503(n), n=2..100);
  • Mathematica
    a[n_] := For[k = 2, True, k++, If[PrimePowerQ[k], If[Mod[k, n] == 1, Return[k]]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, May 18 2018 *)

A242298 Once a number in this sequence is divisible by all numbers 1 to m, subsequent terms are constrained to have the same property; choose the smallest permissible number that is greater than the previous term.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 300, 360, 420, 840, 1680, 2520, 5040, 7560, 10080, 12600, 15120, 17640, 20160, 22680, 25200, 27720, 55440, 83160, 110880, 138600, 166320, 194040, 221760, 249480, 277200, 304920, 332640, 360360, 720720, 1441440
Offset: 1

Views

Author

J. Lowell, May 10 2014

Keywords

Comments

All terms from A095848 belong to this sequence.

Examples

			After 6, none of 7,8,9,10 or 11 are in the sequence since they are not divisible by 1,2 and 3 as 6 is. 12 is a term, but is now divisible by 1,2,3 and 4, adding a new constraint on subsequent terms.
After 24, 30 is not in the sequence because 24 is divisible by all numbers from 1 to 4 and 30 is not divisible by 4. But 36, which is divisible by all of 1 through 4, is a term.
As an irregular table, the n-th row consists of all numbers divisible by A051451(n) but not by A051451(n+1). - _Tom Edgar_, May 22 2014

Links

Crossrefs

Cf. A003418, A051451, A080765, A368777.

Programs

  • PARI
    consecd(a) = {d = divisors(a); for (i=2, #d, if (d[i] - d[i-1] > 1, return(i-1));); return(a);}
findnext(a) = {nconsd = consecd(a); na = a + 1; while (consecd(na) < nconsd, na++); na;}
lista(nn) = {a = 1; print1(a, ", "); for (n=1, nn, a = findnext(a); print1(a, ", "););} \\ Michel Marcus, May 11 2014
  • PARI
    first(n) = {
	my(res = vector(n), step = 1, oldm = 1, newm = 1);
	res[1] = 1;
	for(i = 2, n,
		while(res[i-1] % (newm+1) == 0,
			newm++;
		);
		if(newm > oldm,
			step = lcm([step, lcm([oldm..newm])]);
			oldm = newm
		);
		res[i] = res[i-1]+step
	);
	res
} \\ David A. Corneth, Jan 28 2024

Formula

a(1) = 1. If n > 1, a(n) = a(n-1) + A368777(a(n-1)). - Hal M. Switkay, Jan 26 2024

Extensions

More terms from Michel Marcus, May 11 2014

A082093 a(n) is the least number m such that (m+n)!/m! = (m+1)*(m+2)*...*(m+n) divides lcm(1,...,m).

Original entry on oeis.org

5, 13, 19, 32, 73, 89, 140, 199, 294, 468, 1072, 1072, 1072, 2161, 2976, 32805, 32806, 65732, 65732, 262153, 262154, 524457, 524640, 4194464, 4194464, 8388640, 8388640, 33554432, 33554432, 67108992, 67109088, 2147483659, 2147484110, 4294967312, 4294967312, 17179869209, 17179869210
Offset: 1

Views

Author

Labos Elemer, Apr 10 2003

Keywords

Comments

From David A. Corneth, Aug 30 2019: (Start)
As (m+1)*(m+2)*...*(m+n) is the product of n consecutive integers, it's divisible by n! and so a(n) >= 2^A011371(n) = A060818(n).
None of m+1..m+n are prime. (End)

Examples

			a(6)=89: lcm(1,...,89) = 718766754945489455304472257065075294400 is divisible by 625757605200 = 90*91*92*93*94*95 = (89+6)!/89! and the quotient is 1148634469597477063638686172.
For n=1 see A080765(1) = A082093(1).

Links

Crossrefs

Cf. A011371, A060818, A080765.

Programs

  • Mathematica
    k = 1; lc = 1; Do[While[lc = LCM[lc, k]; Mod[lc, (n + k)!/k! ] != 0, k++ ]; Print[{n, k}], {n, 0, 50}] (* Robert G. Wilson v, Apr 12 2006 *)

Extensions

a(16)-a(19) from Robert G. Wilson v, Apr 12 2006
a(20)-a(23) from Vaclav Kotesovec, Aug 30 2019
a(24)-a(37) from David A. Corneth, Aug 30 2019

A118562 Least number k such that binomial(2k,k) is divisible by all squares to n squared but not (n+1) squared, or 0 if impossible.

Original entry on oeis.org

1, 3, 5, 15, 0, 23, 89, 95, 0, 123, 0, 215, 0, 0, 1117, 943, 0, 2003, 0, 0, 0, 3455, 0, 1439, 0, 7846, 0, 7916, 0, 14735, 13103, 0, 0, 0, 0, 23711, 0, 0, 0, 24049, 0, 44857, 0, 0, 0, 44711, 0, 47594, 0, 0, 0, 77021, 0, 0, 0, 0, 0, 195765, 0, 381398, 0, 0, 374435, 0, 0
Offset: 1

Views

Author

Robert G. Wilson v, Nov 23 2005

Keywords

Comments

a(5)=0 because any number squared which would divide binomial(2k,k) would also be divided by 6^2 since 6=2*3.

Examples

			a(3)=5 because binomial(10,5)=252 which is divisible by the squares of 1, 2 & 3 but not 4 squared.
a(70)=385823.

Links

Crossrefs

Cf. A059097, A111869, A080765, A024619.

Programs

  • Mathematica
    f[n_] := Block[{k = 1, b = Binomial[2n, n]}, While[Mod[b, k^2] == 0, k++ ]; k - 1]; t = Table[0, {100}]; Do[ a = f[n]; If[a < 101 &t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 38000}] (* or *)
expoPF[k_, n_] := Module[{s = 0, x = n}, While[x > 0, x = Floor[x/k]; s += x]; s]; expoCF[k_, n_] := Min[expoPF[ #[[1]], n]/#[[2]] & /@ FactorInteger@k]; f[n_] := Module[{k = 2}, While[ expoCF[k, 2n] >= 2(1 + expoCF[k, n]), k++ ]; k-1]; t = Table[0, {100}]; Do[ a = f[n]; If[a < 101 &t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 400000}]; t

Formula

a(n)=0 iff n is a member of A080765: m such that m+1 divides lcm(1..m).
a(n-1)=0 iff n-1 is a member of A024619: Numbers that are not powers of primes.

A249051 The smallest integer > 1 of exactly n consecutive integers divisible respectively by the first n natural numbers (A000027), or 0 if no such number exists.

Original entry on oeis.org

2, 3, 7, 13, 0, 61, 421, 841, 0, 2521, 0, 27721, 0, 0, 360361, 720721, 0, 12252241, 0, 0, 0, 232792561, 0, 5354228881, 0, 26771144401, 0, 80313433201, 0, 2329089562801, 72201776446801, 0, 0, 0, 0, 144403552893601, 0, 0, 0, 5342931457063201, 0
Offset: 1

Views

Author

Robert G. Wilson v, Oct 30 2014

Keywords

Comments

For all n > 1 and a(n) # 0, a(n) == 1 (mod p#), where p# are the primorial numbers (A034386).
When a(n) is not 0, a(n) = A075059(n).
a(n) = 0 when n is a member of A080765.

Examples

			a(3) = 7 because the smallest k such that 1|k, 2|k+1, 3|k+2, and 4 does not divide k+3 is 7.
a(4) = 13 because the smallest k such that 1|k, 2|k+1, 3|k+2, 4|k+3, and 5 does not divide k+4 is 13.

Crossrefs

Cf. A075059, A034386, A080765.

Programs

  • Mathematica
    f[n_] := Block[{lcm = LCM @@ Range@ n}, If[ lcm == LCM @@ Range[n + 1], 0, lcm + 1]]; Array[ f, 42] (* Robert G. Wilson v, Nov 13 2014 *)

Extensions

a(5) corrected (0, not 181) by Jon Perry, Nov 05 2014
Sequence corrected by Robert G. Wilson v, Nov 13 2014

A119944 First differences of A003418(n) = lcm(1..n).

Original entry on oeis.org

0, 1, 4, 6, 48, 0, 360, 420, 1680, 0, 25200, 0, 332640, 0, 0, 360360, 11531520, 0, 220540320, 0, 0, 0, 5121436320, 0, 21416915520, 0, 53542288800, 0, 2248776129600, 0, 69872686884000, 72201776446800, 0, 0, 0
Offset: 0

Views

Author

Wolfdieter Lang, Jul 20 2006

Keywords

Comments

a(n)=0 iff n=0 or n = n(k) = A080765(k), k >= 1.

Programs

  • Mathematica
    Differences[FoldList[LCM,1,Range[40]]] (* Harvey P. Dale, Jan 09 2016 *)

Formula

a(n) = A003418(n+1) - A003418(n), n >= 0. A003418(0):=1.
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