T. D. Noe - cp's OEIS frontend

A365144 Numbers having each digit once and whose 4th power has each digit four times.

5702631489, 7264103985, 7602314895, 7824061395, 8105793624, 8174035962, 8304269175, 8904623175, 8923670541, 9451360827, 9785261403, 9804753612, 9846032571

Offset: 1

T. D. Noe, Nov 09 2011

Currently same terms as A114260, but that sequence has more terms to follow. - Ray Chandler, Aug 23 2023

			5702631489 is a term since its 4th power 1057550783692741389295697108242363408641 contains four 5's, four 7's, four 0's and so on.

Cf. A050278 (pandigital numbers), A199630, A199631, A199633. Subsequence of A114260.

t = Select[Permutations[Range[0, 9]], #[[1]] > 0 &]; t2 = Select[t, Union[DigitCount[FromDigits[#]^4]] == {4} &]; FromDigits /@ t2 (* T. D. Noe, Nov 08 2011 *)

A241202 Beginning of a polynomial relation of degree n in n+2 terms in the first half of Pascal's triangle. See A241201.

Original entry on oeis.org

1, 2, 26, 9, 149, 489

Offset: 1

T. D. Noe, Apr 21 2014

Is this sequence finite?

Cf. A008865 (binomial(n,k) has 3 consecutive terms in a linear relation).
Cf. A062730 (3 terms in arithmetic progression in Pascal's triangle).
Cf. A241199, A241200 (similar, but quadratic).

t = Table[k = 1; While[b = Binomial[k, Range[0, k/2]]; d = Differences[b, n + 1]; ! MemberQ[d, 0], k++]; {k, Position[d, 0, 1, 1][[1, 1]] - 1}, {n, 6}]; Transpose[t][[2]]

A241201 a(n) is the least r such that there are n+2 consecutive increasing terms in the r-th row of Pascal's triangle (binomial(r,*)) which satisfy a polynomial of degree n.

Original entry on oeis.org

7, 14, 62, 31, 339, 1022

Offset: 1

T. D. Noe, Apr 21 2014

Old definition: "Numbers k such that n+2 consecutive terms of binomial(n,k) satisfy a polynomial relation of degree n for some k in the range 0 <= k <= n/2.".

Is this sequence finite?

			a(1) = 7 because the 3 terms 7, 21, 35 are linear.

Cf. A008865 (binomial(n,k) has 3 consecutive terms in a linear relation).
Cf. A062730 (3 terms in arithmetic progression in Pascal's triangle).
Cf. A241199, A241200 (similar, but quadratic).
Cf. A241202 (position of the first of terms).

t = Table[k = 1; While[b = Binomial[k, Range[0, k/2]]; d = Differences[b, n + 1]; ! MemberQ[d, 0], k++]; {k, Position[d, 0, 1, 1][[1, 1]] - 1}, {n, 6}]; Transpose[t][[1]]

Definition clarified by Don Reble, Dec 14 2020

A241200 For the n in A241199, the index of the first of 4 terms in binomial(n,k) that satisfy a quadratic relation.

Original entry on oeis.org

2, 4, 9, 12, 19, 23, 32, 37, 48, 54, 67, 74, 89, 97, 114, 123, 142, 152, 173, 184, 207, 219, 244, 257, 284, 298, 327, 342, 373, 389, 422, 439, 474, 492, 529, 548, 587, 607, 648, 669, 712, 734, 779, 802, 849, 873, 922, 947, 998, 1024, 1077, 1104, 1159, 1187

Offset: 1

T. D. Noe, Apr 17 2014

This value of k appears to approach n/2 as n grows larger.

			Binomial(14,k) = (1, 14, 91, 364, 1001, 2002, 3003, 3432) for k = 0..7. The 4 quadratic terms begin at k = 2.

Colin Barker, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A008865 (binomial(n,k) has 3 consecutive terms in a linear relation).
Cf. A062730 (3 terms in arithmetic progression in Pascal's triangle).
Cf. A241199 (the values of n).

t = {}; Do[b = Binomial[n, Range[0, n/2]]; d = Differences[b, 3]; If[MemberQ[d, 0], AppendTo[t, Position[d, 0, 1, 1][[1, 1]] - 1]], {n, 3000}]; t
LinearRecurrence[{1,2,-2,-1,1},{2,4,9,12,19},60] (* Harvey P. Dale, Dec 18 2022 *)

Vec(x*(x^2-2)*(x^2+x+1)/((x-1)^3*(x+1)^2) + O(x^100)) \\ Colin Barker, Apr 29 2015

a(n) = (-11-5*(-1)^n-2*(-15+(-1)^n)*n+6*n^2)/16. G.f.: x*(x^2-2)*(x^2+x+1) / ((x-1)^3*(x+1)^2). - Colin Barker, Apr 18 2014 and Apr 29 2015

The terms appear to satisfy a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), with initial terms 2, 4, 9, 12, 19. - T. D. Noe, Apr 18 2014

A241199 Numbers n such that 4 consecutive terms of binomial(n,k) satisfy a quadratic relation for 0 <= k <= n/2.

Original entry on oeis.org

14, 19, 31, 38, 54, 63, 83, 94, 118, 131, 159, 174, 206, 223, 259, 278, 318, 339, 383, 406, 454, 479, 531, 558, 614, 643, 703, 734, 798, 831, 899, 934, 1006, 1043, 1119, 1158, 1238, 1279, 1363, 1406, 1494, 1539, 1631, 1678, 1774, 1823, 1923, 1974, 2078, 2131

Offset: 1

T. D. Noe, Apr 17 2014

From Robert Israel, Apr 28 2015: (Start)
Numbers n >= 14 such that 3*n + 7 is a square.
This is because
C(n,i+3) - 3*C(n,i+2) + 3*C(n,i+1) - C(n,i) = n!/((n-i)!*(i+3)!) * g(n,i)
where g(n,i) = (n-3-2*i) * ((n-3-2*i)^2 - 3*n - 7). (End)

			Binomial(14,k) = (1, 14, 91, 364, 1001, 2002, 3003, 3432) for k = 0..7. The 4 terms beginning with 91 equal 182 - 273*x + 182*x^2 for x = 1..4.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Sequence A241200 gives the position of the first of the 4 terms. Sequence A008865 gives the terms greater than 2 for which 3 consecutive terms satisfy a linear relation.
A014206 is a related sequence. - Avi Friedlich, Apr 28 2015
Cf. A062730 (3 terms in arithmetic progression in Pascal's triangle).

map(k -> (3*k^2+8*k+3,3*k^2+10*k+6),[$1..100]); # Robert Israel, Apr 28 2015

Select[Range[2500], MemberQ[Differences[Binomial[#, Range[0, #/2]], 3], 0] &]
LinearRecurrence[{1,2,-2,-1,1},{14,19,31,38,54},50] (* Harvey P. Dale, Oct 29 2017 *)

Vec(-x*(6*x^4-3*x^3-16*x^2+5*x+14)/((x-1)^3*(x+1)^2) + O(x^100)) \\ Colin Barker, Apr 29 2015

a(n)=(6*n^2+42*n+55-(-1)^n*(2*n+7))/8 \\ Charles R Greathouse IV, Apr 15 2016

a(n) = (55-7*(-1)^n-2*(-21+(-1)^n)*n+6*n^2)/8. G.f.: -x*(6*x^4-3*x^3-16*x^2+5*x+14) / ((x-1)^3*(x+1)^2). - Colin Barker, Apr 18 2014 and Apr 29 2015
The terms appear to satisfy a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), with initial terms 14, 19, 31, 38, 54. - T. D. Noe, Apr 18 2014
Numbers are of the form A200182(3n+1) and A200182(3n-1). - Avi Friedlich, Apr 25 2015
a(2*k-1) = 3*k^2 + 8*k + 3, a(2*k) = 3*k^2 + 10*k + 6. - Robert Israel, Apr 28 2015

A241198 Denominator of new minima of phi(p-1)/(p-1), where phi is Euler's totient function and p = prime(n).

Original entry on oeis.org

1, 2, 3, 15, 35, 77, 1463, 1309, 1001, 4147, 2093, 19019, 17017, 39767, 35581, 323323, 10023013, 1339481, 676039, 20957209, 2800733, 86822723

Offset: 1

T. D. Noe, Apr 17 2014

The values of p are in A241196. The numerator is in A241197.

Cf. A008330 (phi(prime(n)-1)), A073918, A241194, A241195.

tMin = {{2, 1}}; Do[p = Prime[n]; tn = EulerPhi[p - 1]/(p - 1); If[tn < tMin[[-1, -1]], AppendTo[tMin, {p, tn}]], {n, 10^7}]; Denominator[Transpose[tMin][[2]]]

a(20)-a(22) from Giovanni Resta, Apr 14 2016

A241197 Numerator of new minima of phi(p-1)/(p-1), where phi is Euler's totient function and p = prime(n).

Original entry on oeis.org

1, 1, 1, 4, 8, 16, 288, 256, 192, 768, 384, 3456, 3072, 6912, 6144, 55296, 1658880, 221184, 110592, 3317760, 442368, 13271040

Offset: 1

T. D. Noe, Apr 17 2014

The values of p are in A241196. The denominator is in A241198.

			In decimal, the minima are about 1, 0.5, 0.333333, 0.266667, 0.228571, 0.207792, 0.196856, 0.195569, 0.191808, 0.185194, 0.183469, 0.181713, 0.180525, 0.173812, 0.172676, 0.171024, 0.165507, 0.165127, 0.163588.

Cf. A008330 (phi(prime(n)-1)), A073918, A241194, A241195.

tMin = {{2, 1}}; Do[p = Prime[n]; tn = EulerPhi[p - 1]/(p - 1); If[tn < tMin[[-1, -1]], AppendTo[tMin, {p, tn}]], {n, 10^7}]; Numerator[Transpose[tMin][[2]]]

a(20)-a(22) from Giovanni Resta, Apr 14 2016

A241196 Primes p at which phi(p-1)/(p-1) reaches a new minimum, where phi is Euler's totient function.

Original entry on oeis.org

2, 3, 7, 31, 211, 2311, 43891, 78541, 120121, 870871, 1381381, 2282281, 4084081, 13123111, 82192111, 106696591, 300690391, 562582021, 892371481, 6915878971, 71166625531, 200560490131

Offset: 1

T. D. Noe, Apr 17 2014

For these p, the numerator and denominator of phi(p-1)/(p-1) are listed in A241197 and A241198. This sequence appears to be related to A073918, the smallest prime which is 1 more than a product of n distinct primes.
By Dirichlet's theorem on primes in arithmetic progressions, for any n there is a prime p such that p-1 is divisible by the primorial A002110(n).  Then phi(p-1)/(p-1) <= Product_{i=1..n} (1 - 1/prime(i)).  Since Sum_{i >= 1} prime(i) diverges, that goes to 0 as n -> infinity.  Thus there are primes with phi(p-1)/(p-1) arbitrarily close to 0. - Robert Israel, Jan 18 2016
5*10^12 < a(23) <= 12234189897931. - Giovanni Resta, Apr 14 2016

R. K. Guy, Unsolved Problems in Number Theory, A2.

Tamiru Jarso, Tim Trudgian, Quadratic residues that are not primitive roots, arXiv:1710.04320 [math.NT], 2017.
Eric Weisstein's World of Mathematics, Euclid Number

Cf. A002110, A008330 (phi(prime(n)-1)), A073918, A241194, A241195.

m:= infinity:
p:= 1:
count:= 0:
while count < 10 do
  p:= nextprime(p);
  r:= numtheory:-phi(p-1)/(p-1);
  if r < m then
     count:= count+1;
     A[count]:= p;
     m:= r;
  fi
od:
seq(A[i],i=1..count); # Robert Israel, Jan 18 2016

tMin = {{2, 1}}; Do[p = Prime[n]; tn = EulerPhi[p - 1]/(p - 1); If[tn < tMin[[-1, -1]], AppendTo[tMin, {p, tn}]], {n, 10^7}]; Transpose[tMin][[1]]

a(20) from Dimitri Papadopoulos, Jan 11 2016

a(21)-a(22) from Giovanni Resta, Apr 14 2016

A241195 Denominator of phi(prime(n)-1)/(prime(n)-1), where phi is Euler's totient function and prime(n) is the n-th prime.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 2, 3, 11, 7, 15, 3, 5, 7, 23, 13, 29, 15, 33, 35, 3, 13, 41, 11, 3, 5, 51, 53, 3, 7, 7, 65, 17, 69, 37, 15, 13, 3, 83, 43, 89, 15, 95, 3, 7, 33, 35, 37, 113, 19, 29, 119, 15, 5, 2, 131, 67, 15, 69, 35, 141, 73, 51, 31, 13, 79, 33, 7, 173, 87, 11

Offset: 1

T. D. Noe, Apr 17 2014

The numerators are in A241194.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000010 (phi), A241194 (numerators).

Cf. A109395, A006093.

[Denominator(EulerPhi(NthPrime(n)-1)/(NthPrime(n)-1)): n in [1..80]]; // Vincenzo Librandi, Apr 06 2015

with(numtheory): A241195:=n->denom(phi(ithprime(n)-1) / (ithprime(n)-1)): seq(A241195(n), n=1..100); # Wesley Ivan Hurt, Apr 06 2015

Denominator[Table[EulerPhi[p - 1]/(p - 1), {p, Prime[Range[100]]}]]

lista(nn) = forprime(p=2, nn, print1(denominator(eulerphi(p-1)/(p-1)), ", ")); \\ Michel Marcus, Jan 03 2015

a(n) = A109395(A006093(n)). - Ridouane Oudra, Mar 24 2025

A241194 Numerator of phi(p-1)/(p-1), where phi is Euler's totient function and p = prime(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 4, 1, 2, 2, 11, 6, 14, 4, 10, 12, 1, 4, 20, 5, 1, 2, 16, 26, 1, 3, 2, 24, 8, 22, 18, 4, 4, 1, 41, 21, 44, 4, 36, 1, 3, 10, 8, 12, 56, 6, 14, 48, 4, 2, 1, 65, 33, 4, 22, 12, 46, 36, 16, 12, 4, 39, 8, 2, 86, 28, 5, 89, 20, 10, 2, 95

Offset: 1

T. D. Noe, Apr 17 2014

The denominators are in A241195. The new minima of phi(p-1)/(p-1) occur at primes listed in A241196. The numerator and denominator of those terms are in A241197 and A241198.

For primes p>2, the fraction phi(p - 1)/(p - 1) has the maximum value = 1/2 if and only if p is in A019434. - Geoffrey Critzer, Dec 30 2014

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 117.

T. D. Noe, Table of n, a(n) for n = 1..10000
P. Erdős, On the density of some sequences of numbers, III., J. London Math. Soc. 13 (1938), pp. 119-127.
Imre Kátai, On distribution of arithmetical functions on the set prime plus one, Compositio Math. 19 (1968), pp. 278-289.
I. J. Schoenberg, Über die asymptotische Verteilung reeller Zahlen mod 1, Mathematische Zeitschrift 28:1 (1928), pp. 171-199.

Cf. A000010, A005596, A008330 (phi(prime(n)-1)), A065485, A241195, A241196, A241197, A241198.

Cf. A076512, A006093.

[Numerator(EulerPhi(NthPrime(n)-1)/(NthPrime(n)-1)): n in [1..80]]; // Vincenzo Librandi, Apr 06 2015

seq(numer(numtheory:-phi(ithprime(i)-1)/(ithprime(i)-1)), i=1..100); # Robert Israel, Jan 11 2015

Numerator[Table[EulerPhi[p - 1]/(p - 1), {p, Prime[Range[100]]}]]

lista(nn) = forprime(p=2, nn, print1(numerator(eulerphi(p-1)/(p-1)), ", ")); \\ Michel Marcus, Jan 03 2015

From Amiram Eldar, Jul 31 2020: (Start)
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{n=1..m} a(n)/A241195(n) = 0.373955... (Artin's constant, A005596).
Asymptotic mean of inverse ratio: lim_{m->oo} (1/m) * Sum_{n=1..m} A241195(n)/a(n) = 2.826419... (Murata's constant, A065485). (End)
a(n) = A076512(A006093(n)). - Ridouane Oudra, Mar 24 2025

cp's OEIS Frontend

User: T. D. Noe

A365144 Numbers having each digit once and whose 4th power has each digit four times.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A241202 Beginning of a polynomial relation of degree n in n+2 terms in the first half of Pascal's triangle. See A241201.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A241201 a(n) is the least r such that there are n+2 consecutive increasing terms in the r-th row of Pascal's triangle (binomial(r,*)) which satisfy a polynomial of degree n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A241200 For the n in A241199, the index of the first of 4 terms in binomial(n,k) that satisfy a quadratic relation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A241199 Numbers n such that 4 consecutive terms of binomial(n,k) satisfy a quadratic relation for 0 <= k <= n/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A241198 Denominator of new minima of phi(p-1)/(p-1), where phi is Euler's totient function and p = prime(n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A241197 Numerator of new minima of phi(p-1)/(p-1), where phi is Euler's totient function and p = prime(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A241196 Primes p at which phi(p-1)/(p-1) reaches a new minimum, where phi is Euler's totient function.

Views

Author

Keywords

Comments

References