A091465 -id:A091465 - cp's OEIS frontend

A056053 a(n) = smallest odd number 2m+1 such that the partial sum of the odd harmonic series Sum_{j=0..m} 1/(2j+1) is > n.

1, 3, 15, 113, 837, 6183, 45691, 337607, 2494595, 18432707, 136200301, 1006391657, 7436284415, 54947122715, 406007372211, 3000011249847, 22167251422541, 163795064320249, 1210290918990281, 8942907496445513, 66079645178783351, 488266205223462461, 3607826381608149807

Offset: 0

Robert G. Wilson v, Jul 25 2000 and Jan 11 2004

nonn

a(2) = 15 and a(3) = 113 are related to the Borwein integrals. Concretely, a(2) = 15 is the smallest odd m such that the integral Integral_{x=-oo..oo} Product_{1<=k<=m, k odd} (sin(k*x)/(k*x)) dx is slightly less than Pi, and a(3) = 113 is the smallest odd m such that the integral Integral_{x=-oo..oo} cos(x) * Product_{1<=k<=m, k odd} (sin(k*x)/(k*x)) dx is slightly less than Pi/2. See the Wikipedia link and the 3Blue1Brown video link below. - Jianing Song, Dec 10 2022

Calvin C. Clawson, "Mathematical Mysteries, The Beauty and Magic of Numbers," Plenum Press, NY and London, 1996, page 64.

Robert G. Wilson v, Table of n, a(n) for n = 0..500
Grant Sanderson, Researchers thought this was a bug (Borwein integrals), 3Blue1Brown video (2022).
Wikipedia, Borwein integral

Cf. A002387, A056054, A091463, A091464, A091465.

Cf. also A092318, A092317, A092315, A281355, A074599, A025547.

s = 0; k = 1; Do[ While[s = N[s + 1/k, 24]; s <= n, k += 2]; Print[k]; k += 2, {n, 1, 11}]

a(n) ~ floor((1/2)*A002387(2n)).
The next term is approximately the previous term * e^2.
a(n) = A092315(n)*2 + 1 = floor(exp(n*2-Euler)/4+1/8)*2+1 for all n (conjectured). - M. F. Hasler, Jan 24 2017
a(n) ~ exp(2*n - A350763) = (1/2)*exp(2*n - gamma), gamma = A001620. - A.H.M. Smeets, Apr 15 2022

Corrected by N. J. A. Sloane, Feb 16 2004
More terms from Robert G. Wilson v, Apr 17 2004
a(17) corrected - see correction in A092315. - Gerhard Kirchner, Jul 25 2020
a(0) prepended by Robert G. Wilson v, Oct 23 2024

A089365 Smallest prime whose product of digits is 2^n.

Original entry on oeis.org

11, 2, 41, 181, 281, 1481, 881, 4481, 18481, 48281, 48481, 228881, 284881, 828881, 884881, 4448881, 4848881, 18848881, 24888881, 48888841, 88884881, 188888881, 888828881, 848888881, 4848488881, 4488888881, 18848888881, 28888884881

Offset: 0

Amarnath Murthy, Nov 07 2003

			a(8) = 24481 and the digital product is 2^8.

Chai Wah Wu, Table of n, a(n) for n = 0..1000

Cf. A088653, A090840, A091465, A090841, A089298.

NextPrim[n_] := Block[{k = n + 1}, While[ ! PrimeQ[k], k++ ]; k]; a = Table[0, {24}]; p = 2; Do[q = Log[2, Times @@ IntegerDigits[p]]; If[q != 0 && IntegerQ[q] && a[[q]] == 0, a[[q]] = p; Print[q, " = ", p]]; p = NextPrim[p], {n, 1, 10^9}]  (* Robert G. Wilson v, Nov 08 2003 *)
For a(8): a = Map[ FromDigits, Join[{0}, #, {1}] & /@ Permutations[{2, 8, 8 }]]; Min[ Select[a, PrimeQ[ # ] & ]] (* Robert G. Wilson v, Nov 08 2003 *)

Edited, corrected and extended by Robert G. Wilson v, Nov 08 2003

a(24)-a(25) corrected by Chai Wah Wu, Aug 15 2017

A088653 Smallest prime whose product of digits is 3^n.

Original entry on oeis.org

11, 3, 19, 139, 199, 1399, 1999, 13999, 99991, 139999, 199999, 1399999, 9999991, 19399999, 19999999, 919999939, 1399939999, 1999993999, 9199999999, 19399999999, 99999199999, 199999939999, 991999999999, 1999399999999

Offset: 0

Robert G. Wilson v, Nov 22 2003

			a(5) = 1399 and the digital product is 3^5.

Chai Wah Wu, Table of n, a(n) for n = 0..1000

Cf. A089365, A090840, A091465, A090841, A089298.

NextPrim[n_] := Block[{k = n + 1}, While[ ! PrimeQ[k], k++ ]; k]; a = Table[0, {18}]; p = 2; Do[q = Log[3, Times @@ IntegerDigits[p]]; If[q != 0 && IntegerQ[q] && a[[q]] == 0, a[[q]] = p; Print[q, " = ", p]]; p = NextPrim[p], {n, 1, 10^9}]
For a(23): a = Map[ FromDigits, Join[{0}, # ] & /@ Permutations[{1, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9 }]]; Min[ Select[a, PrimeQ[ # ] & ]]
For a(11): a = Map[ FromDigits, Permutations[{2, 6, 8, 8, 8, 9, 9, 9, 9, 9}]]; Min[ Select[a, PrimeQ[ # ] & ]]

A090840 Smallest prime whose product of digits is 5^n.

Original entry on oeis.org

11, 5, 11551, 15551, 1551551, 15551551, 1155555151, 1555551551, 11555555551, 1155155555551, 555555515551, 555555555551, 5555555555551, 555155555555551, 51555555551555551, 51555555555555551, 1155555555555555551, 15551555555555555551, 1155515555555555555551

Offset: 0

Robert G. Wilson v, Dec 09 2003

			a(4) = 1551551 because its digital product is 5^4, and it is prime.

Michael S. Branicky, Table of n, a(n) for n = 0..998

Cf. A089365, A088653, A091465, A090841, A089298.

a:= proc(n) local k, t; for k from 0 do t:= min(select(isprime,
      map(x-> parse(cat(x[])), combinat[permute]([1$k, 5$n]))));
      if tAlois P. Heinz, Nov 05 2021

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; a = Table[0, {18}]; p = 2; Do[q = Log[5, Times @@ IntegerDigits[p]]; If[q != 0 && IntegerQ[q] && a[[q]] == 0, a[[q]] = p; Print[q, " = ", p]]; p = NextPrim[p], {n, 1, 10^9}]
For a(13); a = Map[ FromDigits, Permutations[{1, 1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5}]]; Min[ Select[a, PrimeQ[ # ] &]]

from sympy import isprime
from sympy.utilities.iterables import multiset_permutations as mp
def a(n):
    if n < 2: return [11, 5][n]
    digits = n + 1
    while True:
        for p in mp("1"*(digits-n-1) + "5"*n, digits-1):
            t = int("".join(p) + "1")
            if isprime(t): return t
        digits += 1
print([a(n) for n in range(19)]) # Michael S. Branicky, Nov 05 2021

a(17) and beyond from Michael S. Branicky, Nov 05 2021

A090841 Smallest prime whose product of digits is 7^n.

Original entry on oeis.org

11, 7, 11177, 1777, 71777, 1777717, 1177717771, 77777177, 7177717777, 1777777777, 71777777777, 1717777777777, 7177777777777, 17777777777777, 17177777777777717, 7717777777777777, 1177777777177777777, 1777777777777777177, 7777177777777777777

Offset: 0

Robert G. Wilson v, Dec 09 2003

			a(6) = 1177717771 because its digital product is 7^6, and it is prime.

Michael S. Branicky, Table of n, a(n) for n = 0..999

Cf. A089365, A088653, A090840, A091465, A089298.

a:= proc(n) local k, t; for k from 0 do t:= min(select(isprime,
      map(x-> parse(cat(x[])), combinat[permute]([1$k, 7$n]))));
      if tAlois P. Heinz, Nov 07 2021

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; a = Table[0, {18}]; p = 2; Do[q = Log[7, Times @@ IntegerDigits[p]]; If[q != 0 && IntegerQ[q] && a[[q]] == 0, a[[q]] = p; Print[q, " = ", p]]; p = NextPrim[p], {n, 1, 10^9}]
For a(8); a = Map[ FromDigits, Permutations[{1, 1, 7, 7, 7, 7, 7, 7, 7, 7}]]; Min[ Select[a, PrimeQ[ # ] &]]

from sympy import isprime
from sympy.utilities.iterables import multiset_permutations as mp
def a(n):
    if n < 2: return [11, 7][n]
    digits = n
    while True:
        for p in mp("1"*(digits-n) + "7"*n, digits):
            t = int("".join(p))
            if isprime(t): return t
        digits += 1
print([a(n) for n in range(19)]) # Michael S. Branicky, Nov 07 2021

a(17) and beyond from Michael S. Branicky, Nov 07 2021

A056054 a(n) = smallest even number 2m such that value of odd harmonic series Sum_{j=0..m} 1/(2j) is > n.

Original entry on oeis.org

8, 62, 454, 3348, 24734, 182760, 1350428, 9978382, 73730824, 544801200, 4025566630, 29745137662, 219788490858, 1624029488844, 12000044999386, 88669005690160, 655180257281000, 4841163675961122, 35771629985782052

Offset: 1

Robert G. Wilson v, Jul 25 2000 and Jan 11 2004

nonn

Numbers 2*m such that floor(f(m)) = floor(f(m-1)) where f(m) = Sum_{j=1..m} ((2*j-1)/(2*j)). Examples:
  floor(f(1))=floor(1/2)=0;
  floor(f(2))=floor(1/2+3/4)=floor(1.25)=1, then 2*2=4 is not in the sequence;
  floor(f(3))=floor(1/2+3/4+5/6)=floor(2.083..)=2, then 2*3=6 is not in the sequence;
  floor(f(4))=floor(1/2+3/4+5/6+7/8)=floor(2.958..)=2, then 2*4=8 is the first term of the sequence. - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Aug 15 2007

Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Plenum Press, NY and London, 1996, page 64.

Cf. A002387, A056053, A091463, A091464, A091465.

Cf. A056054.

s = 0; k = 2; Do[ While[s = N[s + 1/k, 24]; s <= n, k += 2]; Print[k]; k += 2, {n, 1, 12}]
(* or assuming that the Mathematica coding in A002387 is correct then *)
b[n_] := Module[{k = Floor[2a[2n]]}, If[ EvenQ[k], k, k + 1]]; Table[ b[n], {n, 19}] (* Robert G. Wilson v, Apr 17 2004 *)

a(n) = 2*A002387(2n).

The next term is approximately the previous term * e^2.

A089298 Smallest prime with digit product 10^n.

Original entry on oeis.org

11, 251, 14551, 155581, 4545551, 45555581, 555555881, 44555555581, 455555558581, 5555555888551, 255555555585881, 14555555558558851, 155555555858885551, 2555555555558885851, 45555555555585855881

Offset: 0

Amarnath Murthy, Oct 30 2003

These numbers may not contain the digits {0, 3, 6, 7 or 9} and must end with the digit 1. Also the number of 2's plus half the number of 4's plus a quarter of the number of 8's must equal the number of 5's which in turn must equal n. - Robert G. Wilson v, Nov 06 2003

			a(4) = 4545551 and the digit product = 10^4.

Cf. A089365, A088653, A090840, A091465, A090841.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; a = Table[0, {10}]; p = 2; Do[ q = Log[10, Times @@ IntegerDigits[p]]; If[q != 0 && IntegerQ[q] && a[[q]] == 0, a[[q]] = p; Print[q, " = ", p]]; p = NextPrim[p], {n, 1, 10^9}]; a

Edited, corrected and extended by Robert G. Wilson v, Nov 06 2003

cp's OEIS Frontend

A056053 a(n) = smallest odd number 2m+1 such that the partial sum of the odd harmonic series Sum_{j=0..m} 1/(2j+1) is > n.

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A089365 Smallest prime whose product of digits is 2^n.

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A088653 Smallest prime whose product of digits is 3^n.

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A090840 Smallest prime whose product of digits is 5^n.

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A090841 Smallest prime whose product of digits is 7^n.

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A056054 a(n) = smallest even number 2m such that value of odd harmonic series Sum_{j=0..m} 1/(2j) is > n.

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A089298 Smallest prime with digit product 10^n.

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