Ely Golden - cp's OEIS frontend

A372496 Integers of the form k^2 + 1, where k >= 1, that are the product of two other integers of the form k^2 + 1, where k >= 1.

10, 50, 170, 290, 325, 442, 962, 1850, 2210, 3250, 5330, 8282, 9802, 12322, 15130, 17425, 17690, 24650, 33490, 44522, 58082, 58565, 64010, 65026, 74530, 94250, 103685, 117650, 145162, 177242, 191845, 214370, 237170, 257050, 305810, 332930, 361202

Offset: 1

Ely Golden, May 03 2024

nonn

This sequence is the sequence of possible c^2 + 1 values of all triples of positive integers (a,b,c) such that (a^2 + 1)*(b^2 + 1) = c^2 + 1.
A001541(n)^2 + 1 is always a term of this sequence for all n > 0. This is because A001541 consists of the x-values of the solutions to the Pell equation x^2 - 2y^2 = 1 which is equivalent to x^2 + 1 = 2*(y^2 + 1) = (1^2 + 1)*(y^2 + 1).
A002061(n)^2 + 1 is always a term of this sequence for all n > 1. This is because A002061 consists of the integers of the form k^2 + k + 1 and (k^2 + k + 1)^2 + 1 = (k^2 + 1)*((k+1)^2 + 1).

			50 is a term since 50 = 7^2 + 1 = 10 * 5 = (3^2 + 1)*(2^2 + 1).

Ely Golden, Table of n, a(n) for n = 1..10667 (terms <= 10^16)

Intersection of A002522 and A135280.

Cf. A001541, A002061.

formQ[k_] := k >= 1 && IntegerQ@Sqrt[k - 1];
prodQ[k_] := AnyTrue[Divisors[k][[2 ;; -2]], formQ[#] && formQ[k/#]&];
okQ[k_] := formQ[k] && prodQ[k];
Select[Range[2, 10^6], okQ] (* Jean-François Alcover, May 10 2024 *)

isok1(k) = issquare(k-1) && (k>1);
isok2(k) = fordiv(k, d, if (isok1(d) && isok1(k/d), return(1)));
isok(k) = isok1(k) && isok2(k); \\ Michel Marcus, May 04 2024

from math import isqrt
def is_perfect_square(n): return isqrt(abs(n))**2 == n
limit = 10**16
sequence_entries = set()
for a in range(1, isqrt(isqrt(limit))+1):
    u = a**2 + 1
    for b in range(a+1, isqrt(limit//u)+1):
        v = b**2 + 1
        if(is_perfect_square(u*v - 1)): sequence_entries.add(u*v)
sequence_entries = sorted(sequence_entries)
for i, j in enumerate(sequence_entries, 1):
    print(i, j)

A372497 Positive integers of the form k^2 - 1 that are the product of two other distinct positive integers of the form k^2 - 1.

Original entry on oeis.org

24, 120, 360, 840, 960, 1680, 3024, 4224, 5040, 7920, 11880, 17160, 22800, 24024, 32760, 36480, 43680, 57120, 70224, 73440, 83520, 93024, 116280, 121800, 143640, 175560, 201600, 212520, 241080, 255024, 303600, 330624, 358800, 421200, 491400, 570024, 591360

Offset: 1

Ely Golden, May 03 2024

nonn

This sequence is the sequence of possible c^2 - 1 values of all triples (a,b,c) of integers > 1 such that (a^2 - 1)*(b^2 - 1) = c^2 - 1.

			120 is a term since 120 = 15*8 = (4^2 - 1)*(3^2 - 1) and 120 = 11^2 - 1.

David A. Corneth, Table of n, a(n) for n = 1..19120 (first 408 terms from Ely Golden, terms <= 10^17)
David A. Corneth, PARI program

Intersection of A005563 and A063066.

Rest[Take[With[{k2=Range[500]^2-1},Select[Union[Times@@@Subsets[k2,{2}]],IntegerQ[Sqrt[#+1]]&]],50]] (* Harvey P. Dale, Apr 20 2025 *)

isok1(k) = issquare(k+1);
isok2(k) = fordiv(k, d, if (isok1(d) && isok1(k/d), return(1)));
isok(k) = isok1(k) && isok2(k); \\ Michel Marcus, May 04 2024

from math import isqrt
def is_perfect_square(n): return isqrt(abs(n))**2 == n
limit = 10**17
sequence_entries = set()
for a in range(2, isqrt(isqrt(limit))+1):
    u = a**2 - 1
    for b in range(a+1, isqrt(limit//u+1)+1):
        v = b**2 - 1
        if(is_perfect_square(u*v + 1)): sequence_entries.add(u*v)
sequence_entries = sorted(sequence_entries)
for i, j in enumerate(sequence_entries, 1):
    print(i, j)

Definition clarified by Harvey P. Dale, Apr 20 2025

A337612 Positive integers m such that A126289^k(m) = m for some positive integer k.

Original entry on oeis.org

1, 6, 10, 15, 20, 21, 35, 38, 42, 45, 52, 58, 63, 66, 68, 74, 76, 77, 78, 84, 85, 88, 92, 99, 110, 116, 117, 124, 130, 133, 143, 146, 153, 164, 171, 187, 189, 198, 208, 224, 228, 232, 238, 246, 247, 255, 261, 266, 268, 272, 273, 279, 282, 284, 285, 304, 312

Offset: 1

Ely Golden, Oct 06 2020

nonn

A126289^k(m) means apply A126289 to m k times.
Equivalently, the numbers that belong to a cycle under the map x -> A126289(x).
There are no primes in this sequence.

			6 is a term since A126289(A126289(6)) = A126289(10) = 6.

Ely Golden, Table of n, a(n) for n = 1..10000
Ely Golden, Python program for (naïvely) generating terms of this sequence

Cf. A052126, A126289, A337609, A337610, A337611.

For any term m, gcd {m, A126289(m), A126289(A126289(m)), ...} = A052126(m).

A337610 Positive integers m such that A126287^k(m) = m for some positive integer k.

Original entry on oeis.org

1, 10, 15, 22, 26, 33, 34, 46, 51, 58, 65, 69, 70, 82, 86, 87, 94, 105, 106, 118, 122, 123, 130, 134, 141, 142, 146, 154, 159, 166, 177, 178, 190, 195, 202, 206, 213, 214, 215, 218, 226, 231, 238, 249, 250, 254, 262, 266, 267, 274, 285, 286, 298, 302, 303, 310

Offset: 1

Ely Golden, Oct 07 2020

A126287^k(m) means apply A126287 to m k times.
Equivalently, the numbers that belong to a cycle under the map x -> A126287(x).
For any term m in this sequence, A126287(A126287(m)) = m.
Supersequence of A017641. Moreover, this sequence (excluding the first term) can be represented as an infinite union of arithmetic progressions.
There are no primes in this sequence.

			10 is a term since A126287(A126287(10)) = A126287(15) = 10.

Ely Golden, Table of n, a(n) for n = 1..10000
Ely Golden, Python program for generating terms of this sequence

Cf. A017641, A126287, A337609, A337611, A337612.

A337609 Positive integers m such that A126286^k(m) = m for some positive integer k.

Original entry on oeis.org

2, 3, 14, 21, 26, 34, 38, 39, 50, 57, 62, 74, 75, 85, 86, 93, 94, 98, 110, 111, 118, 122, 129, 134, 142, 146, 147, 154, 158, 165, 170, 182, 183, 194, 201, 202, 206, 214, 218, 219, 230, 235, 237, 242, 254, 255, 266, 273, 274, 278, 286, 290, 291, 298, 302, 309

Offset: 1

Ely Golden, Oct 07 2020

A126286^k(m) means apply A126286 to m k times.
Equivalently, the numbers that belong to a cycle under the map x -> A126286(x).
For any term m in this sequence, A126286(A126286(m)) = m.
Supersequence of A017545. Moreover, this sequence can be represented as an infinite union of arithmetic progressions.
2 and 3 are the only primes in this sequence.

			3 is a term since A126286(A126286(3)) = A126286(2) = 3.

Ely Golden, Table of n, a(n) for n = 1..10000
Ely Golden, Python program for generating terms of this sequence

Cf. A017545, A126286, A337610, A337611, A337612.

A337611 Positive integers m such that A126288^k(m) = m for some positive integer k.

Original entry on oeis.org

2, 3, 6, 10, 14, 20, 22, 26, 28, 38, 44, 46, 52, 76, 78, 88, 94, 102, 105, 114, 116, 117, 136, 138, 152, 171, 186, 187, 195, 207, 212, 247, 248, 266, 282, 284, 285, 296, 304, 322, 333, 354, 366, 369, 387, 402, 403, 407, 414, 423, 425, 426, 430, 437, 442, 468

Offset: 1

Ely Golden, Sep 05 2020

nonn

A126288^k(m) means apply A126288 to m k times.
Equivalently, the numbers that belong to a cycle under the map x -> A126288(x).
2 and 3 are the only primes in this sequence.

			3 is a term since A126288(A126288(3)) = A126288(2) = 3.

Ely Golden, Table of n, a(n) for n = 1..7144
Ely Golden, Python program for (naïvely) generating terms of this sequence

Cf. A337609, A337610, A337612, A126288, A052126.

gpf(n) = vecmax(factor(n)[,1]);
f(n) = if (n==1, 2, n*gpf(n+1)/gpf(n)); \\ A126288
incycle(n, list) = {my(v=Vec(list)); #select(x->(x==n), v);}
cycle(n) = {my(list = List(), repeat=1); while(repeat, n = f(n); if (incycle(n, list), repeat=0); listput(list, n);); list;}
isok(n) = {my(list = cycle(n)); incycle(n, list);} \\ Michel Marcus, Sep 08 2020

For any term m, gcd {m, A126288(m), A126288(A126288(m)), ...} = A052126(m).

A336587 Smallest nonnegative integer containing the n-th letter of the Hebrew alphabet (in Hebrew using masculine numbers), or -1 if no such integer exists.

Original entry on oeis.org

0, 4, 1000000000000000000000000000000000000000000000000000000000000000, 1, 3, 3, -1, 1, 1000000000000, 2, -1, 3, 2, 2, 0, 4, 0, -1, 1000000000000000, 4, 2, 9

Offset: 1

Ely Golden, Jul 26 2020

This sequence assumes the use of the short scale for naming large numbers. It is the same whether or not 10^9 is called "ביליון" (billion) or "מיליארד" (milliard).

Final forms of the letters are considered the same as the normal forms. There are no numbers with ז (zayin), כ (kaf), or צ (tsadi) in their names. ג (gimel) appears only in vocabulary transliterated into Hebrew based on Landon Curt Noll's latin-based power of 1000 naming system and not in everyday vocabulary (hence why a(3) = 10^63).

Ely Golden, Arbitrary precision Hebrew number naming program in Java (from A298199)
Ely Golden, Spellings for numbers in A336587
Landon Curt Noll, The English Name of a Number.

Cf. A336586, A111098, A147876.

A336586 Smallest nonnegative integer containing the n-th letter of the Hebrew alphabet (in Hebrew using feminine numbers), or -1 if no such integer exists.

Original entry on oeis.org

0, 4, 1000000000000000000000000000000000000000000000000000000000000000, 1000000000000000, 8, 3, -1, 1, 1000000000000, 2, -1, 3, 2, 8, 0, 4, 0, -1, 1000000000000000, 4, 2, 1

Offset: 1

Ely Golden, Jul 26 2020

This sequence assumes the use of the short scale for naming large numbers. It also assumes that 10^9 is called "ביליון" (billion); if 10^9 is instead called "מיליארד" (milliard) then a(4) = 10^9 rather than 10^15.

Final forms of the letters are considered the same as the normal forms. There are no numbers with ז (zayin), כ (kaf), or צ (tsadi) in their names. ג (gimel) appears only in vocabulary transliterated into Hebrew based on Landon Curt Noll's latin-based power of 1000 naming system and not in everyday vocabulary (hence why a(3) = 10^63).

Ely Golden, Arbitrary precision Hebrew number naming program in Java (from A298199)
Ely Golden, Spellings for numbers in A336586
Landon Curt Noll, The English Name of a Number.

Cf. A336587, A111098, A147876.

A309871 Numbers n for which 18n+1, 18n+5, 18n+7, 18n+11, 18n+13 and 18n+17 are primes.

Original entry on oeis.org

892, 2432, 156817, 806697, 822937, 1377022, 1389412, 1418007, 1619642, 1753552, 2017732, 2058647, 2329302, 2554142, 2703347, 3058772, 3135107, 3326522, 3391797, 3723457, 4126867, 4132782, 4171422, 4411837, 4610252, 6378487, 6440087, 6878987, 6897782, 6991547

Offset: 1

Ely Golden, Aug 21 2019

nonn

Ely Golden, Table of n, a(n) for n = 1..1001

Cf. A123986, A056956, A007811, A123985.

tot[n_] := Select[Range[n], CoprimeQ[#, n] &]; m = 18; t = tot[m]; aQ[n_] := AllTrue[m * n + t, PrimeQ]; Select[Range[10^6], aQ] (* Amiram Eldar, Aug 22 2019 *)

x = 1
for i in range(5000000):
    if (18*i+1 in Primes()
    and 18*i+5 in Primes()
    and 18*i+7 in Primes()
    and 18*i+11 in Primes()
    and 18*i+13 in Primes()
    and 18*i+17 in Primes()):
        print(str(x)+" "+str(i))
        x += 1

A298199 Number of letters in the feminine Hebrew name of n, excluding spaces.

Original entry on oeis.org

3, 3, 4, 4, 4, 3, 2, 3, 5, 3, 3, 7, 8, 8, 8, 7, 6, 7, 9, 7, 5, 9, 10, 10, 10, 9, 8, 9, 11, 9, 6, 10, 11, 11, 11, 10, 9, 10, 12, 10, 6, 10, 11, 11, 11, 10, 9, 10, 12, 10, 6, 10, 11, 11, 11, 10, 9, 10, 12, 10, 4, 8, 9, 9, 9, 8, 7, 8, 10, 8, 5, 9, 10, 10, 10, 9, 8, 9, 11, 9, 6

Offset: 0

Ely Golden, Jan 14 2018

Corrected version of A027684.

			a(3) = 4, since "שלוש" (shalosh) has four letters.

Ely Golden, Table of n, a(n) for n = 0..10000
Ely Golden, Arbitrary precision Hebrew number naming program in Java
Index entries for sequences related to number of letters in n

Cf. A298200.

cp's OEIS Frontend

User: Ely Golden

A372496 Integers of the form k^2 + 1, where k >= 1, that are the product of two other integers of the form k^2 + 1, where k >= 1.

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A372497 Positive integers of the form k^2 - 1 that are the product of two other distinct positive integers of the form k^2 - 1.

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A337612 Positive integers m such that A126289^k(m) = m for some positive integer k.

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A337610 Positive integers m such that A126287^k(m) = m for some positive integer k.

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A337609 Positive integers m such that A126286^k(m) = m for some positive integer k.

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A337611 Positive integers m such that A126288^k(m) = m for some positive integer k.

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PARI

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A336587 Smallest nonnegative integer containing the n-th letter of the Hebrew alphabet (in Hebrew using masculine numbers), or -1 if no such integer exists.

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A336586 Smallest nonnegative integer containing the n-th letter of the Hebrew alphabet (in Hebrew using feminine numbers), or -1 if no such integer exists.

Views

Author

Keywords

Comments

Links

Crossrefs

A309871 Numbers n for which 18n+1, 18n+5, 18n+7, 18n+11, 18n+13 and 18n+17 are primes.

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